The Free Particle: V (x) = 0 everywhere
We introduce
(Same as inside of the innite square well, where the potential is zero.) For reasons that will be
clear later, we will write the solution in exponential form instead of sin or cos:
There are no boundary
A. John Mallinckrodt
http:/www.csupomona.edu/~ajm/materials/animations/packets.html
The following movies show wave packets with various combinations of phase and group velocity.
Each wave form is the sum of ve sinusoids with the general format
8
Movie 1:
The delta-function potential
The Dirac delta function: It is an innitely high, innitesimally narrow spike whose area is 1.
Property of the delta function:
This is the most important property of the delta function:
f (x) (x a)dx = f (a)
6
We consider a pot
The nite square well
Bound states
Step 1: Solve Schrdinger equation for all regions
1
Step 2: Apply boundary conditions such that and d /dx are continuous at a and a.
Note: V (x) is even wave functions are either even or odd. Therefore, we only need to im
The nite square well
Bound states
Step 1: Solve Schrdinger equation for all regions
1
Exercise 6 (homework problem)
Consider a step potential
(a) Calculate the reection coefcient for E < V0 and comment on the result.
(b) Calculate the reection coefcient f
Problem solving
Problem 1
A particle of mass m in the harmonic oscillator potential starts out in the state
(x, 0) = A 1 2
m
x
h
2
m 2
h
e 2 x
for some constant A.
(a) What is the expectation value of the energy?
(b) At some later time T the wave function
Formalism of quantum mechanics
So far we have found some interesting properties of simple quantum systems. We now consider the
more general formalism of quantum theory that brings together what we have already discovered
in particular cases.
Quantum theor
Generalized statistical interpretation
The generalized statistical interpretation, together with the Schrdinger equation, is the foundation
of quantum mechanics. It states: If you measure an observable Q on a particle in a state (x, t )
you will get one o
Review
The Quantum Harmonic Oscillator:
'
$
Potential:
1
V (x) = m 2 x2
2
The ground (lowest) solution of the time-independent Schrdinger equation
for the harmonic oscillator is:
m 1/4 m x2
h
e 2
h
1
E0 = h
2
0 (x) =
To nd all other functions we can use n
The Harmonic Oscillator: General Problem
Classical harmonic oscillator: mass m attached to a spring of force constant k.
Hookes law:
Quantum problem:
Want to solve the Schrdinger equation with this potential:
There are two methods to solve it. We will sta
Lecture 4
February 16, 2012
1. The innite square well
A particle in this potential is completely free, except at the two ends, where an innite force
prevents it from escaping.
Lets solve the Schrdinger equation!
First, we seek stationary states
We need to
Homework 2
Problems 2.2, 2.4, 2.5, 2.6, 2.7; Bonus: Problem 1.13
Answers:
2.4
a
2
x=
1
1
3 2( n)2
x2 = a2
p =0
p2 =
n h
a
1
2
3 (n )2
x =
a
2
p =
2.5
2
n h
a
1
(a) A =
2
2
1
x + sin
x e3i t
(b) (x, t ) = ei t sin
a
a
a
(c) x =
(d) p =
2.6
a
32
1 2 cos(3
Homework 3
Problems 2.36, 2.10, 2.12, 2.13 (d) only, 2.14; Bonus: Problem 2.49
Answers:
n2 2 h2
2m(2a)2
2.36
En =
2.10
1 m
2 (x) =
2 h
1/4
x =0
(need to prove)
p =0
2.12
m 2
2m 2
h
x 1 e 2 x
h
(need to prove)
x2 = n +
1
2
p2 = n +
1
mh
2
T=
h
m
1
1
n+
h
Homework 5
Due on April 3
Problems 2.30, 2.34, 2.35, 2.40; Bonus: Problem 2.51
Answers:
1
,
a + 1/
e a cos la
F=
a + 1/
2.30
D=
2.34
(a), (b) see Exercise 6
4 E E V0 ( E E V0 )2
(d) T =
V02
2.35
(a) R = 1/9
(c) T = 8/9
2.40
(a) Three bound states
Bonus An
Homework 6
Due on April 10
Problems 3.5, 3.7, 3.13, 3.14, 3.27, 3.37; Bonus: Problem 3.28
Answers:
d
dx
d
dx
1
(d) (a+ ) =
(ip + m x)
2mh
=
3.5
(a)
3.27
(c) Total probability of getting a1 is 337 = 0.5392
625
0
ict /h
1
(a) |S(t ) = e
0
i sin(bt /h)
0
(
Homework 7
Due on April 17
Problems 3.22, 3.23 (dont need to normalize functions), 4.1, 4.2, 4.3; Bonus: Problem 3.35
Answers:
3.22
3.23
4.2
1 0 2i
(c) A = 2i 0 4
1 0 2i
Eigenvalues are E = 2 . Looking eigenvectors in the form | = c1 |1 + c2 |2
for
yield
Lecture 1
February 7, 2012
1. Quantum mechanics is different from anything else youve learned so far
Up to now, if you were asked to determine the location of an object given its mass and the force
exerted upon it, you would proceed as something like this
Lecture 2
February 9, 2012
Review
The Schr dinger equation:
o
#
h2 2
ih
+V
=
t
2m x 2
"
!
Probability rules for continuous variables:
'
Pa,b =
b
a
$
(x)dx is the probability that x lies between a and b.
(x) is the probability density
(x)dx = 1
x=
x (
Lecture 3
February 14, 2012
1. To gain information about a quantum mechanical system, we use "operators"
Lecture 3, Page 1
Expectation value of momentum p:
p =m
dx
= ih
dt
dx
x
Lecture 3, Page 2
We say that the operator x represents position and the opera
Quantum mechanics in three dimensions
Schrdinger equation in spherical coordinates
How do we generalize the Schrdinger equation to three dimensions?
We previously used
Thus,
Therefore, the Schrdinger equation in three dimensions is
The probability to nd a