physics 3 for electrical engineering
DESCRIPTION
Ben Gurion University of the Negev. www.bgu.ac.il/atomchip , www.bgu.ac.il/nanocenter. Physics 3 for Electrical Engineering. Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin. Week 8. Quantum mechanics – raising and lowering operators, - PowerPoint PPT PresentationTRANSCRIPT
Physics 3 for Electrical EngineeringPhysics 3 for Electrical Engineering
Ben Gurion University of the Negevwww.bgu.ac.il/atomchip, www.bgu.ac.il/nanocenter
Lecturers: Daniel Rohrlich, Ron Folman Teaching Assistants: Daniel Ariad, Barukh Dolgin
Week 8. Quantum mechanics – raising and lowering operators ,1D harmonic oscillator • harmonic oscillator eigenvalues and eigenfunctions • matrix representation • motion of a minimum-uncertainty wave packet • 3D harmonic oscillator • classical limit Sources: Merzbacher (2nd edition) Chap. 5 Sects. 1-4;
Merzbacher (3rd edition) Chap. 5 Sects. 1, 3 and Chap. 10 Sect. 6; Tipler and Llewellyn, Chap. 6 Sect. 5.
Let’s see and compare how these two different methods apply to the quantum harmonic oscillator.
We already stated that Schrödinger’s wave equation, with its continuous solutions, and Heisenberg’s matrix algebra, with its quantum jumps, are equivalent.
The 1D harmonic oscillator
x0
The 1D harmonic oscillator
x0
x0
x0
x0
x0
x0
t1
t3
t4
t5
t6
t2
The 1D harmonic oscillator
x0
x0
x0
x0
x0
F(t6)
F(t4)
F(t2)
F(t3)
x0
t1
t3
t4
t5
t6
t2
)(ψ22
)(ψ
2
1)(
)()(
22
22
2
xxk
dx
d
mxE
kxxV
xVdx
dkxxF
The 1D harmonic oscillator
This system is a model for many systems, e.g. molecules made of two atoms.
x0
)(ψ22
)(ψ
2
1)(
)()(
22
22
2
xxk
dx
d
mxE
kxxV
xVdx
dkxxF
x0
The 1D harmonic oscillator
Any system with a potential minimum at some x = x0 may behave like a harmonic oscillator at low energies:
V(x) = V(x0) + (x–x0)2 V′′(x0) + ….
x0
)(ψ22
)(ψ
2
1)(
)()(
22
22
2
xxk
dx
d
mxE
kxxV
xVdx
dkxxF
x0
The 1D harmonic oscillator
Also, a mode of the electromagnetic field of frequency ν behaves like a 1D harmonic oscillator of frequency ν. Its energy levels nhν correspond to n photons of frequency ν.
)(ψ22
)(ψ
2
1)(
)()(
22
22
2
xxk
dx
d
mxE
kxxV
xVdx
dkxxF
The 1D harmonic oscillator
Schrödinger’s equation:
x0
Solving Schrödinger’s equation Schrödinger’s way:
Define a new variable , where
In terms of ξ , Schrödinger’s equation is
Solutions: Try where H(ξ) is a polynomial. ψ is a solution when H(ξ) is one of the Hermite polynomials: H0(ξ) = 1, H1(ξ) = 2ξ, H2(ξ) = 4ξ2–2, H3(ξ) = 8ξ3–12ξ, ….
, 0ψ22
2
d
d
./ mkxm
.2
where
E
, )(ψ 2/2 He
Solving Schrödinger’s equation Heisenberg’s way:
Define new variables
where
Let’s prove that
Try also to prove (for any three operators ) that
, ˆ
ˆ2
ˆ , ˆ
ˆ2
ˆ
m
pix
ma
m
pix
ma
./ mk
. ˆ ˆ , ˆ , 1 ˆ , ˆ , 2
1ˆˆ ˆ
aaHaaaaH
. ˆˆ,ˆˆ,ˆˆ ˆ,ˆˆ BCACBACBA
ˆ,ˆ,ˆ CBA
Raising and lowering operators
To prove:
. ˆ ˆ , ˆ , 1 ˆ , ˆ , 2
1ˆˆ ˆ
aaHaaaaH
. ˆ ˆ2
2
ˆ
2 ˆ , ˆ
2 ˆ
22
ˆ
2
1ˆˆ
ˆˆ
2
2
1ˆˆ
222
222
Hxm
m
p
pxixm
m
p
m
pix
m
pix
maa
Raising and lowering operators
To prove:
. ˆ ˆ , ˆ , 1 ˆ , ˆ , 2
1ˆˆ ˆ
aaHaaaaH
. 1 2 2
ˆ , ˆ 2 2
ˆ
ˆ , ˆ
ˆ 2
ˆ , ˆ
ii
xpi
m
pix
m
pix
maa
Raising and lowering operators
To prove:
. ˆ ˆ , ˆ , 1 ˆ , ˆ , 2
1ˆˆ ˆ
aaHaaaaH
. ˆ ˆˆˆ )1 ˆˆ(ˆ
ˆˆˆ )ˆ ˆ(ˆ
ˆ , 2
1ˆˆ ˆ , ˆ
aaaaaaa
aaaaaa
aaaaH
Suppose is an eigenstate of with eigenvalue E. Then
using Therefore, is an eigenstate of
with eigenvalue . We call a raising operator.
