Download Chemistry 218 October 8, 2002

Chemistry 218
Problem set 1
October 8, 2002
R. Sultan
Photoelectric Effect
1. An experiment was done on the emission of photoelectrons from a sodium
surface by light of different wavelengths.
The following values for the retarding potentials at which the photocurrent
was reduced to zero were obtained:
λ(Å)
ε (V)
3125
2.128
3650
1.595
4047
1.215
4339
1.025
5461
0.467
Make an appropriate plot of the above data and calculate:
a) the threshold frequency and
b) Planck’s constant.
2. The work function for the metal Cesium is 2.14 eV.
a) Determine the maximum wavelength of light that can produce a
photoelectric current from Cs metal.
b) If light with wavelength 50 nm shorter than that calculated in (a)
strikes a Cs metal surface, what will be the velocity of the ejected
electrons?
c) Do we generate a photocurrent by irradiating green light on a piece of
Cesium? Explain.
3. Derivation of the Schrödinger Equation in One-Dimension
The general equation of motion for a wave is given by:
 2
1  2

x 2 v 2 t 2
where v is the velocity of the wave.
By postulating that  x, t  =  0 ( x)e it (where ω = 2πν  angular frequency),
and by using DeBroglie’s relation, derive the time-independent Schrödinger
equation (in one-dimensional space).
Hint: K.E = ½ mv2 = E – V(x); E = total energy
V(x) = potential energy
4. Free Particle
A free particle is one which has no interaction with any other particle in the
universe. This imposes that V(x) = 0 in the Schrödinger equation.
a) Write the Schrödinger equation for the free particle system in one
dimension.
b) By trying as a possible solution of the S. equation, the wave function
 =Ceikx (where k is a constant and C is the normalization factor),
arrive at an expression of the energy E in terms of k.
c) By analogy to the classical expression of the kinetic energy, show that
the linear momentum is p = k (remember this important relation!).
d) k is known as the wave vector. Determine the relation between k and
the wavelength  .
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e) In part (b), you showed that  = Ce i ( 2mE ) x /  (verify).
Then show that the boundary condition requires that E  0.
f) Try to normalize  . Comment on the result.
g) What can you say about the quantization of energy in the case of the
free particle?
5. DeBroglie’s relation/wave-particle Duality
Calculate the DeBroglie wavelength for:
a) An electron moving with a speed v = 5  103 ms 1.
b) A ball of mass 200 g moving with a speed 30 ms 1.
Comment on the significance of your calculation in each case.
NOTE: Electromagnetic spectrum given on p. 6 of Atkins (6th edition).