Homework 4 Solutions
Physics 486
Fall 2013
Problem 1.
p
i
a. The first excited state by noting a 0 = 21 . We write the raising operator as a = m
x m
p).
2~ (
Now acting a on the given state:
r
m
i
m 1/4 mx2 /2~
x
p
e
a 0 =
2~
m
~
r
m 1/4 m
2
i
x
i~
emx
Fall 2012
P486 Checkpoint 13 Answers
1. If two spin 1/2 particles are combined into a bound state, there are four possible spin
states for the bound state system. Describe them. Hence what are the possible values
for the total spin of the system? And what
Physics 486 - Fall 2012
Discussion 6 Problem Solutions
A particle of mass m is conned inside a SHO potential with classical angular frequency .
(a) Calculate x2 and p2 for n , the n-th stationary state of the SHO. Recall that
h
a+ + a
2m
h
m
a+ a
p=i
2
a
Physics 486 - Fall 2012
Discussion 8 Problem Solutions
A two-state system has a Hamiltonian which can be written in some basis as
H=K
2
2i
2i
,
5
where K > 0 is a constant.
(a) Calculate the allowed energies E1 and E2 , where E1 < E2 .
H | = E |
K
2K E
2i
Physics 486 - Fall 2012
Discussion 9 Problem Solutions
Problem 1
A two-state system has the Hamiltonian
1
0
H=K
0
,
1
1
0
, and |2 =
. Here, |1 has energy
0
1
E1 = K, and |2 has energy E2 = K. Consider also another operator A associated with an observable
Physics 486 - Fall 2012
Discussion 7 Problem Solutions
A particle of mass m inside a SHO potential with classical angular frequency has the initial wave function
2
(x, 0) = Ax3 emx
/2
h
,
for some constant A, which need not be calculated.
m
x.
h
(a) Rewri
Physics 486 - Fall 2012
Discussion 1 Problem Solutions
1. In class, we derived the Planck blackbody spectrum:
hf
8f 2
df
3
hf /kT 1
c e
u(f, T ) df =
(1)
(a) Show that the emitted power is proportional to T 4 . Show that the Stefan-Boltzmann constant is
=
Physics 486 - Fall 2012
Discussion 2 Problem Solutions
A particle conned in an innite square well of width a has the initial wave function
2
2
sin
x
a
a
(x, 0) =
,
0 x a,
as shown in the gure. We can write (x, 0) as a superposition of stationary states.
(
Physics 486 - Fall 2012
Discussion 3 Problem Solutions
A free particle of mass m has the wave function at t = 0
(x, 0) = e|x| ,
where > 0.
(a) Find (k), the momentum space wave function at t = 0.
1
(k) =
2
=
2
=
2
(x, 0) eikx dx
0
ex eikx dx +
e+x eikx d
Fall 20112
P486 Checkpoint 11 Answers
1. Describe the strategy for solving the radial equation for the hydrogen atom. What
are the similarities and dierences with the strategy for the analytic solution for
the SHO?
The methods used to analytically solve t
Fall 2012
P486 Checkpoint 10 Answers
1. How is the angular equation in 2D related to the eigenvalue equation for the
angular momentum operator, Lz ? And how is the angular equation in 3D related
to the eigenvalue equation for the angular momentum operator
Fall 2012
P486 Checkpoint 01 Answers
1. Read lecture notes, Ch. II-III.1 and Griths (Ch. 2.1-2).
Why is it necessary to normalize the wave function of a quantum particle?
Explain.
The statistical interpretation of quantum mechanics says that the modulus s
Fall 2012
P486 Checkpoint 3 Answers
1. Read lecture notes, Ch. III.2-3 and Griths (Ch. 2.4-5).
What is a wave packet? Describe how Plancherels theorem is used to describe the wave
function of a free particle.
A wave packet is a normalizable combination of
Fall 2012
P486 Checkpoint 5 Answers
1. Why is the simple harmonic oscillator (SHO) such an important problem?
Any smooth potential with a minimum can be approximated in the vicinity of the minimum (assuming
the second derivative does not vanish) by a harm
Fall 2012
P486 Checkpoint 4 Answers
1. Read lecture notes, Ch. III.3-4 and Griths (Ch. 2.5-6).
Interpret Eq. [2.124] in Griths. What does this equation tell us about the derivative
of the wave function if the potential energy function is nite? And what do
Fall 2012
P486 Checkpoint 8 Answers
1. Consider a hermitian operator with a discrete eigenvalue spectrum. Describe the properties of the eigenfunctions of such operators.
