PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 7 - Solution
Problem 1)
An atom of mass 4.109 eV/c2 has its position measured within 2 nm accuracy. Assume that it is in a
Gaussian wave packet state afterwards. How much time
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 2 - Solutions
Problem 1)
a) Do functions defined on the interval [0L] and that vanish at the end points x = 0 and x = L
form a vector space?
Answ.: Yes. This is just a special
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 3 - Solution
Problem 1)
"0 0 1%
$
'
Consider the matrix = $ 0 0 0 ' .
$1 0 0'
#
&
i)
ii)
iii)
Is it hermitian?
Answ.: Yes it is identical to its adjoint (swapping rows and col
PHYSICS 621 - Fall Semester 2012 - ODU
Graduate Quantum Mechanics - Problem Set 1 - Solutions
Problem 1)
E = Tkin + V ( x ) = H ( p, x ) =
p2
+ mgx .
2m
H p
= =v=x
p m
H
= mg = F = p
x
Problem 2)
r
r
A(r, , z ) = b A = b, Ar = 0, Az = 0
2
2
A
rA
1
1
B =
1-D Translations
Consider the operator
T(x) |x = |x + x
Obviously this operator represents a translation in the x direction by some
distance x.
For an innitesimal shift, 0, we would have T( ) |x = |x + . Applying
this translation operator to an arbitrary
1
Harmonic Oscillator (HO)
2
p
1
The classical Hamiltonian for the HO is given by H = 2m + 2 kx2 . The frequency
( ) of the oscillation is independent of the amplitude. It is given by = k /m.
2
P2
Therefore the QMHamiltonian could be written as H = 2m + m
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 4 - Solution
Problem 1)
Assuming a particle is described with the usual cartesian coordinates (x,y,z) and momenta (px, py, pz).
Write down the x, y and z components of the ang
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 5 - Solution
Problem 1)
An operator A, corresponding to a physical observable, has two normalized eigenstates 1 and 2
with non-degenerate eigenvalues a1 and a2, respectively.
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 10 - Solution
Problem 1)
Using the matrix elements of the operator Lx in the subspace for l = 1 (see Shankar p. 327-328, in
particular the 3rd block in the matrix 12.5.23), sh
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 11 - Solution
Problem 1)
Starting with the ground state !,!,! () (Eq. 13.1.27, Shankar p. 357) of the hydrogen atom in
configuration space, calculate the corresponding wave fu
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 9 - Solution
Problem 1)
A particle in 2 dimensions is described by a wave function (x,y). We can make a variable substitution
to circular (cylindrical) variables (x,y) (r, ) b
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 8 - Solution
Problem 1)
Consider a harmonic oscillator which is in an initial state a|n> + b|n+1> at t=0 , where a, b are real
numbers with a2 + b2 = 1. Calculate the expectat
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics - Problem Set 6 - Solution
Problem 1)
The normalized wave function ( x, t ) satisfies the time-dependent Schrdinger equation for a free
particle of mass m, moving along the x-axis (in 1 dim
PHYSICS 621 - Fall Semester 2013 - ODU
Graduate Quantum Mechanics Final Exam - Solution
Problem 1)
2 2
The eigenstates of the particle in a box problem have eigenvalues of En = n
. Like all bound
2 mL2
states, the expectation value for the momentum operat