1
Name: ____________________________
Lecture Problem Set #4
Due Monday, Sep. 28, 2009
(10 points)
1. The following questions regard the quantum mechanical harmonic oscillator.
a) Write the wavefunctions and sketch the n = 1, n = 2, and n = 3 solutions to the particle in a one
dimensional box. Then write the wavefunctions and sketch the
ν
= 0,
ν
= 1 and
ν
= 2 solutions
to the harmonic oscillator with mass, m, and spring constant, k. Write the corresponding energy
levels for each state. Use your pencil to shade the classically forbidden regions of the
wavefunctions. Write the explicit normalization constants. Express your wavefunctions in
terms of the variable, x.
b) Calculate the positions of the classical turning points and the nodes of the quantum mechanical
harmonic oscillator in the states with
ν
= 0,
ν
= 1 and
ν
= 2. Express you answers in terms of
()
1
24
km
α
==. If no nodes exist in a given state, indicate that this is so.
c) Suppose the energy of a system is 10 (in arbitrary units) when it is in its lowest quantum state.
What will be the energy of the next highest state if the system is a i) harmonic oscillator? ii)
particle in a one-dimensional box?
d) Write down the formula for the average value of the force on the quantum mechanical
harmonic oscillator when
ν
= 0. Recall that the force operator is ˆ
f
kx
=
−. Evaluate the
expression by sketching the functions and discussing the value on the basis of their symmetry.
e) Use the symmetry of the potential energy operator, 2
1
2
Vkx=, to determine whether the average
potential energy of the quantum mechanical harmonic oscillator is zero for the state,
ν
= 0.
2. Assume that the vibration of a 12C16O molecule can be approximated as a harmonic oscillator with
reduced mass 12
12
mm
mm
μ
=+ and force constant k = 1856.4 N/m.
a) Calculate the vibrational frequency of the molecule (substituting the reduced mass,
μ
, for m).
b) Calculate the zero-point energy and the energy of the state with
ν
= 5.
c) Indicate whether the following quantities will increase or decrease if 14C18O is considered
instead: Justify your answers.
i) zero-point energy
ii) vibrational transition frequency, ΔE = E
ν
– E
ν
–1
iii) classical turning points