Homework: Show that is a lowering operator, i.e.
ψ H
. ψˆ ψ ˆ ˆˆ ψˆˆ aEaHaaH
.ˆ ˆ , ˆ aaH ψˆa H
E a
a
, ψˆ ψ ˆ ˆˆ ψˆˆ aEaHaaH
Suppose is an eigenstate of with eigenvalue E. Then
using Therefore, is an eigenstate of
with eigenvalue . We call a raising operator.
The harmonic oscillator must have a ground state – call it ψ0(x) or – with minimum energy. For we have which means
and therefore
ψ H
.ˆ ˆ , ˆ aaH ψˆa H
E a
, ψˆ ψ ˆ ˆˆ ψˆˆ aEaHaaH
0 00ˆ a
, )(ψ )(ψ ˆ
ˆ0 00 xdx
d
mxx
m
pix
. )(ψ 2/ 00
2 xmeAx
0
Since and , the ground-state energy
is , and the energy of the state , defined by
is
From
we learn that Thus we call the number
operator. Since we can write .
Likewise, since , we can
write
2
1ˆˆ ˆ
aaH 00ˆ a
20
E n
,0ˆ )!( 2/1 n
ann . 2
1
nEn
nnnEnHnaa n 2
1ˆ 2
1ˆˆ
. ˆˆ nnnaa ˆˆ aa
nnaan ˆˆ 1 ˆ nnna
1 1ˆˆ ˆ ˆ nnaannaan
.11 ˆ nnna
Since and , the ground-state energy
is , and the energy of the state , defined by
is
Normalization:
So the ground state normalization is Then for all n, (Prove it!)
2
1ˆˆ ˆ
aaH 00ˆ a
20
E n
,0ˆ )!( 2/1 n
ann
.
||
||
|)(ψ|1
20
/ 20
20
2
mA
dxeA
dxx
xm
.
|| 40 m
A . 1 nn
. 2
1
nEn
What are the lowest eigenstates of the harmonic oscillator?
Note that the eigenfunctions ψn(x) are even or odd in x. Why?
32
3
1 )(ψˆ
6
1 )(ψ
12
2 )(ψˆ
2
1 )(ψ
2
)(ψˆ )(ψ
)(ψ
2/ 34/3
1/40
3
3
2/ 24/1
1/20
2
2
2/ 4/3
1/4
1/2
01
2/ 4/1
0
2
2
2
2
xm
xm
xm
xm
exxmm
xax
exmm
xax
xem
xax
em
x
Harmonic oscillator eigenvalues and eigenfunctions
This Figure is taken from here.
Harmonic oscillator eigenvalues and eigenfunctions
This Figure is taken from here.
If the harmonic oscillator represents a mode of the electromagnetic field, an energy level ½ represents n photons each having energy , plus additional “zero-point energy” of per mode.
Harmonic oscillator eigenvalues and eigenfunctions
) ( nEn
2
Matrix representation
In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since ½ if m = n and vanishes otherwise, we can represent as an infinite matrix:
,ˆ and ˆeven and etc., ,ˆ ,ˆ ,ˆ pxaaH
) ( ˆ nnHm H
...............
...7000
...0500
...0030
...0001
ˆ2H
Matrix representation
In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since if m = n–1 and vanishes otherwise, we can represent as an infinite matrix:
,ˆ and ˆeven and etc., ,ˆ ,ˆ ,ˆ pxaaH
nnam ˆa
...............
...0000
...3000
...0200
...0010
a
Matrix representation
In the basis of harmonic-oscillator eigenvectors, we can represent the operators as matrices. Since if m = n+1 and vanishes otherwise, we can represent as an infinite matrix:
,ˆ and ˆeven and etc., ,ˆ ,ˆ ,ˆ pxaaH
1ˆ nnama
...............
