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Unformatted text preview: Numerical Solution of a Potential–Final Project 1 Introduction The purpose is to determine the lowest order wave functions of and energies a potential which describes the vibrations of molecules fairly well. Consider the vibrations of diatomic HBr (Hydrogen Bromide). It is a gas room temperature and is useful in several respects [1] For simplicity, we will consider Bromine nucleus fixed (it has nearly 80 times the mass of hydrogen) and consider the oscillations of the hydrogen nucleus. The potential for the hydrogen nucleus has the following phenomenological form V ( x ) = V o (1 e x/a ) 2 , (1) where x is the displacement of the hydrogen nucleus from equilibrium. For V o = 50¯ h 2 /Ma 2 a graph of the potential is shown below. We will solve for the energies and wave functions in this potential. For small fluctuations around the equilibrium (i.e. small quantum number n ) a harmonic oscillator approximation is valid, and the energies are given by the simple harmonic oscillator (see below). However for larger fluctuations around equilibrium (large quantum number n ) the shape of the potential matters and the harmonic approximation is poor. The graph below shows the energies for this potential and the energies in a corresponding harmonic approximation. One sees that as n becomes large the harmonic approximation starts to fail. For actual hydrogen bromide the parameters are: a ’ 1 . 1 a o ’ . 58 ˚ A and V o ’ 4 . 83eV ’ 433¯ h 2 /Ma 2 [2]. Because V o is rather large (relative to ¯ h 2 /Ma 2 ), the energies are quite close to the harmonic approximation until the quantum number is rather large. We will therefore use V o ’ 50¯ h 2 /Ma 2 in our analytical and numerical work unless otherwise directed. 2 Analysis 1. For what range of energies do we expect discrete energies and for what range do we expect continuous energies.. Describe classically what would happen if the nucleus were to be given a kick (e.g. by light) of energy E in the discrete case and the continuous case. Why is V o called the dissociation energy? 2. Near the minimum, we can approximate the full potential with a simple harmonic potential. Show that near x = 0 V ( x ) ’ 1 2 kx 2 , (2) where k ’ 2 V o /a 2 . Show that allowed energies of the molecular vibrations are approx imately E = ¯ hω o ( n + 1 2 ) , (3) and determine ω o in terms of V o and a . 11 1 2 3 4 5 10 20 30 40 50 x/a V(x) / [h 2 /2M (2 π a) 2 ] Potential Potential Harmonic Approx Figure 1: The potential and its harmonic oscillator approximation. The horizontal lines show the allowed energies for the full potential and the the allowed energies in the harmonic oscillator approximation 3. Show that the classical turning points of the potential occur at x ∓ = a log 1 ± p E/V o ....
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This note was uploaded on 01/25/2012 for the course PHYSICS 251 taught by Professor Staff during the Fall '11 term at SUNY Stony Brook.
 Fall '11
 Staff

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