Introduction to coupled oscillation and eigenmodes (Ch. 2-4 of Unit 4 )

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

4 13 2

⎛⎝⎜

⎞⎠⎟

Lecture 21 Wed. 10.31.2018

A running collection of links to course-relevant sites and articles

AJP article on superball dynamics AIP publications

AAPT summer reading These are hot off the presses:

Slightly Older ones:

“Relawavity” and quantum basis of Lagrangian & Hamiltonian mechanics:

AMOP Ch 0 Space-Time Symmetry - 2019Seminar at Rochester Institute of Optics, Auxiliary slides, June 19, 2018

Comprehensive Harter-Soft Resource Listing 2014 AMOP

2018 AMOP

UAF Physics YouTube channel 2017 Group Theory for QM

Classical Mechanics with a Bang!

Principles of Symmetry, Dynamics, and SpectroscopyQuantum Theory for the Computer Age

Modern Physics and its Classical Foundations 2018 Adv Mechanics

Classes“Texts”Physics Web Resources

Wave–particle duality of C60 moleculesOptical vortex knots – One Photon at a Time

Sorting ultracold atoms in a three-dimensional optical lattice in a realization of Maxwell’s demon - Kumar-Nature-Letters-2018Synthetic three-dimensional atomic structures assembled atom by atom - Berredo-Nature-Letters-2018

2-CW laser wave - BohrIt Web AppLagrangian vs Hamiltonian - RelaWavity Web App

Neat external material to start the class:

LearnIt Physics Web Applications

https://modphys.hosted.uark.edu/testing/markup/AnalyItBJS.htmlAnalyIt Web Application, posted 10/22/2018 in our testing area:

http://thearmchaircritic.blogspot.com/2011/11/punkin-chunkin.htmlhttp://www.sussexcountyonline.com/news/photos/punkinchunkin.htmlhttp://www.twcenter.net/forums/showthread.php?358315-Shooting-range-for-medieval-siege-weapons-Anybody-knowshttps://modphys.hosted.uark.edu/markup/TrebuchetWeb.htmlhttps://modphys.hosted.uark.edu/markup/TrebuchetWeb.html?scenario=MontezumasRevengehttps://modphys.hosted.uark.edu/markup/TrebuchetWeb.html?scenario=SeigeOfKenilworthhttps://modphys.hosted.uark.edu/pdfs/Journal_Pdfs/Trebuchet-SciAm_273_66_July_1995_chevedden1.pdf‘Simple’ Pendulum Sim: https://modphys.hosted.uark.edu/markup/PendulumWeb.html‘Cycloid’ Pendulum: https://modphys.hosted.uark.edu/markup/CycloidulumWeb.htmlGoogle search on: "Satelite view of Patricia" (Images)Physics Girl Channel - Fun with Vortex Rings in the Pool: https://www.youtube.com/watch?v=72LWr7BU8AoiBall demo - Quasi-periodicity: https://youtu.be/_jntDtULxDc

Older Links from Lectures 14-20

https://modphys.hosted.uark.edu/markup/CoulItWeb.html?scenario=SynchrotronMotionhttps://modphys.hosted.uark.edu/markup/CoulItWeb.html?scenario=SynchrotronMotion2Mechanical Analog to EM Motion (YouTube video) - https://youtu.be/hTd5FTJ-vRkCoulIt Web Simulation: Bound-state motion in parabolic coordinatesCoulIt Web Simulation: Bound-state motion in hyperbolic coordinatesOscillIt Web App: Simulations of various types of resonance: 18, 27, 31, 35, 38, 39Smith Charthttp://nobelprize.org/

Links to supplement Lecture 21BoxIt Web App: A-Type w/Cosine AB-Type w/Cosine A-Type w/Freq ratios AB-Type 2:1 Freq ratioWiki on Pafnuty Chebyshev

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.2 Coupled pendulumsFig. 3.3.3 One 2-dimensional coupled oscillator

2D harmonic oscillators

x=x1

k1 k12

y=x2

k2m1 m2

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.2 Coupled pendulums Fig. 3.3.3 One 2-dimensional coupled oscillator

2D harmonic oscillator energy

T = 1

2m1 !x1

2 + 12

m2 !x22

2D HO kinetic energy T(v1, v2)

x=x1

k1 k12

y=x2

k2m1 m2

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.3 One 2-dimensional coupled oscillator

V = 12

k1x12 + 1

2k2x2

2 + 12

k12 x1 − x2( )2

= 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22

2D HO potential energy V(x1, x2)

T = 1

2m1 !x1

2 + 12

m2 !x22

2D HO kinetic energy T(v1, v2) Lagrangian L=T-V

Fig. 3.3.2 Coupled pendulums

x=x1

k1 k12

y=x2

k2m1 m2

2D harmonic oscillator energy

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.3 One 2-dimensional coupled oscillator

V = 12

k1x12 + 1

2k2x2

2 + 12

k12 x1 − x2( )2

= 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22

2D HO potential energy V(x1, x2)

T = 1

2m1 !x1

2 + 12

m2 !x22

2D HO kinetic energy T(v1, v2) Lagrangian L=T-V

ddt

∂T∂ !x1

⎛

⎝⎜⎞

⎠⎟= m1!!x1 = F1 = − ∂V

∂ x1= − k1 + k12( )x1 + k12x2

ddt

∂T∂ !x2

⎛

⎝⎜⎞

⎠⎟= m2!!x2 = F2 = − ∂V

∂ x2= k12x1 − k2 + k12( )x2

2D HO Lagrange equations

Fig. 3.3.2 Coupled pendulums

x=x1

k1 k12

y=x2

k2m1 m2

2D harmonic oscillator equations

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.3 One 2-dimensional coupled oscillator

V = 12

k1x12 + 1

2k2x2

2 + 12

k12 x1 − x2( )2

= 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22

2D HO potential energy V(x1, x2)

T = 1

2m1 !x1

2 + 12

m2 !x22

2D HO kinetic energy T(v1, v2) Lagrangian L=T-V

ddt

∂T∂ !x1

⎛

⎝⎜⎞

⎠⎟= m1!!x1 = F1 = − ∂V

∂ x1= − k1 + k12( )x1 + k12x2

ddt

∂T∂ !x2

⎛

⎝⎜⎞

⎠⎟= m2!!x2 = F2 = − ∂V

∂ x2= k12x1 − k2 + k12( )x2

2D HO Lagrange equations

m1 0

0 m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

!!x1

!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO Matrix operator equations

Fig. 3.3.2 Coupled pendulums

x=x1

k1 k12

y=x2

k2m1 m2

2D harmonic oscillator equations

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.3 One 2-dimensional coupled oscillator

V = 12

k1x12 + 1

2k2x2

2 + 12

k12 x1 − x2( )2

= 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22

2D HO potential energy V(x1, x2)

T = 1

2m1 !x1

2 + 12

m2 !x22

2D HO kinetic energy T(v1, v2) Lagrangian L=T-V

ddt

∂T∂ !x1

⎛

⎝⎜⎞

⎠⎟= m1!!x1 = F1 = − ∂V

∂ x1= − k1 + k12( )x1 + k12x2

ddt

∂T∂ !x2

⎛

⎝⎜⎞

⎠⎟= m2!!x2 = F2 = − ∂V

∂ x2= k12x1 − k2 + k12( )x2

2D HO Lagrange equations

m1 0

0 m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

!!x1

!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Matrix operator notation:

