Riley hobson bence · Problem 27.27

Group theory

Mathematics — Methods · numerical-methods-schrodinger-equation

Problem

The Schr\"odinger equation for a quantum mechanical particle of mass mm moving in a one-dimensional harmonic oscillator potential V(x)=kx2/2V(x) = kx^2/2 is −ℏ22md2ψdx2+kx2ψ2=Eψ.-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + \frac{kx^2\psi}{2} = E\psi. For physically acceptable solutions, the wavefunction ψ(x)\psi(x) must be finite at x=0x = 0, tend to zero as x→±∞x \to \pm\infty and be normalised, so that ∫∣ψ∣2 dx=1\int |\psi|^2\,dx = 1. In practice, these constraints mean that only certain (quantised) values of EE, the energy of the particle, are allowed. The allowed values fall into two groups: those for which ψ(0)=0\psi(0) = 0 and those for which ψ(0)≠0\psi(0) \neq 0. Show that if the unit of length is taken as [ℏ2/(mk)]1/4[\hbar^2/(mk)]^{1/4} and the unit of energy is taken as ℏ(k/m)1/2\hbar(k/m)^{1/2}, then the Schr\"odinger equation takes the form d2ψdy2+(2E′−y2)ψ=0.\frac{d^2\psi}{dy^2} + (2E' - y^2)\psi = 0. Devise an outline computerised scheme, using Runge--Kutta integration, that will enable you to: (a) determine the three lowest allowed values of EE; (b) tabulate the normalised wavefunction corresponding to the lowest allowed energy. You should consider explicitly: (i) the variables to use in the numerical integration; (ii) how starting values near y=0y = 0 are to be chosen; (iii) how the condition on ψ\psi as y→±∞y \to \pm\infty is to be implemented; (iv) how the required values of EE are to be extracted from the results of the integration; (v) how the normalisation is to be carried out.

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