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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 it (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 . 1/ 2 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).