Theory of the Motion of Microparticles in a Field of Potential Forces

Abstract

In this chapter we shall consider some simple problems of atomic mechanics concerning the motion of particles in the field of potential forces. Generally speaking, if the forces are independent of time, the basic problem of atomic mechanics is that of finding the stationary states of the system. For in this case, according to (30.8), an arbitrary state ψ(x, t) can be represented as a superposition of stationary states with constant amplitudes c _n :

$$\psi \left( {x,\,t} \right) = \sum\limits_n {{c_n}{\psi _n}\left( {x,\,t} \right),} $$

(46.1)

$${\psi _n}\left( {x,\,t} \right) = {\psi _n}\left( x \right) \cdot {e^{ - i{E_n}t/\hbar }},$$

(46.2)

where the Ψ _n (x) are wave functions of stationary states and the E _n the corresponding energy values. The wave functions Ω_n(x) are the eigenfunctions of the energy operator H and are determined, according to (30.4), from Schrödinger’s equation for stationary states:

$$H\psi = E\psi .$$

(46.3)