M i !!x = − K i x

2D HO Matrix operator equations

Fig. 3.3.2 Coupled pendulums

x=x1

k1 k12

y=x2

k2m1 m2

2D harmonic oscillator equations

x = 0

k k k

x = 0

1 12 2

1 2

m1 m2

θ1θ2

κ

ℓℓ

mm

1

2

.

x = 01

x =

02

m

Fig. 3.3.1 Two 1-dimensional coupled oscillators

Fig. 3.3.3 One 2-dimensional coupled oscillator

V = 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22

= 12

x K x where: K =k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2)

T = 12

m1 !x12 + 1

2m2 !x2

2

= 12!x M !x

2D HO kinetic energy T(v1, v2) Lagrangian L=T-V

ddt

∂T∂ !x1

⎛

⎝⎜⎞

⎠⎟= m1!!x1 = F1 = − ∂V

∂ x1= − k1 + k12( )x1 + k12x2

ddt

∂T∂ !x2

⎛

⎝⎜⎞

⎠⎟= m2!!x2 = F2 = − ∂V

∂ x2= k12x1 − k2 + k12( )x2

2D HO Lagrange equations

m1 0

0 m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

!!x1

!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Matrix operator notation:

M i !!x = − K i x

2D HO Matrix operator equations

Fig. 3.3.2 Coupled pendulums

x=x1

k1 k12

y=x2

k2m1 m2

2D harmonic oscillator equations

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Each coordinate is rescaled to symmetrize mass factors on -terms.

New constants Kij have pseudo-reciprocity symmetry for a special scale factor ratio:

Matrix equations and reciprocity symmetry

F1F2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

m1!!x1m2!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

κ11 κ12κ 21 κ 22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

General form of 2D-HO equation of motion has force matrix components: κ11 = k1 + k11, κ 22 = k2 + k22

Off-diagonal force constants satisfy Reciprocity Relations: −κ12= k12=

∂ F1∂ x2

=- ∂ 2 V∂ x2∂ x1

=- ∂ 2 V∂ x1∂ x2

=∂ F2∂ x1

=k21=−κ 21

Rescaling and symmetrization

x1,x2( )

q1 = s1x1,q2 = s2x2( )

!!q j

− m1s1!!q1 =κ11

q1s1

+κ12q2s2

−!!q1 =κ11m1

q1 +κ12s1m1s2

q2 ≡ Κ11q1 +Κ12q2

− m2s2!!q2 =κ12

q1s1

+κ 22q2s2

−!!q2 =κ12s2m2s1

q1 +κ 22m2

q2 ≡ Κ21q1 +Κ22q2

Κ21 =κ12s2m2s1

= Κ12 =κ12s1m1s2

= −k12m1m2

Κ11 =κ11m1

= k1 + k12m1

Κ22 =κ 22m2

= k2 + k12m2

s2s1

=m2m1

Caution is advised since such forced symmetry may give modes with imaginary frequency.

Diagonal constants Kjj are not affected by scaling. To be equal requires: κ11m1

= κ 22m2

or: κ11κ 22

= m1m2

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

2D harmonic oscillator equation solutions 1. May rewrite equation in acceleration matrix form:

!!x1

!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

m1 0

0 m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

k1 + k12m1

−k12m1

−k12m2

k2 + k12m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

M i !!x = −K i x

!!x = −A x where: A = M−1 i K

2. Need to find eigenvectors of acceleration matrix such that:

Then equations decouple to:

A en = εn en =ωn

2 en e1 , e2 ,...

!!en = −A en = −εn en = −ωn2 en where εn is an eigenvalue

and ωn is an eigenfrequency

2D harmonic oscillator equation solutions 1. May rewrite equation in acceleration matrix form:

!!x1

!!x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

m1 0

0 m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= −

k1 + k12m1

−k12m1

−k12m2

k2 + k12m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

M i !!x = −K i x

!!x = −A x where: A = M−1 i K

2. Need to find eigenvectors of acceleration matrix such that:

Then equations decouple to:

A en = εn en =ωn

2 en e1 , e2 ,...

!!en = −A en = −εn en = −ωn2 en where εn is an eigenvalue

and ωn is an eigenfrequency

To introduce eigensolutions we take a simple case of unit masses (m1=1=m2)

So equation of motion is simply: !!x = −K x

Eigenvectors are in special directions where is in same direction as !!x = −K x

x

x = en

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

V = 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22 = 1

2x K x = 1

2x i K i x = x1 x2( ) k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2) quadratic form defines layers of elliptical V-contours

XX11((EEaasstt))

XX22((NNoorrtthh))

ε++((NNoorrtthhEEaasstt))

--XX22((SSoouutthh))

--XX11((WWeesstt))

--ε++((SSoouutthhWWeesstt))

ε--((NNoorrtthhWWeesstt))

--ε--((SSoouutthhEEaasstt))

Fig. 3.3.4 Plot of potential function V(x1,x2) showing elliptical V(x1,x2)=const. level curves.

What direction is the same as ??

x = en

K x XX11

XX22UU--

UU++

xx

FAST axis

SLOW axis(a) PE Contours

V = 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22 = 1

2x K x = 1

2x i K i x = x1 x2( ) k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2) quadratic form defines layers of elliptical V-contours

XX11((EEaasstt))

XX22((NNoorrtthh))

ε++((NNoorrtthhEEaasstt))

--XX22((SSoouutthh))

--XX11((WWeesstt))

--ε++((SSoouutthhWWeesstt))

ε--((NNoorrtthhWWeesstt))

--ε--((SSoouutthhEEaasstt))

Fig. 3.3.4 Plot of potential function V(x1,x2) showing elliptical V(x1,x2)=const. level curves.

What direction is the same as ?? Not most directions!

x = en

K x XX11

XX22UU--

UU++

xx

FAST axis

SLOW axis(a) PE Contours

x

x

K x K x

Force:F=-∇V =M•a=-K•x

V = 12

k1 + k12( )x12 − k12x1x2 +

12

k2 + k12( )x22 = 1

2x K x = 1

2x i K i x = x1 x2( ) k1 + k12 −k12

−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2) quadratic form defines layers of elliptical V-contours

XX11((EEaasstt))

XX22((NNoorrtthh))

ε++((NNoorrtthhEEaasstt))

--XX22((SSoouutthh))

--XX11((WWeesstt))

--ε++((SSoouutthhWWeesstt))

ε--((NNoorrtthhWWeesstt))

--ε--((SSoouutthhEEaasstt))

Fig. 3.3.4 Plot of potential function V(x1,x2) showing elliptical V(x1,x2)=const. level curves.

