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American Journal of Physics

Nodes of wavefunctions

Marco Moriconi

Published Feb 12, 2007• 14 citations
Publisher

Abstract

We give a simple argument to show that the nth wavefunction for the one-dimensional Schrödinger equation has n−1 nodes. We also show that if n1<n2, then between two consecutive zeros of ψn1, there is a zero of ψn2.
DOI
10.1119/1.2404960

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9. One-dimensional Schrödinger equation and its solutions

De Gruyter eBooks•Aug 4, 2014•0 citations
We give a simple argument to show that the nnnth wavefunction for the one-dimensional Schrödinger equation has n−1n-1n−1 nodes. We also show that if n1<n2n_1 < n_2n1​<n2​, then between two consecutive zeros of ψn1ψ_{n_1}ψn1​​, there is a zero of ψn2ψ_{n_2}ψn2​​.
Abstract This work extended the simple diffusion QMC method of solving the Schrödinger equation for a wavefunction ψ; to that of solving for a correction to a trial wavefunction ψo such that ψ= ψo+ . The method may be applied in the manner of successive corrections to obtain very high accuracies. As described, the method is limited to solutions for systems with fixed nodes. The method was illustrated for the cases of the particle-in-a-box and the hydrogen atom. For the latter, successive corrections improved the accuracy in energy by about a factor of ten with each iteration. For higher dimensional systems, difficulties in the fitting procedure may be expected to limit the applicability of the method.
We analyze a one dimensional quantum system consisting of a test particle interacting with two harmonic oscillators placed at the positions a1 and a2, with a1>0 and ∣a2∣>a1, in the two possible situations: a2>0 and a2<0. At time zero, the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. Under suitable assumptions on the physical parameters of the model, we consider the time evolution of the wave function and we compute the probability Pn1n2−(t) [Pn1n2+(t)] that both oscillators are in the excited states labeled by n1 and n2>0 at time t>∣a2∣v0−1 when a2<0 (a2>0). We prove that Pn1n2−(t) is negligible with respect to Pn1n2+(t) up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott [“The wave mechanics of α-ray tracks,” Proc. R. Soc. London, Ser. A 126, 79 (1929)], where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schrödinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schrödinger equation for the three-particle system, with no reference to the wave packet collapse postulate.