CHEM 3420 Physical Chemistry II, Spring 2022
Homework 2
Due: March 26th, 2022
Please finish the homework before the due date, late homework assignments will not be
accepted. This homework assignment covers Topic 2: The Quantum Theory of Motion. There
are 100 points in total for each assignment.
1. Determine the linear momentum and kinetic energy of a free proton described by the
− ikx
wavefunction e with 𝑘 = 10 𝑚−1. [15]
2. Calculate the energy separations in joules, kilojoules per mole, and electronvolts, and as a
wavenumber, of an electron in a one-dimensional nanoparticle in a box of length 1.5 nm
between the levels:
(a) n=3 and n=2, [10]
(b) n=8 and n=7. [10]
3. Suppose that a proton of an acidic hydrogen atom is confined to an acid that can be
represented by a barrier height 1.0 eV and length 200 pm. Calculate the probability that a
proton with energy of 0.75 eV can escape from the acid. [15]
4. An electron in a carbon nanotube of length L is described by the wavefunction:
2 1/2
𝜋𝑥
𝜓(𝑥) = ( ) sin ( )
𝐿
𝐿
Compute the expectation value of the kinetic energy of the electron. [15]
5. Calculate the zero-point energy of a harmonic oscillator consisting of a rigid CO molecule
absorbed to a metal surface by a bond of force constant 237.8 N m-1. Hint: mass of CO =
4.651 𝑥 10−26 𝑘𝑔. [15]
6. What is the relation between the mean kinetic and potential energies of an electron in a
hydrogen atom? Hint: The potential is Columbic. [10]
7. Confirm that the wavefunction for a particle on a ring ml ( ) =
eiml
( 2 )
1/ 2
ml = 0, 1, 2,...
is normalized. [10]
1
0