What direction is the same as ?? Not most directions! Only extremal axes work. (major or minor axes)

x = en

K x XX11

XX22UU--

UU++

xx

FAST axis

SLOW axis(a) PE Contours

x

x

K x K x

Force:F=-∇V =M•a=-K•x

V = 12

k + k12( )x12 − k12x1x2 +

12

k + k12( )x22 = 1

2x K x = 1

2x i K i x = x1 x2( ) k + k12 −k12

−k12 k + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2) quadratic form defines layers of elliptical V-contours (Here: k1 =k= k2)

XX11

XX22UU--

UU++

xx

FF== --∇∇VV ∇∇VVFAST axis

SLOW axis(a) PE Contours and gradients

(b) Symmetric UU++ Coordinate

SLOW Mode

XX11

XX22UU--

UU++

U+XX11

XX22

XX11pphhaassoorrUU++

pphhaassoorr

UU++is SLOW

phasor

XX11

XX22UU--

UU++

UU--

XX11

XX22

XX22pphhaassoorr(90° turned)

XX11pphhaassoorr

UU--pphhaassoorr

UU--is FAST

phasor

XX22XX22pphhaassoorr

XX22XX22pphhaassoorr

0°

In-phase

mode

180°

Out-of-phase

mode

XX22pphhaassoorr(90° turned)

(c) Anti-symmetric UU-- Coordinate

FAST Mode

XX11((EEaasstt))

XX22((NNoorrtthh))

ε++((NNoorrtthhEEaasstt))

--XX22((SSoouutthh))

--XX11((WWeesstt))

--ε++((SSoouutthhWWeesstt))

ε--((NNoorrtthhWWeesstt))

--ε--((SSoouutthhEEaasstt))

Fig. 3.3.4 Plot of potential function V(x1,x2) showing elliptical V(x1,x2)=const. level curves.

ε++ aaxxiiss

ε-- aaxxiiss

PPootteennttiiaallEEnneerrggyyVV((xx11 ,, xx22 ))

ε

XX11((EEaasstt))

--XX22((SSoouutthh))

XX22((NNoorrtthh))

--XX11((WWeesstt))

Fig. 3.3.5 Topography lines of potential function V(x1,x2) and orthogonal ε+ and ε− normal mode slopes

What direction is the same as ?? Not most directions! Only extremal axes work. (major or minor axes)

x = en

K x

With Bilateral symmetry (k1 =k= k2) the extremal axes lie at ±45°

NW-SE “Advanced slope”

NE-SW “Beginner slope”

Force:F=-∇V =M•a=-K•x

V = 12

k + k12( )x12 − k12x1x2 +

12

k + k12( )x22 = 1

2x K x = 1

2x i K i x = x1 x2( ) k + k12 −k12

−k12 k + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟

x1

x2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D HO potential energy V(x1, x2) quadratic form defines layers of elliptical V-contours (Here: k1 =k= k2)

XX11

XX22UU--

UU++

xx

FF== --∇∇VV ∇∇VVFAST axis

SLOW axis(a) PE Contours and gradients

(b) Symmetric UU++ Coordinate

SLOW Mode

XX11

XX22UU--

UU++

U+XX11

XX22

XX11pphhaassoorrUU++

pphhaassoorr

UU++is SLOW

phasor

XX11

XX22UU--

UU++

UU--

XX11

XX22

XX22pphhaassoorr(90° turned)

XX11pphhaassoorr

UU--pphhaassoorr

UU--is FAST

phasor

XX22XX22pphhaassoorr

XX22XX22pphhaassoorr

0°

In-phase

mode

180°

Out-of-phase

mode

XX22pphhaassoorr(90° turned)

(c) Anti-symmetric UU-- Coordinate

FAST Mode

XX11((EEaasstt))

XX22((NNoorrtthh))

ε++((NNoorrtthhEEaasstt))

--XX22((SSoouutthh))

--XX11((WWeesstt))

--ε++((SSoouutthhWWeesstt))

ε--((NNoorrtthhWWeesstt))

--ε--((SSoouutthhEEaasstt))

Fig. 3.3.4 Plot of potential function V(x1,x2) showing elliptical V(x1,x2)=const. level curves.

What direction is the same as ?? Not most directions! Only extremal axes work. (major or minor axes)

x = en

K x

Force:F=-∇V =M•a=-K•x

BoxIt (Beating) Web Simulation (A=1, B=-0.1, C=0, D=1) with Comparison Cosine wave (T=2π)

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M)

First step in finding eigenvalues: Solve secular equation

where:

First step in finding eigenvalues: Solve secular equation

where:

Secular equation has n-factors, one for each eigenvalue.

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( )! ε − εn( )

0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5

0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

First step in finding eigenvalues: Solve secular equation

where:

Secular equation has n-factors, one for each eigenvalue.

Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.

Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( )! ε − εn( )

0 = M − ε11( ) M − ε21( )! M − εn1( )

0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5

0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)

0 00 0

⎛⎝⎜

⎞⎠⎟= 4 1

3 2⎛⎝⎜

⎞⎠⎟

2

− 6 4 13 2

⎛⎝⎜

⎞⎠⎟+ 5 1 0

0 1⎛⎝⎜

⎞⎠⎟

0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

First step in finding eigenvalues: Solve secular equation

where:

Secular equation has n-factors, one for each eigenvalue.

Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.

Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( )! ε − εn( )

0 = M − ε11( ) M − ε21( )! M − εn1( )

0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5

0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)

0 00 0

⎛⎝⎜

⎞⎠⎟= 4 1

3 2⎛⎝⎜

⎞⎠⎟

2

− 6 4 13 2

⎛⎝⎜

⎞⎠⎟+ 5 1 0

0 1⎛⎝⎜

⎞⎠⎟

0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M

Replace jth HC-factor by (1) to make projection operators . p1 = (1)(M − 5⋅1) = 4 − 5 13 2 − 5

⎛⎝⎜

⎞⎠⎟= −1 1

3 −3⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1)(1) = 4 −1 13 2 −1

⎛⎝⎜

⎞⎠⎟= 3 1

3 1⎛⎝⎜

⎞⎠⎟

p1 = 1 ( ) M − ε21( )! M − εn1( )p2 = M − ε11( ) 1 ( )! M − εn1( ) "pn = M − ε11( ) M − ε21( )! 1 ( )

First step in finding eigenvalues: Solve secular equation

where:

Secular equation has n-factors, one for each eigenvalue.

Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.

Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( )! ε − εn( )

0 = M − ε11( ) M − ε21( )! M − εn1( )

0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5

0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)

0 00 0

⎛⎝⎜

⎞⎠⎟= 4 1

3 2⎛⎝⎜

⎞⎠⎟

2

− 6 4 13 2

⎛⎝⎜

⎞⎠⎟+ 5 1 0

0 1⎛⎝⎜

⎞⎠⎟

0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M

Replace jth HC-factor by (1) to make projection operators . p1 = (1)(M − 5⋅1) = 4 − 5 13 2 − 5

⎛⎝⎜

⎞⎠⎟= −1 1

3 −3⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1)(1) = 4 −1 13 2 −1

⎛⎝⎜

⎞⎠⎟= 3 1

3 1⎛⎝⎜

⎞⎠⎟

p1 = 1 ( ) M − ε21( )! M − εn1( )p2 = M − ε11( ) 1 ( )! M − εn1( ) "pn = M − ε11( ) M − ε21( )! 1 ( )

ε j ≠ εk ≠ ...

