PHY373. Homework #2
Due September 16 by 10 am
NOTATION: O t
dx (x, t)O(x, t)
1. We solved the Schrdinger equation for the standard and symmetric innite
o
wells using sin(kx) and cos(kx) solutions. A general solution of the form
(x) = Aeikx + Beikx
is, of
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 10 Solutions
Matthias Ihl
12/05/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
The potential for this
PHY373
Homework 10
April 14, 2011
(1) Write down all sets of possible quantum numbers (excluding spin) for
an electron in a hydrogen atom for values of the principle quantum number
n = 1, 2, and 3.
(2) By many orders of magnitude, strong optical transitio
PHY373
Homework 9
April 7, 2011
(1) Starting with the coordinate transformations
r 2 = x2 + y 2 + z 2
cos = z/r
tan = y/x
(1)
(2)
(3)
you can take derivatives to get the nine relations needed to get the spherical polar
forms of Lx , Ly , and Lz . For exam
PHY373
Homework 1
January 20, 2011
(1) The potential for a simple harmonic oscillator is given by
1
V (x) = kx2 .
2
As we will study in detail later, the ground state wave function is a Gaussian:
(x) = A exp(x2 /a2 )
where a is a constant.
(1.1) Use the
CALCULUS OF VARIATIONS
MA 4311 LECTURE NOTES
I. B. Russak
Department of Mathematics
Naval Postgraduate School
Code MA/Ru
Monterey, California 93943
July 9, 2002
c 1996 - Professor I. B. Russak
1
Contents
1 Functions of n Variables
1.1 Unconstrained Minimu
PHY373. Homework #9
Due 11/20/15 by 10 am
1. Calculate the probability that an electron in the ground state of the hydrogen atom will be
found in the interval 0.9a0 < 5 < 1.1a0 , where a0 is the Bohr radius.
2. Check that when l = n 1, the radial wave fun
PHY373. Homework #1
Due September 9 by 10 am
NOTATION: A
A2 A
2
1. Given the wave function
for x 0
0
2
2
Ax (x c) for 0 x c
(x) =
0
for c x
(a) Find the normalization constant A
(b) Evaluate X , X 2 and X
(c) Evaluate P , P 2 and P
(d) Compute X P . Can
PHY373. Homework #3
Due September 25 by 10 am
1. A particle is known to be in the ground state of a simple harmonic oscillator.
What is the probability that a measurement of its position will nd it outside
the classically allowed region?
2. Using raising(
PHY373. Homework #4
Due 9/30/15 by 10 am
1. Find the eigenvalues and eigenfunctions of the potential:
0 for
V0 for
V (x) =
for
0<x<a
a<x<b
x > band x < 0
2. Prove the following:
(a) [X n , P ] = i nX n1
(b) [f (X), P ] = i
df (X)
dX
(c) [X, XP ] = [X, P
PHY373. Homework #8
Due 11/4/15 by 10 am
1. Show that the angular momentum operators Lx , Ly and Lz are represented in spherical coordinates by
cos cot
= i cos
sin cot
= i
Lx = i
Ly
Lz
sin
and verify that the operator L2 is represented in spherical
PHY373. Homework #7
Due 10/28/15 by 10 am
1. A tunneling microscope can be approximated by a square barrier as shown in the picture.
Compute the transmission coecient and show that in the limit qd
1 it can be approximated by e2qd , where d is the width of
Lecture 8
Angular momentum
75
76
8.1
LECTURE 8. ANGULAR MOMENTUM
Introduction
Now that we have introduced three-dimensional systems, we need to introduce into our
quantum-mechanical framework the concept of angular momentum.
Recall that in classical mecha
FRENCH REVOLUTION
The “Age of Rousseau"
(The Republic)
-: Creation of the Republic
- Execution of Louis XVI
- Committee of Public Safety
«- Reign of Terror
- Thermidorian Reaction
- Concordat of 1801
- War of the 2lqld Coalitio
brannan (slb4343) Rotational Motion Test smith (51716)
This print-out should have 34 questions.
Multiple-choice questions may continue on
the next column or page nd all choices
before answering.
10.0 points
Four equal masses m are so small they can
be tre
PHY373
Homework 8
March 31, 2011
(1) Work out the following commutation relations for all combinations i, j =
x, y, z
[ri , rj ]
[ri , pj ]
[pi , rj ]
[pi , pj ]
(2) More commutators, same combinations
[Li , rj ]
[Li , pj ]
[
]
Lz , r2
[
]
Lz , p2
(3) Mor
PHY373
Useful Formulae
Reection coecient at a potential step (E > V ):
B
k1 k2
=
A
k1 + k2
r=
One dimensional time dependent Schrdinger equation:
o
where k1,2 =
2
2
h
(x, t) + V (x, t) = (x, t)
h
2m x2
t
2m(E V1,2 )/
h
Transmission through a barrier (tunn
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 9 Solutions
Matthias Ihl
11/28/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
The correctly normalized
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 8 Solutions
Matthias Ihl
11/19/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
To calculate the expecta
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 7 Solutions
Matthias Ihl
11/09/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
For the angular part, we
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 6 Solutions
Matthias Ihl
10/29/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
(a) H:
= a,
1
|a =
2
S
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 5 Solutions
Matthias Ihl
10/19/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
We know that
f |(AB )g =
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 4 Solutions
Matthias Ihl
10/13/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
1
Problem 1
For = 1/2:
+
+
dx| (x)|2
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 3 Solutions
Matthias Ihl
09/30/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
We will frequently work in God-given
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 2 Solutions
Matthias Ihl
09/10/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
We will frequently work in God-given
PHY 373 Modern Physics II: Quantum
Mechanics, Homework Set 1 Solutions
Matthias Ihl
09/15/2007
Note: I will post updated versions of the homework solutions on my homepage: http:/zippy.ph.utexas.edu/~msihl/teaching.html
We will frequently work in God-given
PHY373
Mathematica Primer
February 1, 2011
Evaluation of many integrals in calculating matrix elements is greatly facilitated by the use of
a symbolic math program like Mathematica or Maple. The Physics computer facility on the 7th
oor of RLM has Mathemat
PHY373
Useful Formulae
Free particle SE solution:
(x) = A expikx +B expikx
de Brglie wave relation:
o
hk
hk
h
=
p
(x, t) = A expik(x 2m t) +B expik(x+ 2m t)
where k = 2mE/
h
One dimensional time dependent Schrdinger equation:
o
Reection coecient at a pot
PHY373
Homework 3
February 3, 2011
(1) Find the transcendental equation for the odd solutions for the particle
in a nite box. Pick a potential well depth deep enough to support at least
two bound (odd) states, make a plot of the equation and nd the energi