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Griffiths proposed a pair of boundary conditions that define a point interaction in one dimensional quantum mechanics. The conditions involve the nth derivative of the wave function where n is a non-negative integer. We re-examine the interaction so defined and explicitly confirm that it is self-adjoint for any even value of n and for n = 1. The interaction is not self-adjoint for odd n > 1. We then propose a similar but different pair of boundary conditions with the nth derivative of the wave function such that the ensuing point interaction is self-adjoint for any value of n. [ABSTRACT FROM AUTHOR]

In nonrelativistic quantum mechanics in one dimension a wave packet can be constructed for which the minimum uncertainty relation for position and momentum, ΔxΔp=ħ/2, holds exactly. The wave function of the wave packet is Gaussian and satisfies the Schrödinger equation for a harmonic oscillator potential. We illustrate a similar situation in relativistic quantum mechanics for the one-dimensional Dirac equation. [ABSTRACT FROM AUTHOR]

In their paper 'Overcoming misconceptions in quantum mechanics with the time evolution operator', Garc'ıa Quijas and Ar'evalo 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. [ABSTRACT FROM AUTHOR]

Comments on the article on the variational principle and the simple properties of the ground-state wave function. Expectation value of the Hamiltonian; Global phase factor.