(Assume distinct e-values here: Non-degeneracy clause)

Each pk contains eigen-bra-kets since: (M-εk1)pk=0 or: Mpk=εkpk=pkM . Mp1 =

4 13 2

⎛⎝⎜

⎞⎠⎟⋅ −1 1

3 −3⎛⎝⎜

⎞⎠⎟= 1· −1 1

3 −3⎛⎝⎜

⎞⎠⎟= 1·p1

Mp2 =4 13 2

⎛⎝⎜

⎞⎠⎟⋅ 3 13 1

⎛⎝⎜

⎞⎠⎟= 5· 3 1

3 1⎛⎝⎜

⎞⎠⎟= 5·p2

pk =j≠k∏ M − ε j1( )

First step in finding eigenvalues: Solve secular equation

where:

Secular equation has n-factors, one for each eigenvalue.

Each ε replaced by M and each εk by εk 1 gives Hamilton-Cayley matrix equation.

Obviously true if M has diagonal form. (But, that’s circular logic. Faith needed!)

An eigenvector of M is in a direction that is left unchanged by M.

εk is eigenvalue associated with eigenvector direction. A change of basis to called diagonalization gives

M ε k = ε k ε k , or: M − ε k1( ) ε k = 0

ε k

ε1 , ε2 ,! εn{ }

ε1 M ε1 ε1 M ε2 ! ε1 M εnε2 M ε1 ε2 M ε2 ! ε2 M εn" " # "

εn M ε1 εn M ε2 ! εn M εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

ε1 0 ! 00 ε2 ! 0" " # "0 0 ! εn

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

ε k

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Only possible non-zero {x,y} if denominator is zero, too!

0=det M − ε ⋅1=det 4 13 2

⎛⎝⎜

⎞⎠⎟−ε 1 0

0 1⎛⎝⎜

⎞⎠⎟=det 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟

M ε = 4 13 2

⎛⎝⎜

⎞⎠⎟

xy

⎛

⎝⎜

⎞

⎠⎟ = ε x

y⎛

⎝⎜

⎞

⎠⎟ or: 4 − ε 1

3 2 − ε⎛

⎝⎜⎞

⎠⎟xy

⎛

⎝⎜

⎞

⎠⎟ =

00

⎛⎝⎜

⎞⎠⎟

Trying to solve by Kramer's inversion:

x =det 0 1

0 2 − ε⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

and y =det 4 − ε 0

3 0⎛⎝⎜

⎞⎠⎟

det 4 − ε 13 2 − ε

⎛

⎝⎜⎞

⎠⎟

detM − ε1 = 0 = −1( )n ε − ε1( ) ε − ε2( )! ε − εn( )

0 = M − ε11( ) M − ε21( )! M − εn1( )

0 = (4 − ε )(2 − ε )−1·3 = 8 − 6ε + ε 2 −1·3 = ε 2 − 6ε + 50 = ε 2 −Trace(M)ε + det(M) = ε 2 − 6ε + 5

0 =M2 − 6M + 51 = (M −1⋅1)(M − 5⋅1)

0 00 0

⎛⎝⎜

⎞⎠⎟= 4 1

3 2⎛⎝⎜

⎞⎠⎟

2

− 6 4 13 2

⎛⎝⎜

⎞⎠⎟+ 5 1 0

0 1⎛⎝⎜

⎞⎠⎟

0 = (ε −1)(ε − 5) so let: ε1 = 1 and: ε2 = 5

detM − ε1 = 0 = −1( )n ε n + a1ε

n−1 + a2εn−2 +…+ an−1ε + an( )

a1 = −TraceM,!, ak = −1( )k diagonal k-by-k minors of ∑ M,!, an = −1( )n det M

Replace jth HC-factor by (1) to make projection operators . p1 = (1)(M − 5⋅1) = 4 − 5 13 2 − 5

⎛⎝⎜

⎞⎠⎟= −1 1

3 −3⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1)(1) = 4 −1 13 2 −1

⎛⎝⎜

⎞⎠⎟= 3 1

3 1⎛⎝⎜

⎞⎠⎟

p1 = 1 ( ) M − ε21( )! M − εn1( )p2 = M − ε11( ) 1 ( )! M − εn1( ) "pn = M − ε11( ) M − ε21( )! 1 ( )

ε j ≠ εk ≠ ...

(Assume distinct e-values here: Non-degeneracy clause)

Each pk contains eigen-bra-kets since: (M-εk1)pk=0 or: Mpk=εkpk=pkM . Mp1 =

4 13 2

⎛⎝⎜

⎞⎠⎟⋅ −1 1

3 −3⎛⎝⎜

⎞⎠⎟= 1· −1 1

3 −3⎛⎝⎜

⎞⎠⎟= 1·p1

Mp2 =4 13 2

⎛⎝⎜

⎞⎠⎟⋅ 3 13 1

⎛⎝⎜

⎞⎠⎟= 5· 3 1

3 1⎛⎝⎜

⎞⎠⎟= 5·p2

pk =j≠k∏ M − ε j1( )

Notice pk commutes with M,..

since M1, M2,..commute with M.

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

Multiplication properties of pj :

Mpk =ε kpk = pkM

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

P1 =(M − 5⋅1)(1− 5)

= 14

1 −1−3 3

⎛⎝⎜

⎞⎠⎟

P2 =(M −1⋅1)(5 −1)

= 14

3 13 1

⎛⎝⎜

⎞⎠⎟

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)(1− 5)

= 14

1 −1−3 3

⎛⎝⎜

⎞⎠⎟

P2 =(M −1⋅1)(5 −1)

= 14

3 13 1

⎛⎝⎜

⎞⎠⎟

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

“Gauge” scale factors that only affect plots

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

⏐y〉or〈y⏐

⏐x〉 or 〈x⏐

“Gauge” scale factors that only affect plots

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 1 00 1

⎛⎝⎜

⎞⎠⎟

⏐y〉or〈y⏐

⏐x〉 or 〈x⏐

“Gauge” scale factors that only affect plots

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

⏐y〉or〈y⏐

⏐x〉 or 〈x⏐

“Gauge” scale factors that only affect plots

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

⏐y〉

⏐x〉 or

Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn

MPk =ε kPk

“Gauge” scale factors that only affect plots

Matrix-algebraic method for finding eigenvector and eigenvalues With example matrix M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn

MPk =ε kPk

M = 4 13 2

⎛⎝⎜

⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ 5 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Matrix and operator Spectral Decompositons M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn

MPk =ε kPk

M = 4 13 2

⎛⎝⎜

⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ 5 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn

Matrix and operator Spectral Decompositons M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn

MPk =ε kPk

M = 4 13 2

⎛⎝⎜

⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ 5 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn

Example:

M50= 4 13 2

⎛⎝⎜

⎞⎠⎟=150 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+550 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟=41 1+3·550 550−1

3·550−3 550+3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

M50= 4 13 2

⎛⎝⎜

⎞⎠⎟=150 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+550 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟=41 1+3·550 550−1

3·550−3 550+3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Matrix and operator Spectral Decompositons M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

Eigen-operators then give Spectral Decomposition of operator M M =MP1 +MP2 + ...+MPn = ε1P1 + ε2P2 + ...+ εnPn

MPk =ε kPk

M = 4 13 2

⎛⎝⎜

⎞⎠⎟= 1P1 + 5P2 = 1 1 1 + 5 2 2 = 1 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟+ 5 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and Functional Spectral Decomposition of any function f(M) of M f (M) == f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn

