Physics 507 Diagnostic
This little test is supposed to give you some feedback if you have the necessary
background knowledge to take this course. It is worth 5 percent of the total
credit you can get for this course, you will get these 5 percent for takin
Homework Solutions Physics 507
Problem 1: Hydrogen polarizability and van der Waals interaction
Study the interaction of a hydrogen atom with an external electric eld by
(a) by perturbation theory, summing over all bound states,
(b) by an approximate trea
Physics 507 Midterm
Problem 1: (10 points) Time-evolution, particle in a box
A particle of mass m bounces elastically between two infinite parallel plane walls
separated by a distance a, i.e. in a potential
0
if |x| < a2 ;
V (x) =
, if x a2 .
The energies
Homework Solutions Physics 507
Problem 17:
Consider the motion of a particle of mass m in the onedimensional potential
V (x) =
V0 X
(x n)
n=
where denotes the Dirac delta-function and is the periodic distance. Prove
that for such a potential, the implici
Homework Solutions Physics 507
Problem 32:
The motion of a particle of mass m in one dimension is described by the
Hamiltonian
p2
+ |x|.
H=
2m
with > 0.
a. Derive the asymptotic behavior of the ground-state wave function as
|x| becomes large.
b. Use a Gau
Homework Solutions Physics 507
Problem 23:
Determine the total cross section for the scattering of slow particles (ka 1)
by a potential V (r) = C(r a).
Solution:
Since ka 1 it is sufficient to consider s-wave ( = 0) scattering. We have
two regions: 0 r <
Homework Solutions Physics 507 Problem 21:
Consider a system of angular momentum = 1. The basis of its state space
is given by cfw_|+i, |0i, |i, which are the eigenstates of the z component of
the angular momentum operator Lz . Let the Hamiltonian for thi
Homework Solutions Physics 507
Problem 35: Transmission and Reflection with spin
In three dimensions, a spin-1/2 particle with mass m moves in the potential
V (x) =
V 0 z
0,
x > 0;
x 0.
a) A beam of particles comes from in the positive x-direction with
e
Physics 507 Final
Problem 1: 10 points
I want to see if you have reviewed the solutions that I gave you:
Consider an electric dipole consisting of two electric charges e and e at a
mutual distance 2a. Consider also a particle of charge e and mass m with a
Homework Solutions Physics 507
Problem 26:
(a) Evaluate, in the Born approximation, the differential cross section for
the scattering of a particle of mass m by a delta-function potential
V (r) = B(r). What is the angular and velocity dependence ?
(b) Fin
Homework Solutions Physics 507
Problem 13:
Does a particle with mass m have bound
tential
+,
V (x) = h2 /ma2 ,
0,
states in the one-dimensional pox<0
0 < x < a?
x>a
If yes, determine the corresponding energies.
Solution:
We assume that there is a bound s
Homework Solutions Physics 507 Problem 29: Warm-up excercise 1
A particle is placed in an infinite one-dimensional well of width a:
V (x) =
0, if 0 < x < a;
, everywhere else.
It is subject to a perturbation W of the form
a
W (x) = aw0 x
2
where w0 is a
Physics 507 Midterm
Problem 1: 10 points
(a) Give the definition of a hermitean operator.
(b) Which of the following operators are hermitean
x,
d
,
dx
ix2 ,
d2
?
dx2
(c) What can you say in general about the non-degenerate eigenfunctions of
H ?
(d) What c
Physics 507 Final
Problem 1: 10 Points
A particle with m moves in an attractiv, spherically symmetric potential
V (r) =
2U
h
(r a)
2m
where U > 0 and a > 0.
(a) Write down the general analytic form of the stationary wave funktionen
for given angular mo
2 FOURIER ANALYSIS AND GENERALISED FUNCTIONS
1.2. Knovirledge assumed of the reader
The book is written for mathematical readers with some interest
in, and general knowledge of, methods of mathematical proof
(particularly of results concerned with limitin
Homework Physics 507
Solutions are due on Mo., Oct. 5, in class.
Problem 5:
Given is a particle with mass m in a one-dimensional innitely deep square
potential well of width a. In a series of measurements of the energy of the 3
lowest levels are found wit
Homework Physics 507
Solutions are due on Fr., Sept 25, in class.
Problem 1: More on the Gaussian wave packet
A Gaussian wave packet at t = 0 is
(x) = N ex
2 /2a2
.
a) Determine N and (k).
b) Calculate the mean values x , x2 , k , and k 2 which are dened
Homework Solutions Physics 507
Problem 5:
Given is a particle with mass m in a one-dimensional innitely deep square
potential well of width a. In a series of measurements of the energy of the 3
lowest levels are found with equal probability.
a) Calculate
Homework Solutions Physics 507
Problem 1:
I want that you review the problem of the particle in a square-well box.
Therefore:
Consider a particle in a potential well. The potential energy V of the
particle is:
, if x < 0;
V (x) = 0, if 0 < x < a;
, if a
Homework Solutions Physics 507 - Fall 2013
Problem 1: Used fall 2013
Calculate the Fourier transforms
1
eikx f (x) dx
f (k) =
2
of the following functions. Without loss of generality, assume > 0 whenever
necessary.
a) f1 (x) =
1
(2
2
b) f2 (x) =
1 |x|/
Homework Solutions Physics 507
Problem 4:
By denition, the Hermitian operator A is called an observable if its orthonormal system of eigenvectors forms a basis in the state space. Two operators A
and B commute when [A, B] = AB BA = 0. Prove the following
Homework Solutions Physics 507
Problem 8:
The Hamiltonian for the one-dimensional harmonic oscillator may be written
in coordinate representation as
x|H|x =
h2 d2
1
(x x ) + Kx2 (x x ).
2
2m dx
2
Calculate the momentum-space representation of H, i.e. p|H
Homework Solutions Physics 507
Problem 12:
An electron moves in a one-dimensional harmonic (oscillator) potential and
under an additional inuence of a constant electric eld.
a) Write the Hamiltonian using the creation and annihilation operators
of the har
Homework Solutions Physics 507
Problem 16: Morse potential
Calculate the bound state energy levels of a particle in the potential
V (x) = V0 [e2x 2ex ]
Use suitable dimensionless variables. A new independent coordinate
exp(x) may also turn out to be usef
Homework Solutions Physics 507
Problem 19:
An electron is attracted to a plane lm of liquid 4 He atoms by a potential
V (r) =
,
if z < 0;
e2 /z, if 0 < z.
( > 0, the surface of the lm is in the x y plane at z = 0.) Calculate
the ground-state energy of the
Homework Solutions Physics 507
Problem 22:
A particle moves in a spherically symmetric potential V (r). Its wave function
is
(r) = (x + y + 3z)f (|r|)
a) Is (r) an eigenfunction of L2 ? If yes, give the eigenvalue.
b) A measurement of Lz is performed. Whi
Homework Solutions Physics 507
Problem 28:
A particle is scattered by a function potential
V (x) = (x).
a) Calculate the transmission and scattering matrices.
b) Calculate the reection and transmission coecients.
c) Can one adjust these parameters so that
Homework Solutions Physics 507
Problem 25: I want you to review the reection/transmission problems
In three dimensions, a spin-1/2 particle with mass m moves in the potential
V (x) =
V0 z
0,
x > 0;
x 0.
a) A beam of particles comes from in the positive x-
Homework Solutions Physics 507 Problem 32:
The motion of a particle of mass m in one dimension is described by the
Hamiltonian
p2
H=
+ |x|.
2m
a. Derive the asymptotic behavior of the ground-state wave function as
|x| becomes large.
b. Use a Gaussian wave