National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
Statistical Physics: September 12, 2012
Solution for the Homework 1
Problem 6.5: Imagine a particle that can be in only three states, with energies 0.05eV, 0 and 0.05eV.
This particle is in equilibrium with a reservoir at 300K.
(a) Calculate the partitio
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 7: 3D Schr
odinger Equation
Problem 1:
A particle is confined inside a rectangular box given by 0 x a, 0 y b and 0 z c.
n y
The solution to the Schrodinger equation is (x, y, z) = A sin nxax sin yb sin nzcz where
2 2
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 2: Postulates of Quantum Mechanics
Question 1: An orthonormal basis of a Hamiltonian operator in four dimensions is defined as
follows:
1i = E 1i , H
2i = 2E 2i , H
3i = 3E 3i , H
4i = 4E 4i .
H
A system i
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 3: Time Evolution, Symmetries, and Conservation Laws
Question 1: An electron is moving freely inside a onedimensional infinite potential box with
walls at x = 0 and x = a. If the electron is initially in the ground s
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 10: 3D Schr
odinger Equation
Problem 1:
r, calculate the
Given the fact that the radial momentum operator is given by pr = i~ 1r r
commutator [
r, pr ] between the position operator r and the radial momentum operator.
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 9: 3D Schr
odinger Equation
Problem 1: Infinite Spherical Well
Consider a particle of mass m and energy E > 0 moving in infinite spherical well of radius a, i.e.
0
for 0 r a,
V (r) =
otherwise.
1. Show that the Schr
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 8: 3D Schr
odinger Equation
Central Potential
Problem 1:
A hydrogen atom can be viewed as two pointcharges a proton and an electron with Coulombs
interacting potential between them. Write the Schrodinger equation for
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 1: Mathematical Framework of Quantum
Mechanics
Question 1: Two vectors in a threedimensional complex vector space are defined
by:
2
1 + 3i
Ai = 7i , Bi = 4
1
8
Let a = 6 + 5i
a) Compute a Ai, a Bi, and a (Ai +
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 4: Orbital Angular Momentum
Problem 1:
A particle in an onedimensional infinite square well potential with walls at x = 0 and x = a is
in a state described the the following wave function at t = 0
x
1
2
3x
(x) = si
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 5: Orbital Angular Momentum
Problem 1:
Given the following angular momentum relations:
+ = L
x + iL
y, L
= L
x iL
y
L
p
+ l, mi = ~ l(l + 1) m(m + 1) l, m + 1i
L
p
l, mi = ~ l(l + 1) m(m 1) l, m 1i
L
(1)
z 
National Institute of Management Sciences, Peshawar
phy
PHY 751

Fall 2016
PH3101: Quantum Mechanics II
Tutorial 6: Generalized Angular Momentum
Problem 1:
(a) Calculate the energy eigenvalues of an axially symmetric rotator and find the degeneracy of
each energy level. The Hamiltonian of an axially symmetric rotator is given by