Download Physics 451 - BYU Physics and Astronomy

Physics 451
Quantum mechanics
Fall 2012
Karine Chesnel
Phys 451
Announcements
Test 1 next week
Mo Sep 24 – Th Sep 27
• Today: Review - Monday: Practice test
Be prepared to present the solution of
your chosen problem during class (~ 5 to 10 min)
Phys 451
EXAM I
• Time limited: 3 hours
• Closed book
• Closed notes
• Useful formulae provided
Review lectures,
Homework
and sample test
Phys 451
EXAM I
1. Wave function, probabilities
and expectation values
2. Time-independent Schrödinger equation
3. Infinite square well
4. Harmonic oscillator
5. Free particle
Phys 451
Review I
What to remember?
Review I
Quantum mechanics
1. Wave function and expectation values
Density of probability
 ( x, t )   ( x, t )

Normalization:


 ( x, t )  1
2
2
Review I
Quantum mechanics
1. Wave function and expectation values

 

p     i
 dx
x 



x 
  xdx
*
*

“Operator” x
“Operator” p

 

Q    Q  x, i
 dx
x 


*
Quiz 9a
What is the correct expression for the operator
T= Kinetic energy?
A.
  



2m  x 
B.
x 2 / 2m
C.
1   

 
2m  x 
2
2
2
2
1

D.
 m 2 x 2
2m x
E.
2

2m x 2
2
Review I
Quantum mechanics
1. Wave function and expectation values
Variance:
x  x  x
2
2
p  p  p
2
2
 x p 
2
2
2
Heisenberg’s
Uncertainty principle
Review I
Quantum mechanics
2. Time-independent Schrödinger equation
2

2
i

V 
2
t
2m x
Here
V ( x)
The potential is independent of time
General solution:
 ( x, t )   ( x) (t )
“Stationary state”
Review I
Quantum mechanics
2. Time-independent Schrödinger equation
2
1 d
1 d 2
i

V
2
 dt
2m  dx
Function of
time only
 (t )  e
Solution:

iE
E
Function of
space only
t
( x, t )   ( x)eiEt /
Stationary state
Quantum mechanics
Review I
2. Time-independent Schrödinger equation
Q
for each
Stationary state
is independent of time
p m v m
d x
0
dt
^
H E
A general solution is

 ( x, t )   cn  n ( x, t )
n 1
where
 n ( x, t )   n ( x)eiEnt /
^
H   cn En
2
n
Review I
Quantum mechanics
3. Infinite square well
d 2
2


k

2
dx
with
0
a
The particle can only exist
in this region
x
k
2mE
Review I
Quantum mechanics
3. Infinite square well
 3 , E3
n 
2
 n
sin 
a
 a

x

Excited states
 2 , E2
Quantization of the energy
Ground state
0
 1 , E1
a
x
n 2 2 2
En 
2
2ma
Review I
Quantum mechanics
3. Infinite square well

 ( x, t )   cn n ( x)e
n 1

 iEn t /

x
n 1
a
  cn sin(n
)e iEnt /

2
n x
cn    n* ( x) ( x, 0)dx 
sin(
)  ( x, 0)dx

a 
a

^
H  n  En n

*
 n   nm
m
^

H   cn En
n 1
2
Quiz 9b
The particle is in this sinusoidal state.
What is the probability of measuring the energy E0 in this state?
A. 0
B. 1
C. 0.5
D. 0.3
0
a
x
E. 1
9
Review I
Quantum mechanics
4. Harmonic oscillator
V(x)
V ( x) 
1 2 1
kx  m 2 x 2
2
2
a 
1
2m 

ip  m x 
• Operator position
• Operator momentum
1

H    a a   
2

x
x
2m
pi
or
m
2
 a  a 
 a  a 
1

H    a a   
2

Review I
Quantum mechanics
4. Harmonic oscillator
Ladder operators:
a 
n 1
Raising operator:
a n  n  1 n1
Lowering operator:
a n  n n1
n 
1
n
 a   0
n!
1

En   n   
2

a
n
 n1
En  
En
En  
Quantum mechanics
Review I
5. Free particle
p2
  E
2m
d 2
2


k

2
dx
 ( x, t )   ( x ) e
1
 ( x, t ) 
2
k
with

 iEt /
 Ae
  ( k )e

i  kx t 
i ( kx t )
dk
2mE
 Be
 i  kx t 
Wave
packet
Quantum mechanics
Review I
Free particle
Method:
1. Identify the initial wave function
 ( x, 0)
2. Calculate the Fourier transform
1
(k ) 
2


 ( x, 0)e ikx dx

3. Estimate the wave function at later times
1
 ( x, t ) 
2

i ( kx t )

(
k
)
e
dk


Quantum mechanics
Review I
5. Free particle
1
 ( x, t ) 
2
Dispersion relation
 (k )

i ( kx t )

(
k
)
e
dk


here
k2

2m
here
Physical interpretation:
• velocity of the each wave at given k:
• velocity of the wave packet:
v phase 
vgroup

k
d

dk
v phase
k

2m
vgroup
k

m