Examples:

M = 4 13 2

⎛⎝⎜

⎞⎠⎟= ± 1 4

1 −41

−43

43

⎛

⎝⎜⎜

⎞

⎠⎟⎟± 5 4

341

43

41

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

M = 4 13 2

⎛⎝⎜

⎞⎠⎟

Last step: make Idempotent Projectors: Pk =

pkε k − εm( )

m≠k∏ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏

Multiplication properties of pj :

p jpk = p j M − εm1( ) = p jM − εmp j1( )m≠k∏

m≠k∏

p jpk = ε jp j − εmp j( )m≠k∏ = p j ε j − εm( )

m≠k∏ =

0 if : j ≠ k

pk ε k − εm( ) if : j = km≠k∏

⎧⎨⎪

⎩⎪

p1 = (M − 5⋅1) = −1 13 −3

⎛⎝⎜

⎞⎠⎟

p2 = (M −1⋅1) = 3 13 1

⎛⎝⎜

⎞⎠⎟

p1p2 =0 00 0

⎛⎝⎜

⎞⎠⎟

Mpk =ε kpk = pkM

(Idempotent means: P·P=P)

PjPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪

Mpk =ε kpk = pkMimplies :MPk =ε kPk = PkM

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Factoring bra-kets into “Ket-Bras:

〈 ε2 |= (3/2 1/2)/k21/21/2

|ε2 〉=k2

1/2-3/2

|ε1 〉=k1

〈 ε1 |= (1/2 -1/2)/k1

| 1 〉 or 〈1 |

| 2 〉or〈2 |

1/4 1/2 3/4 5/41 3/2

-1/2

-1

-3/2

1/4

1/2

3/4Eigen-bra-ketprojectorsof matrix:

M= 4 13 2

The Pj are Mutually Ortho-Normal as are bra-ket 〈εj⏐and⏐εj〉 inside Pj’s

ε1 ε1 ε1 ε2ε2 ε1 ε2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

...and the Pj satisfy a Completeness Relation: 1= P1 + P2 +...+ Pn

=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐

P1 + P2 =1 00 1

⎛⎝⎜

⎞⎠⎟

= ε1 ε1 + ε2 ε2

= 1 00 1

⎛⎝⎜

⎞⎠⎟

{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness

{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness

x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .

ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j

⏐y〉or〈y⏐

⏐x〉 or 〈x⏐

Orthonormality vs. Completeness

“Gauge” scale factors that only affect plots

Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.

PjPk = δ jkPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪1= P1 +P2 +...+Pn

Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.

PjPk = δ jkPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪1= P1 +P2 +...+Pn

1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk

{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness

{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness

x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .

ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j

State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.

Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.

PjPk = δ jkPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪1= P1 +P2 +...+Pn

1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk

{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness

{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness

x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .

ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j

State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.

Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.

PjPk = δ jkPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪1= P1 +P2 +...+Pn

1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk

x y = δ (x, y) = ψ 1(x)ψ *1(y)+ψ 2 (x)ψ *

2 (y)+ ..

However Schrodinger wavefunction notation ψ(x)=〈x⏐ψ〉 shows quite a difference...

Dirac δ-function

{⏐x〉,⏐y〉}-orthonormality with {⏐ε1〉,⏐ε2〉}-completeness

{⏐ε1〉,⏐ε2〉}-orthonormality with {⏐x〉,⏐y〉}-completeness

x y = δ x,y = x 1 y = x ε1 ε1 y + x ε2 ε2 y .

ε i ε j = δ i, j = ε i 1 ε j = ε i x x ε j + ε i y y ε j

State vector representations of orthonormality are quite similar to representations of completeness. Like 2-sides of the same coin.

Orthonormality vs. Completeness vis-a`-vis Operator vs. StateOperator expressions for orthonormality appear quite different from expressions for completeness.

PjPk = δ jkPk =0 if : j ≠ kPk if : j = k

⎧⎨⎪

⎩⎪1= P1 +P2 +...+Pn

1=⏐ε1〉〈ε1⏐+⏐ε2〉〈ε2⏐+...+⏐εn〉〈εn⏐|εj〉〈εj⏐εk〉〈εk⏐=δjk⏐εk〉〈εk⏐ or: 〈εj⏐εk〉=δjk

x y = δ (x, y) = ψ 1(x)ψ *1(y)+ψ 2 (x)ψ *

2 (y)+ ..

ε i ε j = δ i, j = ...+ψ *i (x)ψ j (x)+ψ 2 (y)ψ *

2 (y)+ ....→ dx∫ ψ *i (x)ψ j (x)

However Schrodinger wavefunction notation ψ(x)=〈x⏐ψ〉 shows quite a difference… ...particularly in the orthonormality integral.

Dirac δ-function

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

A Proof of Projector Completeness (Truer-than-true by Lagrange interpolation)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1.

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) .

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

x2= xm

2Pm x( )m=1

N∑

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line,

x2= xm

2Pm x( )m=1

N∑

x1

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line,

x2= xm

2Pm x( )m=1

N∑

x1 x1 x2

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.

x2= xm

2Pm x( )m=1

N∑

x1 x1 x2 x1 x2 x2

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.

x2= xm

2Pm x( )m=1

N∑

Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.

All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1.

x2= xm

2Pm x( )m=1

N∑

Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.

P1 + P2 = j≠1∏ M − ε j1( )j≠1∏ ε1 − ε j( ) + j≠1

∏ M − ε j1( )j≠1∏ ε2 − ε j( ) =

M − ε21( )ε1 − ε2( ) +

M − ε11( )ε2 − ε1( ) =

M − ε21( )− M − ε11( )ε1 − ε2( ) =

−ε21+ ε11ε1 − ε2( ) = 1 (for all ε j )

All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1. Completeness is truer than true as is seen for N=2.

x2= xm

2Pm x( )m=1

N∑

Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

A Proof of Projector Completeness (Truer-than-true)Compare matrix completeness relation and functional spectral decompositions

with Lagrange interpolation formula of function f(x) approximated by its value at N points x1, x2,… xN.

1=P1+P2 +...+Pn = Pkεk∑ =

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑ f (M) = f (ε1)P1 + f (ε2 )P2 + ...+ f (εn )Pn = f (ε k )Pkεk∑ = f (ε k )

M − εm1( )m≠k∏

ε k − εm( )m≠k∏εk

∑

Each polynomial term Pm(x) has zeros at each point x=xj except where x=xm. Then Pm(xm)=1. So at each of these points this L-approximation becomes exact: L(f(xj))= f(xj) . If f(x) happens to be a polynomial of degree N-1 or less, then L(f(x))= f(x) may be exact everywhere.

1= Pm x( )

m=1

N∑

x= xmPm x( )

m=1

N∑

One point determines a constant level line, two separate points uniquely determine a sloping line, three separate points uniquely determine a parabola, etc.

