Current misconception concerning the time evolution operator

P. C. Garcia Quijas, L. M. Arévalo Aguilar

Recently, there have been many efforts to use the research techniques developed in the field of physics education research to improve the teaching and learning of quantum mechanics. In particular, part of this research is focusing on misconceptions held by students. For instance, a set of misconceptions is associated with the concept of stationary states. In this paper, we argue that a possible way to remove these is to solve the Schrödinger equation using the evolution operator method (EOM), and stress the fact that to find stationary states is only the first step in solving that equation. The EOM consists in solving the Schrödinger equation by direct integration, i.e. Ψ( x , t ) = U ( t )Ψ( x , 0), where is the time evolution operator, and Ψ( x , 0) is the initial state. We apply the evolution operator method in the case of the harmonic oscillator.

Ronald A. Lovett

Deng-yuan Chen

L. M. Arévalo Aguilar, P. C. Garcia Quijas

We contend that the statement attributing to us the implication that the wave functions ?I(x, t) and ?II(x, t) are different cannot not be logically supported. On the contrary, throughout our paper ?Overcoming misconceptions in quantum mechanics with the time evolution operator? (Garcia Quijas and Ar?valo Aguilar 2007 Eur. J. Phys. 28 147) we constructed an argument to give support to the statement that the two wave functions ?I(x, t) and ?II(x, t) are the same. In other words, in our paper we really did appreciate that methods I and II lead to identical results.

F. Masafumi Toyama, Yukihisa Nogami

In their paper ‘Overcoming misconceptions in quantum mechanics with the time evolution operator’, García Quijas and Arévalo Aguilar (2007 Eur. J. Phys. 28 147) examined the time-dependent wave function of a particle in the one-dimensional harmonic oscillator potential using two different methods. The two wave functions that the authors obtained through the methods have different analytical expressions. The authors showed numerically that the two wave functions lead to the same probability density. When the real parts of the wave functions are compared, however, they are different in their details. That was puzzling because both wave functions are supposed to be solutions of the same time-dependent Schrödinger equation with the same initial condition. We point out that the two wave functions are actually identical. We show this analytically.

S. Balasubramanian