Physics 451
Quantum mechanics I
Fall 2012
Karine Chesnel
Phys 451
Announcements
Monday Sep 3: NO CLASS (Holiday)
Homework
• Homework 1: Today Aug 31st by 7pm
Pb 1.1, 1.2, 1.3
• Homework 2: Group presentations Sep 5th
• Homework 3: F Sep 7th by 7pm
Pb 1.4, 1.5, 1.7, 1.8
Help sessions: T Th 3-6pm
Phys 451
Announcements
Next class Sep 5th
Group presentations
Will count as homework 2, 20 points plus 5 quiz points for presenting
Famous scientist who contributed to the foundation
of Quantum Mechanics
Einstein
Schrödinger
Heisenberg
Dirac
Pauli
Born
Bohr
Planck
De Broglie
Quantum mechanics
Probabilities
Discrete variables
Examples of discrete distributions:
Distribution of scores
in a class
Age pyramid
for a certain population
(Utah, 2000)
Quantum mechanics
Probabilities
Discrete variables
Distribution of the system
Probability for a given j:
N ( j)
N ( j)
P( j ) 
N

Average value of j:
j   jP( j )
j 0
Average value of
a function of j
Average value

f ( j )   f ( j ) P( j )
j 0
“Expectation” value
Example:
number of siblings
for each student
in the class
Quantum mechanics
Quiz 2a
What is the definition of the variance?
2
A.
j
B.
j2
C.
j2  j2
D.
j  j
E.
j
2
2
2
 j2
Quantum mechanics
Probabilities
Discrete variables
j  j  j
The deviation:
j  0
Variance
   j   j  j
2
The standard deviation
2

2
j
2
 j
2
2
Quantum mechanics
Probabilities
Discrete variables
The variance defines how wide/narrow a distribution is
intensity
brightness
Distribution of scores
in a class
Wide: large 
Spectral analysis
of a photograph
Narrow: small 
Quantum mechanics
Probabilities
Continuous variables
The probability of finding
the particle in the segment dx
The density of probability:
P   ( x)dx
 ( x)
b
Probability to find the particle
between positions a and b:
  ( x).dx
a

Normalization:
  ( x).dx  1

Quantum mechanics
Probabilities
Continuous variables

x 
Average values:
 x ( x)dx


f ( x) 

f ( x). ( x )dx

Variance:
  x  x
2
2
2
  f ( x)  f ( x)
2
2
2
Quantum mechanics
Connection to
Wave function
Density of probability (now function of space and time):
 ( x, t )   ( x, t )
2

Normalization:

 ( x, t ) dx  1
2

Solutions  ( x, t ) have to be normalizable:
- needs to be square-integrable
Quantum mechanics
Quiz 2b
Is this wave function square integrable or not?
 ( x, t )  Aeit kx 
A. YES
B. NO
Quantum mechanics
Normalization of
Wave function

Normalization:

 ( x, t ) dx  1
2

Can  stay normalized in time?
If  satisfies the Schrödinger equation and is normalizable, then indeed



d
2
   ( x, t ) dx   0
dt  
