Boundary Conditions and the Meaning of Wave Groups in Wave Mechanics

Abstract

Reasons are given for believing that the boundary conditions at a discontinuity in the potential ($V$) in wave mechanics may be ${\ensuremath{\varphi}}_{1}={\ensuremath{\varphi}}_{2}{\ensuremath{\mu}}^{\frac{1}{2}}$ and $\frac{{\ensuremath{\varphi}}_{1}d{\ensuremath{\varphi}}_{1}}{\mathrm{dx}}=\frac{{\ensuremath{\varphi}}_{2}d{\ensuremath{\varphi}}_{2}}{\mathrm{dx}}$. Here $\ensuremath{\varphi}$ is the wave function and $\ensuremath{\mu}$ the ratio of the group velocities. With these boundary conditions the waves are either completely transmitted or completely reflected according as the energy $E$ is greater or less than $V$. The particles therefore need not be supposed to enter the region in which $V>E$ where their velocities would be imaginary and their kinetic energies negative. It is suggested that the group of waves associated with a particle should be identified with the particle and that the energy and momentum of the particle are the energy and momentum of the group of waves. When the group is stopped or absorbed it contracts to small dimensions and so becomes a particle.