Download Sep 17 - BYU Physics and Astronomy

Physics 451
Quantum mechanics I
Fall 2012
Sep 17, 2012
Karine Chesnel
Quantum mechanics
Announcements
• Homework this week:
Tuesday Sep 18 by 7pm:
HW # 6 pb 2.10, 2.11, 2.12, 2.13, 2.14
Thursday Sep 20 by 7pm:
HW # 7 pb 2.19, 2.20, 2.21, 2.22
• Monday: Review- Practice test
Plan to work on your selected problem
with your group and prepare the solution
to be presented in class (~ 5 to 7 min)
Quantum mechanics
No student assigned to the following transmitters
17A79020
1E71A9C6
Please register your i-clicker at the class website!
Quantum mechanics
Ch 2.3
Harmonic oscillator
V(x)
Solving the Schrödinger equation:
1  2
2
p   m x    E

2m 
Expressing the Hamiltonian
a 
in terms of convenient operators:
1

H    a a   
2

or
x
1
2m 

ip  m x 
1

H    a a   
2

Quantum mechanics
Ch 2.3
Harmonic oscillator
Ladder operators:
Raising operator:
a n  n  1 n1
Lowering operator:
a n  n n1
In this definition, the states  n
are normalized
a 
n 1
a
n
 n1
En  
En
En  
Quantum mechanics
Quiz 8a
What will be the final state of the particle
after applying this operator?
(ignore the coefficients)
a a  a   a   n
A.  n
B.  n1
C.  n 2
D.  n1
E.  n 3
2
3
Quantum mechanics
Ch 2.3
Harmonic oscillator
Stationary states
The ground state is given by the condition
a 0 
1
 ip  m x  0  0
2m 
m
 m   2
 0  x  
 e
 
1/4
Ground energy
a 0  0
E0 
1

2
x2
1
n
n 
 a   0
n!
1

En   n   
2

Quantum mechanics
Ch 2.3
Harmonic oscillator
Stationary states
m
 m   2
 0  x  
 e
 
1/4
n 
1
n
 a   0
n!
x2
The stationary states are orthonormal
*

 m n   nm
Hermite polynomials
Pb 2.10
Building  0  1 and  2
Checking the orthogonality
Quantum mechanics
Ch 2.3
Harmonic oscillator
Energy levels
V(x)
1

En   n   
2

x
Quantum mechanics
Ch 2.3
Harmonic oscillator
Expressing x, p and H in terms of ladder operators:
a 
1
2m 

ip  m x 
• Operator position
x
• Operator momentum
pi
• Operator Hamiltonian
2m
m
2
 a  a 
 a  a 
1
1


H    a a      a a  
2
2


Quantum mechanics
Ch 2.3
Harmonic oscillator
V(x)
Expectation values
x 
a

2m

 a

p i
m
2
a
 a


V 
1

m 2 x 2 
2
4
T 
1 2

p 
2m
4
^

 a  a 
 a  a 
H   cn En
n 1
2
2
2

x

 a    a 
2
2m
 a    a 
2
2m
2
2
 a a  a a
 a a  a a
Pb 2.11, 2.12
Quantum mechanics
Quiz 8b
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