P1 + P2 = j≠1∏ M − ε j1( )j≠1∏ ε1 − ε j( ) + j≠1

∏ M − ε j1( )j≠1∏ ε2 − ε j( ) =

M − ε21( )ε1 − ε2( ) +

M − ε11( )ε2 − ε1( ) =

M − ε21( )− M − ε11( )ε1 − ε2( ) =

−ε21+ ε11ε1 − ε2( ) = 1 (for all ε j )

All distinct values ε1≠ε2≠...≠εN satisfy ΣΡk=1. Completeness is truer than true as is seen for N=2.

However, only select values εk work for eigen-forms MΡk= εkΡk or orthonormality ΡjΡk=δjkΡk.

x2= xm

2Pm x( )m=1

N∑

Lagrange interpolation formula→Completeness formula as x→M and as xk →εk and as Pk(xk) →Ρk

L f (x)( ) = f (xk )·k=1

N∑ Pk (x) where: Pk (x) =

Πj≠k

Nx − x j( )

Πj≠k

Nxk − x j( )

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.

ε1 = 21 −2

1( ), ε2 = 23

21( ){ } , ε1 = 2

1

−23

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, ε2 = 21

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

1 −21

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.

ε1 = 21 −2

1( ), ε2 = 23

21( ){ } , ε1 = 2

1

−23

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, ε2 = 21

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

1 −21

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x K x x K y

y K x y K y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

ε1 K ε1 ε1 K ε2

ε2 K ε1 ε2 K ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

21 −2

1

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⋅ 4 13 2

⎛

⎝⎜⎞

⎠⎟ ⋅ 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= 1 00 5

⎛

⎝⎜⎞

⎠⎟

Use Dirac labeling for all components so transformation is OK

ε1 = 21 −2

1( ), ε2 = 23

21( ){ } , ε1 = 2

1

−23

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, ε2 = 21

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

1 −21

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x K x x K y

y K x y K y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

ε1 K ε1 ε1 K ε2

ε2 K ε1 ε2 K ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

21 −2

1

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⋅ 4 13 2

⎛

⎝⎜⎞

⎠⎟ ⋅ 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= 1 00 5

⎛

⎝⎜⎞

⎠⎟

Use Dirac labeling for all components so transformation is OK

ε1 1 ε1 2

ε2 1 ε2 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

1 ε1 1 ε2

2 ε1 2 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

ε1 1 ε1 ε1 1 ε2

ε2 1 ε1 ε2 1 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

21 −2

1

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⋅ 21

21

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= 1 00 1

⎛

⎝⎜⎞

⎠⎟

Check inverse-d-tran is really inverse of your d-tran.

ε1 = 21 −2

1( ), ε2 = 23

21( ){ } , ε1 = 2

1

−23

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, ε2 = 21

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

(ε1,ε2 )← (1,2) d−Tran matrix (1,2)← (ε1,ε2 ) INVERSE d−Tran matrix

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

1 −21

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

, x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Given our eigenvectors and their Projectors. Diagonalizing Transformations (D-Ttran) from projectors

P1 =(M − 5⋅1)

(1− 5)= 1

41 −1−3 3

⎛⎝⎜

⎞⎠⎟= k1

21

−23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗

21 −2

1( )k1

= ε1 ε1

P2 =(M −1⋅1)

(5 −1)= 1

43 13 1

⎛⎝⎜

⎞⎠⎟

= k221

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⊗ 23

21( )

k2

= ε2 ε2

Load distinct bras 〈ε1| and 〈ε2| into d-tran rows, kets |ε1〉 and |ε2〉 into inverse d-tran columns.

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x K x x K y

y K x y K y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

ε1 K ε1 ε1 K ε2

ε2 K ε1 ε2 K ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

21 −2

1

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⋅ 4 13 2

⎛

⎝⎜⎞

⎠⎟ ⋅ 2

121

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= 1 00 5

⎛

⎝⎜⎞

⎠⎟

Use Dirac labeling for all components so transformation is OK

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟⋅

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

ε1 1 ε1 ε1 1 ε2

ε2 1 ε1 ε2 1 ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

21 −2

1

23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

⋅ 21

21

−23

21

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

= 1 00 1

⎛

⎝⎜⎞

⎠⎟

Check inverse-d-tran is really inverse of your d-tran. In standard quantum matrices inverses are “easy”

ε1 x ε1 y

ε2 x ε2 y

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

†

=x ε1

*y ε1

*

x ε2*

y ε2*

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

x ε1 x ε2

y ε1 y ε2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

−1

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

Analyzing 2D-HO beats and mixed mode eigen-solutions

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0 = (K − 9)(K −11) = (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

10 −11 −1−1 10 −11

⎛⎝⎜

⎞⎠⎟

9 −11=

1 +1+1 1

⎛⎝⎜

⎞⎠⎟

2

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

10 − 9 −1−1 10 − 9

⎛⎝⎜

⎞⎠⎟

11− 9=

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

2

Eigen-projectors Pk

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

Analyzing 2D-HO beats and mixed mode eigen-solutions

= (K − 9)(K −11)= (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

10 −11 −1−1 10 −11

⎛⎝⎜

⎞⎠⎟

9 −11=

1 +1+1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 21 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

10 − 9 −1−1 10 − 9

⎛⎝⎜

⎞⎠⎟

11− 9=

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 2−1/ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 -1 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 1 / 2 +1 / 2 ( ), ε2 = 1 / 2 -1 / 2 ( ) Eigenbra vectors:

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

Analyzing 2D-HO beats and mixed mode eigen-solutions

= (K − 9)(K −11) = (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

10 −11 −1−1 10 −11

⎛⎝⎜

⎞⎠⎟

9 −11=

1 +1+1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 21 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

10 − 9 −1−1 10 − 9

⎛⎝⎜

⎞⎠⎟

11− 9=

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 2−1/ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 -1 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 1 / 2 +1 / 2 ( ), ε2 = 1 / 2 -1 / 2 ( ) Eigenbra vectors:

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

x(t) = ε1 ε1 x(0) e−iω1t + ε2 ε2 x(0) e−iω2t

x1(t)x2(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε1 x(0) e−iω1t + −1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε2 x(0) e−iω2t

Analyzing 2D-HO beats and mixed mode eigen-solutions

Mixed mode dynamics

= (K − 9)(K −11) = (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

qε1qε2 ‹ε2|x(0)›=-1/√2

‹ε1|x(0)›= 1/√2

Beatcos(ω2-ω1)t /2

Carriercos(ω2+ω1)t /2

sin(ω2-ω1)t /2

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

10 −11 −1−1 10 −11

⎛⎝⎜

⎞⎠⎟

9 −11=

1 +1+1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 21 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

10 − 9 −1−1 10 − 9

⎛⎝⎜

⎞⎠⎟

11− 9=

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 2−1/ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 -1 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 1 / 2 +1 / 2 ( ), ε2 = 1 / 2 -1 / 2 ( ) Eigenbra vectors:

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

Fig. 3.3.9 Beats in weakly coupled symmetric oscillators with equal mode magnitudes.