...0300
...0020
...0001
...0000
a
The normalized harmonic-oscillator eigenvectors themselves are the basis vectors:
. ... ,
...
1
0
0
0
3 ,
...
0
1
0
0
2 ,
...
0
0
1
0
1 ,
...
0
0
0
1
0
The transpose of a matrix is written and defined by
.
.........
...
...ˆ ,
.........
...
...ˆ
db
ca
Mdc
ba
M T
M TM
:ˆ ˆˆˆ jiT
ijT MiMjjMiM
The adjoint of a matrix is written and defined by
M M
:ˆ ˆˆˆ **jiij MiMjjMiM
.
.........
...**
...**ˆ ,
.........
...
...ˆ
db
ca
Mdc
ba
M
The adjoint of a matrix is written and defined by
Any observable is self-adjoint, i.e.
“ is Hermitian” and “ is self-adjoint” mean the same thing.
M M
.
.........
...**
...**ˆ ,
.........
...
...ˆ
db
ca
Mdc
ba
M
A
A A
.ˆˆ AA
:ˆ ˆˆˆ **jiij MiMjjMiM
The adjoint of a matrix is written and defined by
Any observable is self-adjoint, i.e.
“ is Hermitian” and “ is self-adjoint” mean the same thing.The raising and lowering operators and are not self-adjoint, but are adjoints of each other:
M M
.
.........
...**
...**ˆ ,
.........
...
...ˆ
db
ca
Mdc
ba
M
A
A A
.ˆˆ AA
:ˆ ˆˆˆ **jiij MiMjjMiM
aa
. ˆˆ , ˆˆ aaaa
are manifestly self-adjoint:pxH ˆ and ˆ ,ˆ
,
............
...500
...030
...001
ˆ2
H
are manifestly self-adjoint:pxH ˆ and ˆ ,ˆ
.
............
...020
...201
...010
2
ˆˆ
2
ˆ
,
............
...020
...201
...010
2
ˆˆ 2
ˆ
miaa
mip
maa
mx
Vectors, too, have adjoints. For any we have
and, for example,
Try to prove, for any two operators , the rule
ψ
, ψψ
.
...
0
0
1
...001
. ˆˆ)ˆˆ( ABBA
BA ˆ,ˆ
Motion of a minimum-uncertainty wave packet
The ground state wave function ψ0(x) is a minimum-uncertainty wave function: we can calculate
In general, the time evolution of any initial wave function Ψ(x,0) can be obtained from the expansion of Ψ(x,0) in the basis of energy eigenstates. If the initial wave function is
for given cn, then the wave function at time t is
. 2
, 2
,
2
px
mp
mx
, )0(0
n
n nc
. )(0
2/
0
/
n
tinn
ti
n
tiEn necenect n
Motion of a minimum-uncertainty wave packet
The period of a classical harmonic oscillator having angular frequency ω is T = 2π /ω . If we add T to t in Ψ(x,t), the wave function does not change, up to an overall phase factor –1:
Schrödinger even found a solution Ψ(x,0) that moves between x = R and x = –R while the probability distribution |Ψ(x,0)|2 keeps its (minimum uncertainty) shape:
. )(0
2/
0
/
n
tinn
ti
n
tiEn necenect n
. )( )( tTt
.
|),(| 2/) cos( 4/1
2 2
tRxmem
tx
Motion of a minimum-uncertainty wave packet
Schrödinger thought his solution might be typical, but it is not. Usually the probability distribution spreads over time. Prove that a free 1D wave packet spreads, that if it is initially
then at time t the wave packet is
Show that Δp = is constant in time, but
, )0,(22 )(4//2/1
)(4/1
)2(xxxpi
x eex
.
2/1 2 4/1)2(
]/2)(4/[)/(/2/
),(
222
xmtix
mtixmtpxxpimtpi
txee
x2
. /)( 2222 mtpxtx
3D harmonic oscillator
The Schrödinger equation for a general 3D harmonic oscillator,
allows for a harmonic potential with different strengths along the three axes.
Show that the eigenfunctions are products of 1D harmonic oscillator eigenfunctions. What are the lowest energies and their degeneracies as a function of kx, ky and kz?
, ψ2222
),,(ψ 22222
z
ky
kx
k
mzyxE zyx
Classical limit
A classical particle in a square well has equal probability to be at any point inside. How about a quantum particle in its ground state? a quantum particle in a highly excited state?
The probability P(x) for classical harmonically oscillating particle to be at any point x is inversely proportional to its speed at that point: P(x) ~ (2E/m – ω2x2)–1/2, where E is total energy.
n = 0 n = 10
.