CHEM 614
2002
HOMEWORK 2
1.
When a lithium crystal is irradiated with light of wavelengths 300 nm and 400 nm
photoelectrons are emitted with the maximum kinetic energies 1.83 eV and 0.8 eV,
respectively. From this information, calculate the Planck constant, the wavelength
corresponding to the threshold frequency 0, and the work function W for lithium.
2.
By directing a 589 nm laser beam toward a well-focused beam of Na atoms and
Na2 molecules, at right angles, it is possible to separate the Na atoms from the Na2
molecules. The wavelength 589 nm corresponds to a strongly allowed electronic
transition in the Na atom, and the atom is therefore most likely to absorb a photon from
the laser beam. The momentum delivered by the photon is essentially big enough to
remove the atom from the beam.
Calculate the magnitude of the momentum delivered by the photon, and the velocity
component that the atom acquires in the direction of the laser beam because of the
momentum transfer. Take the atomic mass of Na to be 23 times that of a proton.
3.
4.
Which of the following are eigenfunctions for d/dx?
(a) x2 (b) exp( -3.4x2) (c) 37 (d) exp (x) (e) sin (ax) (f) cos (4x) + i sin (4x)
Ascertain that the expression
En  n 2 h 2 / 8mL2
has the proper dimensions.
5.
Evaluate the probability for finding a particle in the middle third of one
dimensional "box" in states n=1, 2, 3.
6.
Estimate the probability for finding a particle in the n=1 state in the line element
 x centered at the midpoint of a one-dimensional "box" if  x =0.01L. How does
this compare to the classical probability?
7.
Use common sense to evaluate the following integral for the particle in a onedimensional box, assuming that  is normalized.

L/5
0
 52 dx
8.
Given that the time-independent equation of motion in one dimension (x) for a
clamped string is
d 2 ( x)
 2 
 
  ( x)
2
dx
  
Find the condition that must be satisfied by a and  in order that
 (x) = A sin (x) + B cos (  x )
is a solution.
2
9.
Write a brief explanatory paragraph on each of the following topics. The
paragraph needs to demonstrate that you understand the main points of the topic.
(a)
operators
(b)
eigenvalue equations
(c)
the Schrödinger equation
(d)
phase modulation of wavefunctions
(e)
boundary conditions
(f)
the energy eigenvalues for the particle in a one-dimensional well