ε1

ε2

x(t) = ε1 ε1 x(0) e−iω1t + ε2 ε2 x(0) e−iω2t

x1(t)x2(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε1 x(0) e−iω1t + −1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε2 x(0) e−iω2t

Analyzing 2D-HO beats and mixed mode eigen-solutions

Mixed mode dynamics

x1(t)x2(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

e−iω1t + e−iω2t

2e−iω1t − e−iω2t

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟= e

−i (ω1+ω2 )2

t

2e−i (ω1−ω2 )

2t+ e

i (ω1−ω2 )2

t

e−i (ω1−ω2 )

2t− e

i (ω1−ω2 )2

t

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

= (K − 9)(K −11)

100% modulation (SWR=0)

= (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

eia + eib

2= e

ia+b2 e

ia−b2 + e

−i a−b2

2

BoxIt (Beating) Web Simulation

qε1qε2 ‹ε2|x(0)›=-1/√2

‹ε1|x(0)›= 1/√2

Beatcos(ω2-ω1)t /2

Carriercos(ω2+ω1)t /2

sin(ω2-ω1)t /2

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 10 −1

−1 10⎛⎝⎜

⎞⎠⎟

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 99 = 0

K1 =ω02 ε1( ) = 9, K2 =ω0

2 ε2( ) = 11,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

10 −11 −1−1 10 −11

⎛⎝⎜

⎞⎠⎟

9 −11=

1 +1+1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 21 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

10 − 9 −1−1 10 − 9

⎛⎝⎜

⎞⎠⎟

11− 9=

1 −1−1 1

⎛⎝⎜

⎞⎠⎟

2

== 1/ 2−1/ 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1 / 2 -1 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 1 / 2 +1 / 2 ( ), ε2 = 1 / 2 -1 / 2 ( ) Eigenbra vectors:

x=x1

k1=9 k12=1

y=x2

k2=9

m1=1 m2=1

Fig. 3.3.9 Beats in weakly coupled symmetric oscillators with equal mode magnitudes.

ε1

ε2

x(t) = ε1 ε1 x(0) e−iω1t + ε2 ε2 x(0) e−iω2t

x1(t)x2(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε1 x(0) e−iω1t + −1/ 2

1 / 2

⎛

⎝⎜⎜

⎞

⎠⎟⎟ε2 x(0) e−iω2t

Analyzing 2D-HO beats and mixed mode eigen-solutions

Mixed mode dynamics

x1(t)x2(t)

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

e−iω1t + e−iω2t

2e−iω1t − e−iω2t

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟= e

−i (ω1+ω2 )2

t

2e−i (ω1−ω2 )

2t+ e

i (ω1−ω2 )2

t

e−i (ω1−ω2 )

2t− e

i (ω1−ω2 )2

t

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟= e

−i (ω1+ω2 )2

t cos(ω 2 − ω1 )t

2

i sin(ω 2 − ω1 )t

2

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= (K − 9)(K −11)

100% modulation (SWR=0)

Note the i phase

= (K − K1)(K − K2 )

Det(K) = 10·10 −1= 99Trace(K) = 10 +10 = 20

eia + eib

2= e

ia+b2 e

ia−b2 + e

−i a−b2

2= e

ia+b2 cos a − b

2⎛⎝⎜

⎞⎠⎟

BoxIt (Beating) Web Simulation

Note: Beat frequency is half-difference of eigenvalues

Videos of Coupled Pendula aided by Overhead Projector

Stronger coupling on the left, illustrated indirectly by a darker looking spring on screen

Launch embedded videos using your browser/App

or ⇐ view on YouTube ⇒ View on YouTubeView on YouTube

BoxIt (Beating) Web Simulation (A=1, B=-0.1, C=0, D=1)

BoxIt (Beating) Web Simulation (A=1, B=-0.1, C=0, D=1)

BoxIt (Beating) Web Simulation (A=1, B=-0.1, C=0, D=1) with frequency ratios

Approximating decimal frequencies ω=α using successive levels of continued fractions.α = n0 +

1

n1 +1

n2 +1

n3 +1

n4 +1!

A0 =α = 3.14159265...

A1 =1

A0 − n0= 7.06...

A2 =1

A1 − n1= 15.99...

A3 =1

A2 − n2= 1.003...

n0 = INT (A0 ) = 3

n1 = INT (A1) = 7

n2 = INT (A2 ) = 15

n3 = INT (A3) = 1

π ≅ = 3.000..

π ≅ 3+ 17= 227

= 3.1428

π ≅ 3+ 1

7 + 115

= 333106

= 3.141509

π ≅ 3+ 1

7 + 115 +1

= 355113

= 3.14159292

Pi (π=3.14159265…) converges rather quickly by cf.

A0 = G = 1.618033989...

A1 =1

A0 − n0= 1.6180...

A2 =1

A1 − n1= 1.6180...

A3 =1

A2 − n2= 1.6180...

n0 = INT (A0 ) = 1

n1 = INT (A1) = 1

n2 = INT (A2 ) = 1

n3 = INT (A3) = 1

G ≅ = 1.000..

G ≅ 1+ 11= 21= 2.000

G ≅ 1+ 1

1+ 11

= 32= 1.500

G ≅ 1+ 1

1+ 11+1

= 53= 1.666...

Not so much for the Golden Mean G=(1+√5)/2=1.618...

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

C202

12

r0

0°

0°

0°

180°

t = 0

1/4

1/2

3/4

0°

45°

-45°

-45°

0°

45°

-45°

|+〉

revivals

or beats

|−〉

|+〉+|−〉√2

|+〉+i|−〉√2

|+〉−i|−〉√2

|+〉−|−〉√2

(φ= 0) (φ= π)(a)

(b)

parity

states

even +45°

odd -45°

localized x

flipped y

L

R

Optical

E(t)Coupled

Pendular12D-HO beats and mixed mode geometry

A “visualization gauge” We hold these two fixed...

t = 1/12

C202

12

r0

0°

0°

0°

180°

t = 0

1/4

1/2

3/4

0°

45°

-45°

-45°

0°

45°

-45°

|+〉

revivals

or beats

|−〉

|+〉+|−〉√2

|+〉+i|−〉√2

|+〉−i|−〉√2

|+〉−|−〉√2

(φ= 0) (φ= π)(a)

(b)

parity

states

even +45°

odd -45°

localized x

flipped y

L

R

Optical

E(t)Coupled

Pendular12D-HO beats and mixed mode geometry

A “visualization gauge” We hold these two fixed...

...and let these two rotate at beat frequency

t = 1/12

C202

12

r0

0°

0°

0°

180°

t = 0

1/4

1/2

3/4

0°

45°

-45°

-45°

0°

45°

-45°

|+〉

revivals

or beats

|−〉

|+〉+|−〉√2

|+〉+i|−〉√2

|+〉−i|−〉√2

|+〉−|−〉√2

(φ= 0) (φ= π)(a)

(b)

parity

states

even +45°

odd -45°

localized x

flipped y

L

R

Optical

E(t)Coupled

Pendular12D-HO beats and mixed mode geometry

A “visualization gauge” We hold these two fixed...

...and let these two rotate at beat frequency

t = 1/12

t = 1/6

t = 1/4

C202

12

r0

0°

0°

0°

180°

t = 0

1/4

1/2

3/4

0°

45°

-45°

-45°

0°

45°

-45°

|+〉

revivals

or beats

|−〉

|+〉+|−〉√2

|+〉+i|−〉√2

|+〉−i|−〉√2

|+〉−|−〉√2

(φ= 0) (φ= π)(a)

(b)

parity

states

even +45°

odd -45°

localized x

flipped y

L

R

Optical

E(t)Coupled

Pendular12D-HO beats and mixed mode geometry

t = 1/12

t = 1/6

A “visualization gauge” We hold these two fixed...

