Physics 451
Quantum mechanics I
Fall 2012
Sep 7, 2012
Karine Chesnel
Phys 451
Announcements
Homework
• Homework 3: F Sep 7th by 7pm
Pb 1.4, 1.5, 1.7, 1.8
• Homework 4: T Sep 11 by 7pm
Pb 1.9, 1.14, 2.1, 2.2
• Homework 5: Th Sep 13 by 7pm
Pb 2.4, 2.5, 2.7, 2.8
Please don’t forget to submit your homework on time!
Help sessions: T Th 3-6pm
Phys 451
Remarks from the TA
after grading the first homework
1. Simplify your answers to their simplest forms.
Don't leave it like x=(1/3-1/5)^(1/2) or x=1-Sigma,
while you already have a value for Sigma.
2. Don't make your "rough sketch" too rough. Label your axes, and draw
the curve nicely. Be a little more professional than the Physics 121 students.
3. Some simple calculus and graphs can be done by hand,
such as a standard Gaussian. Don't rely entirely on Mathematica.
4. Don't write too compactly. Leaving enough space in your writing not
only benefits the TA but also helps yourself when you go back and check.
5. Write your CID instead of your name.
Quantum mechanics
Quiz 3a
Evolution of Y in time?
“If the wave function is normalized at a time t,
it is then normalized at any time.”
A. True
B. False
Quantum mechanics
Probabilities &
Wave function
Density of probability (now function of space and time):
 ( x, t )  Y ( x, t )
2

Normalization:

Y ( x, t ) dx  1
2

Solutions Y ( x, t ) have to be normalizable:
- needs to be square-integrable
Quantum mechanics
Expectation values
Probabilities
Quantum Mec.
Y ( x, t )
 ( x)
Density of probability:
2
Average position x:

x 


x  ( x)dx
x 
 x Y ( x, t )
2
dx


Average value for f(x):

f ( x) 



f ( x)  ( x)dx
f 


f ( x) Y ( x, t ) dx
2
Quantum mechanics
Expectation values

f 

f Y ( x, t ) dx
2


f 

Y * ( x, t ) f Y ( x, t )dx

The expectation value is the average of all the measurements
of the quantity f on a ensemble of identically prepared particles
Differentiation between expectation value and most probable value
See pb 1.4 and pb 1.5
Quantum mechanics
Expectation values
Evolution of <x> in time?
d x

d
2
 v 
x Y ( x, t ) dx

dt
dt 
Expectation value
for the velocity

i
* Y
v   Y
dx
m 
x

Expectation value
for the momentum
Y
p  m v  i  Y
dx
x

*
Schroedinger
equation
Quantum mechanics
Expectation values
Generalization

 

p   Y  i
 Ydx
x 



x 
 Y xYdx
*
*

“Operator” x
“Operator” p

 

Q   Y Q  x, i
 Ydx
x 


*
Quantum mechanics
Expectation values
Examples
• Kinetic energy:
1 2 p2
T  mv 
2
2m


2
2
 
* 1 
* 
T Y
Y
Ydx
 i
 Ydx  
2

2m 
x 
2m 
x

• Angular momentum:
2
Lrp
and so on..
Quantum mechanics
Ehrenfest’s theorem
V
 
dt
x
d p
Equivalent to Newton’s second law
ma  F
See pb 1.7
Quantum mechanics
Effect of potential offset?
V+V0
V
Y
? (picking an extra phase)
Q
? (no effect)
See pb 1.8