Download Chemistry 218 November 4, 2002

Chemistry 218
Problem set 4
November 4, 2002
R. Sultan
1. Calculate the expectation value < x 2 > for the harmonic oscillator. Verify your
1   2 


result with:  x    v   
2   mk 

1/ 2
1

  v   2 .
2


2. In Physics, the angular momentum L about a point is defined as
  


L = r  p where p is the linear momentum at a point r . The origin of the

vector r is taken at the point about which the angular momentum is to be
calculated. Write the equation for each component of angular momentum Lx ,
L y and L z in terms of x, y and z and the components of linear momentum p x ,
2
3.
4.
5.
6.
7.
p y and p z .
Derive the quantum mechanical operators for the three components of angular
momentum. For the classical expression, see Exercise 1.
Write down the eigenvalue equation for L̂2 and L̂ z operating on the spherical
harmonics Yl ,ml ( ,  ) of the hydrogen atom. Give the eigenvalues.
Draw a clear vector model representation of all the possible orientations of the

angular momentum vector L for l = 3.
A review of the “particle on a ring” problem:
Consider the problem of a particle of mass m moving along a circular path of
radius r.
a. By drawing a schematic representation of the particle, show that the
problem can be reduced to a single variable problem.
The problem is known to have a solution of the form   Ae iml , where
ml = 0, ±1, ±2, …….
b. Find the normalization factor A.
c. The hamiltonian operator in circular polar coordinates is

2 1 d 2
H  ( ) 2
2m r d 2
Find an expression for the total energy of the particle. What is the value of
the potential energy?

d. The angular momentum along the z – axis is Jˆ z  i
.

Find the eigenvalue J z of the angular momentum.
e. Deduce an expression for the energy in terms of J z and I, the moment of
inertia of the particle. Compare your result with the classical expression of
the rotational energy.
In Quantum mechanics, the components of the angular momentum vector
̂
operator L are the operators L̂x , L̂ y , L̂ z .
a. Using the definitions of those operators (see Exercise 2), show that
[ L̂x , L̂ y ] = iLˆ z . This is a very good exercise on sequential operators.
The RAISING and LOWERING operators ( L̂ and L̂ respectively), are
defined by the following expressions:
L̂ = L̂x + iLˆ y
L̂ = L̂ – iLˆ

x
y
b. Given that [ L̂x , L̂ y ] = iLˆ z , express the operators   L̂ L̂ and
B̂  L̂ L̂ in terms of L̂2 and L̂ z .
c. Show that the spherical harmonics Yl ,ml ( ,  )   l ,ml ( ) ml ( ) (of the H-
atom) are eigenfunctions of the operators  and B̂ , determine the
eigenvalues in terms of quantum numbers l and ml.
d. Now find an expression of L̂2 in terms of L̂ , L̂ and L̂ z .
Hint: First express L̂x and L̂ y (part a.) in terms of L̂ and L̂ .
e. Given that:
1/ 2
Lˆ Y  l (l  1)  m(m  1) Y
 l ,m
l , m 1
Lˆ Yl ,m  l (l  1)  m(m  1) Yl ,m1
1/ 2
and using the result obtained in d., show that:
Lˆ2Yl ,m  l (l  1) 2Yl ,m