Eigenfunctions of a hermitian operator always have real eigenvalues. Eigenfunctions
Fall 2012
P486 Checkpoint 6 Answers
1. What are Hermite polynomials? Describe their properties.
Hermite polynomials describe the form of the polynomials (terminating power series) for the various
energy levels of the SHO. They are denoted Hn (x) and are c
Fall 2012
P486 Checkpoint 9 Answers
1. Consider the time-independent Schrodinger equation in three dimensions.
How does separation of variables help us nd a solution? Explain.
The three-dimensional time-independent Schrodinger equation is a complicated pa
Fall 2012
P486 Checkpoint 12 Answers
1. Describe the Stern-Gerlach experiment. What does it measure, and how is
the measurement related to angular momentum? Explain.
The Stern-Gerlach experiment is a procedure which can be used to separate particles
based
Physics 486 - Fall 2012
Discussion 4 Problem Solutions
Suppose (x, t) is the position space wave function at time t. Recall that (p, t), the momentum space wave
function at time t is the Fourier transform of (x, t).
1
2
h
1
(x, t) =
2
h
h
(x, t) eipx/ dx
Physics 486 - Fall 2012
Discussion 5 Problem Solutions
Problem 1
A particle of mass m and energy E > 0 approaches a potential drop V0 .
V (x) =
0,
for x < 0,
V0 , for x > 0,
where V0 > 0.
(a) Starting from
(x) =
eik1 x + A eik1 x , for x < 0,
B eik2 x ,
f
Homework 8 Solutions
Physics 486
Fall 2013
Problem 1 (Griffiths 3.7).
with the same eigenvalue, say , so
a. By definition, we can say f (x) and g(x) are eigenfunctions of Q
Qf (x) = f (x) and Qg(x) = g(x). Then take some arbitrary linear combination af (
Homework 1 Solutions
Physics 486
Fall 2013
Problem 1.
a. Since R = 4c E , we must evaluate the integral:
Z
c
E df
R =
4
0
Z
hf
c
8f 2
=
df
4 0
c3 ehf /kB T 1
Z
f3
2h
df
=
c2 0 ehf /kB T 1
We make a change of variables to a dimensionless quantity hf /kB
Homework 6 Solutions
Physics 486
Fall 2013
Problem 1.
We can find the force on the wall as:
F
E
x
n 2 2 ~2
x 2mx2
2 2 2
n ~
mx3
=
=
=
Using n = 2, the acceleration is then immediately found from the force as:
a=
F
4 2 ~2
=
M
mM x3
Problem 2 (Griffiths
Homework 11 Solutions
Physics 486
Fall 2013
Problem 1.
a. The eigenstates will simply be the product of the two one-dimensional harmonic oscillators:
r
r
r
m
1
m
mx
my
(x x2 +y y 2 )
1/4 2~
n1 ,n2 = p
Hn1
(x y ) e
x Hn2
y
~
~
2n1 +n2 (n1 )!(n2 )! ~
This i
Homework 9 Solutions
Physics 486
Fall 2013
Problem 1 (Griffiths
3.12).
R
We have hxi = (x, t)x(x, t) dx. Then, we write the Fourier transform of :
Z
1
(x, t) =
eipx~ (x, t) dp
2~
Then, we can write:
hxi =
=
=
Z
Z Z
0
1
eip x/~ (p0 , t) dp0 x
eipx/~ (x
HW 2 Solutions
Physics 486
Problem 1.
a. We show x
and p do not commute by evaluating x
p and px
:
(x)
x
= ih ihx
x
= x ih
x
= ihx
x
= ih
px
x
p
Since these results differ, x
and p do not commute.
b. We can evaluate a commutator by evaluating it on a s
Homework 14 Solutions
Physics 486
Fall 2013
Problem 1 (Griffiths 4.11).
R
a. We can normalize the wavefunction by taking |R|2 dV = 1. Calculating from equation 4.82:
Z
|R20 |2 r2 dr
1 =
0
Z 2
c0
r 2 r/a 2
=
1
e
r dr
2a
2a
0
c 2 Z
z 2 z 2
0
a3
1
e z dz
Homework 3 Solutions
Physics 486
Fall 2013
Problem 1.
First, we find the time dependent wave function (x, t), by applying the time evolution operator, eiHt/~ ,
to the time independent state. So:
(x, t)
=
eiHt/~ (1 + 2 )
=
1 eiE1 t/~ + 2 eiE2 t/~
We then e
Homework 10 Solutions
Physics 486
Fall 2013
Problem 1 (Griffiths 3.12).
a. We can find the matrix elements by noting the following:
h1|H0 |1i = E1
h1|H0 |2i = 0
h2|H0 |1i = 0 h2|H0 |1i = E2
We can then immediately read off the matrix for H0 , as:
0 = E1