Physics 451
Quantum mechanics I
Fall 2012
Sep 21, 2010
Karine Chesnel
Quantum mechanics
Announcements
Homework this week:
• Thursday Sep 20 by 7pm:
HW # 7 pb 2.19, 2.20, 2.21, 2.22
• Friday 21: Review - Monday 24: Practice test 1
Plan to work on your selected problem
with your group and prepare the solution
to be presented in class (~ 5 to 7 min)
Test 1: Mon Sep 24 – Th Sep 27
Note from the TA about homework
1. Answer the problems completely. A lot of the problems
have multiple parts. For example, they first ask you to do
the derivation, and then ask for a qualitative description,
and finally let you give an analog to something.
Don't just do the math and forget everything else.
2. Use precise terminology to describe phenomena.
For example, in problem 2.2 of Homework 4, you are
supposed to comment on the concavity/divergence of the
function. Those are the terms I am looking for.
Don't write something like "the function dies at infinity".
That is a vague expression and it is also unprofessional.
Muxue Liu
Quantum mechanics
Quiz 9a
a 
1
2m 

ip  m x 
Since the operators a+ and a- are shifting the stationary states
from one level to another,
and since the stationary states are all orthogonal,
the expectations values for x and p on any state will ALWAYS be zero!
A. True
B. False
Pb 2.13
Quantum mechanics
Ch 2.3
Harmonic oscillator
Solving the Schrödinger equation the direct way!
(analytic method)
2


d2
 V ( x)   E

2
 2m dx

d 2 1
2 2


m

x   E
2
2m dx
2
2
V(x)
x
Quantum mechanics
Ch 2.3
Harmonic oscillator
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
x

m
General solution
x
    h   e
 2 /2
Expanding h in power series

h    a0  a1  a2 2  a3 3  ...   a j j
j 0
Quantum mechanics
Ch 2.3
Harmonic oscillator
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
x
d 2
2


K 
2
d


K
Is equivalent to:
a j 2
2 j 1 K

aj
 j  1 j  2
Recursion formula
2E

Quantum mechanics
Ch 2.3
Harmonic oscillator
V(x)
Solving the Schrödinger equation the direct way!
(analytic method)
x
Final solution:
 m 
 n    




1/4
1
2n n !
H n   e
 2 /2
Hermite polynomials
Quantum mechanics
Ch 2.3
Harmonic oscillator
Quantum mechanics
Ch 2.3
Harmonic oscillator
n=100
Quantum mechanics
Quiz 9b
“In quantum mechanics,
the energy of a particle is always quantized”
A. True
B. False
Quantum mechanics
Ch 2.4
Free particle
V = 0 everywhere
p2
  E
2m
Quantum mechanics
Ch 2.4
Free particle
d 2

 E
2
2m dx
2
d 2
2


k

2
dx
General Solution
k
with
 ( x)  Ae  Be
ikx
Complete wave function
 ( x, t )   ( x ) e
 iEt /

2mE
 ikx
 Ae  Be
ikx
 ikx
e
 iEt /
Quantum mechanics
Ch 2.4
Free particle
Wave function represents a physical wave:
( x, t )   ( x)eiEt /  Aeikx eiEt /  Beikx eiEt /
 Aei kxt   Bei kxt 
wave travelling
in the (-x) direction
with speed v
wave travelling
in the (+x) direction
with speed v
with

E
k2

2m
Velocity of the phase

k
V 
k 2m
Quantum mechanics
Ch 2.4
Free particle
Talking about velocity
• Velocity of the phase
v phase

k
 
k 2m
• Analogy with classical velocity
vclassical
p
k
 
m m
vclassical  2v phase
(using the de Broglie formulae)
Quantum mechanics
Ch 2.4
Free particle
Normalization
• A single wave for a given E is NOT a physical solution!


e
 i 2
dx  

• A superposition of waves IS normalizable!
 ( x, t ) 
1
2


 ( k )ei ( kx t ) dk

superposition
(summation)
dispersion
function
Individual
waves
Wave
packet
Quantum mechanics
Ch 2.4
Free particle
 ( x, t ) 
1
2
1
 ( x, 0) 
2
Plancherel’s
theorem
 (k ) 
1
2




 ( k )ei ( kx t ) dk
Wave
packet


ikx

(
k
)
e
dk


 ( x, 0)e  ikx dx

Extension from discrete sum to continuous integration
Fourier
transform
Inverse Fourier
transform
Pb 2.20
Quantum mechanics
Ch 2.4
Free particle
Method:
1. Identify the initial wave function
 ( x, 0)
2. Calculate the Fourier transform
 (k ) 
1
2


 ( x, 0)e  ikx dx

3. Estimate the wave function at later times
 ( x, t ) 
1
2


 ( k )ei ( kx t ) dk

Pb 2.21, 2.22
Quantum mechanics
 ( x, 0)
Quiz 9c
-a
A particle is in a given initial state (x,0)
what will be the shape of the Fourier transform (k)?
A.
-/a
B.
(k )
(k )
/a
C.
k
k
-/a
a
x
(k )
/a
k
Quantum mechanics
Ch 2.4
Free particle
 ( x, t ) 
Dispersion relation
 (k )
1
2


 ( k )ei ( kx t ) dk

where
k2

2m
here
Physical interpretation:
• velocity of the each wave at given k:
v phase 
• velocity of the wave packet:
vgroup

k
d

dk
v phase
k

2m
vgroup
k

m