...and let these two rotate at beat frequency

t = 1/4

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0The K secular equation Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)The K secular equation

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

Eigen-projectors Pk

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

2D harmonic oscillator equations Lagrangian and matrix forms and Reciprocity symmetry

2D harmonic oscillator equation eigensolutions Geometric method

Matrix-algebraic eigensolutions with example M= Secular equation Hamilton-Cayley equation and projectors Idempotent projectors (how eigenvalues⇒eigenvectors) Operator orthonormality and Completeness (Idempotent means: P·P=P) Spectral Decompositions Functional spectral decomposition Orthonormality vs. Completeness vis-a`-vis Operator vs. State Lagrange functional interpolation formula Diagonalizing Transformations (D-Ttran) from projectors

2D-HO eigensolution example with bilateral (B-Type) symmetry Mixed mode beat dynamics and fixed π/2 phase

2D-HO eigensolution example with asymmetric (A-Type) symmetry Initial state projection, mixed mode beat dynamics with variable phase

ANALOGY: 2-State Schrodinger: i!∂t|Ψ(t)〉=H|Ψ(t)〉 versus Classical 2D-HO: ∂2tx=-K•x Hamilton-Pauli spinor symmetry (ABCD-Types)

4 13 2

⎛⎝⎜

⎞⎠⎟

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

1 ⋅x(0) = (P1 + P2 ) 10

⎛⎝⎜

⎞⎠⎟= 2

3

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ 2

3 21 ( ) 1

0⎛⎝⎜

⎞⎠⎟+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ -2

1 23 ( ) 1

0⎛⎝⎜

⎞⎠⎟

Spectral decomposition of initial state x(0)=(1,0):

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

1 ⋅x(0) = (P1 + P2 ) 10

⎛⎝⎜

⎞⎠⎟= 2

3

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ 2

3 21 ( ) 1

0⎛⎝⎜

⎞⎠⎟+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ -2

1 23 ( ) 1

0⎛⎝⎜

⎞⎠⎟

= 23

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(23 )+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(-21 )

Spectral decomposition of initial state x(0)=(1,0):

(Note projection onto eigen-axes)

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

ε2

q1 t( ) = 32

cos2t, q2 t( ) = − 12

cos4t⎛

⎝⎜⎞

⎠⎟

1 ⋅x(0) = (P1 + P2 ) 10

⎛⎝⎜

⎞⎠⎟= 2

3

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ 2

3 21 ( ) 1

0⎛⎝⎜

⎞⎠⎟+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ -2

1 23 ( ) 1

0⎛⎝⎜

⎞⎠⎟

= 23

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(23 )+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(-21 )

Spectral decomposition of initial state x(0)=(1,0):

(Note projection of x(0) onto eigen-axes)

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

ε1

q1(0)=√3/2 q2(0)=-1/2

x(0) = 10

⎛⎝⎜

⎞⎠⎟

K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

Fig. 3.3.6 Normal coordinate axes, coupled oscillator trajectories and equipotential (V=const.) ovals for an integral 1:2 eigenfrequency ratio (ω0(ε1)=2.0, ω0(ε2)= 4.0) and zero initial velocity.

ε2

Using derives a parabolic trajectory!

q1 t( ) = 32

cos2t, q2 t( ) = − 12

cos4t⎛

⎝⎜⎞

⎠⎟

cos4t = 2cos2 2t −1

q2 t( ) = − 122cos2 2t + 1

2= − 4

3q1 t( )⎡⎣ ⎤⎦

2 + 12

1 ⋅x(0) = (P1 + P2 ) 10

⎛⎝⎜

⎞⎠⎟= 2

3

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ 2

3 21 ( ) 1

0⎛⎝⎜

⎞⎠⎟+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ -2

1 23 ( ) 1

0⎛⎝⎜

⎞⎠⎟

= 23

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(23 )+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(-21 )

Spectral decomposition of initial state x(0)=(1,0):

(Note projection of x(0) onto eigen-axes)

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

ε1

q1(0)=√3/2 q2(0)=-1/2

x(0) = 10

⎛⎝⎜

⎞⎠⎟

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K =K11 K12K12 K22

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

k1 + k12 −k12−k12 k2 + k12

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 7 −3 3

−3 3 13

⎛

⎝⎜⎜

⎞

⎠⎟⎟x=x1

k1=7-3√3 k12=3√3

y=x2

k2=13-3√3

m1=1 m2=1

K 2 −Trace(K)K + Det(K) = K 2 − 20K + 64 = 0 = (K − 4)(K −16)

K1 =ω02 ε1( ) = 4, K2 =ω0

2 ε2( ) = 16,

The K secular equation

Eigenvalues Kk and squared eigenfrequencies ω0(εk)2

P1 =

K11 − K2 K12

K12 K22 − K2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K1 − K2=

7 −16 −3 3−3 3 13−16

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4 −16=

9 +3 3+3 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

3 33 1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4= 3 / 2

1 / 2

⎛

⎝⎜

⎞

⎠⎟ 3 / 2 1 / 2 ( ) = ε1 ε1

P2 =

K11 − K1 K12

K12 K22 − K1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

K2 − K1=

7 − 4 −3 3−3 3 13− 4

⎛

⎝⎜⎜

⎞

⎠⎟⎟

16 − 4=

3 −3 3−3 3 9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

12

=

1 − 3− 3 3

⎛

⎝⎜⎜

⎞

⎠⎟⎟

4=

−1/ 23 / 2

⎛

⎝⎜

⎞

⎠⎟ −1/ 2 3 / 2 ( ) = ε2 ε2

Eigen-projectors Pk

ε1 = 3 / 2 1 / 2 ( ), ε2 = −1/ 2 3 / 2 ( ) Eigenbra vectors:

ε2

Using derives a parabolic trajectory!

q1 t( ) = 32

cos2t, q2 t( ) = − 12

cos4t⎛

⎝⎜⎞

⎠⎟

cos4t = 2cos2 2t −1

q2 t( ) = − 122cos2 2t + 1

2= − 4

3q1 t( )⎡⎣ ⎤⎦

2 + 12

1 ⋅x(0) = (P1 + P2 ) 10

⎛⎝⎜

⎞⎠⎟= 2

3

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ 2

3 21 ( ) 1

0⎛⎝⎜

⎞⎠⎟+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟⊗ -2

1 23 ( ) 1

0⎛⎝⎜

⎞⎠⎟

= 23

21

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(23 )+

-21

23

⎛

⎝⎜⎜

⎞

⎠⎟⎟

(-21 )

Spectral decomposition of initial state x(0)=(1,0):

(Note projection of x(0) onto eigen-axes)

Spectral decomposition of 2D-HO mode dynamics for lower symmetry

Det(K) = 7·13− 27 = 91− 27 = 64Trace(K) = 7 +13 = 20

ε1

q1(0)=√3/2 q2(0)=-1/2

x(0) = 10

⎛⎝⎜

⎞⎠⎟

Example of a Tschebycheff Polynomial order 2

Pafnuty Chebyshev

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