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Physics Concepts

Classical Mechanics Study of how things move Newton’s laws Conservation laws Solutions in different reference frames (including

rotating and accelerated reference frames) Lagrangian formulation (and Hamiltonian form.) Central force problems – orbital mechanics Rigid body-motion Oscillations lightly Chaos

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2

Mathematical Methods

Vector Calculus Differential equations of vector quantities Partial differential equations More tricks w/ cross product and dot product Stokes Theorem “Div, grad, curl and all that”

Matrices Coordinate change / rotations Diagonalization / eigenvalues / principal axes

Lagrangian formulation Calculus of variations “Functionals” and operators Lagrange multipliers for constraints

General Mathematical competence

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3

Correlating Classical and Quantum Mechanics

Correspondence Principle In the limit of large quantum numbers, quantum

mechanics becomes classical mechanics. First formulated by Niels Bohr, one of the

leading quantum theoreticiansWe will illustrate with Particle in a box Simple harmonic oscillator

Equivalence principle is useful Prevents us from getting lost in “quantum

chaos”. Allows us to continue to use our classical

intuition as make small systems larger. Rule of thumb. System size>10 nm, use

classical mechanics.

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4

1-D free particle

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Classical Lagrangian and Hamiltonian for free 1-D particle

2 2

2 2

2

,

2 2

;

2

T V

T V Total Energy

for free particle

p pE E

m m

p i E ix t

E im x t

L

H

H

H

Schroedinger’s equation for free particle

( )0

i kx te

5

Hydrogen Atom

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Classical Lagrangian and Hamiltonian

2 2 2 1 2

0

2 21 2

20

2 21 2

20

1 1( )

2 4

1

2 2 4

1

2 2 4

r

r

q qr r

r

p q q

r r

p q q

r r

rel

rel

L

H

22 1 2

0

1

2 4

q qE i

r t

H

Schroedinger’s equation for hydrogen

6

Hydrogen Atom

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2 2

1 22

0

1

2 2 4r op opP L q q

E ir r t

H

Schroedinger’s equation for hydrogen

22

22

2 2

1

1 1sin

sin sin

r op

op

P rr r

L

7

Particle in a box

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2 2

22

2sinn

V Em x

nx

L L

2 2 221 2 3

2

34

27

26 71

8

6.62 10 sec

10

16 1.66 10 ( )

2.06 10 10

' .

n n nE h

mL

h j

Let L nm

m kg Oxygen

E joules eV

Thus ideal gas law doesn t need quantum mech

8

Particle in a box

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21 2

21

( ) .( ) 2

classical

Emv E v

mC

P x constv x E

m

no match between quantum and classical

probability

Averaged quantum probability approaches classical constant probability.

* 2

2sin

2sin

n

n n

nx

L Ln

xL L

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Simple harmonic oscillator (SHO)

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2 2 2

2

1 1 2

2 21

( )( ) 2

classical

E kmv kx E v x

m mC

P xv x E k

xm m

10

Expectation values

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Bra-ket notation and Matrix formulation of QM

All wave functions may be written as linear combination of eigenfunctions.

Thus effect of operator can be replaced by a matrix showing effect of operator on each eigenfunction.

All QM operators (p, L, H) have real eigenvalues – They are “Hermitian” operators*

T

T

E functional formalism

E matrix formalism

Exactly like

T I

H

H

*" " "" ketBra

11

Expectation values

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Bra-ket notation and Matrix formulation of QM

All wave functions may be written as linear combination of eigenfunctions.

Thus effect of operator can be replaced by a matrix showing effect of operator on each eigenfunction.

All QM operators (p, L, H) have real eigenvalues – They are “Hermitian” operators*

*

( * )T T

E functional formalism

E matrix formalism

Exactly like

T I except is complex conjugate of

H

H

*" " "" ketBra

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Spin Matrix

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*

*

1 0 0

0 0 0

0 0 1

z

z

z

S functional formalism

S matrix formalism

S

z

z

S

S

13

Wind up

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Classical mechanics is valid for

In other words … almost all of human experience and endeavor.

Use it well!

30,000 /

10

v km s

r nm