Quantum mechanical equations of change. I
Joseph O. Hirschfelder
Simple, but completely general, equations of change are derived for a property, χ†𝒪Ψ of a quantum mechanical system, where χ (r,s,t) and Ψ (r,s,t) are any two wavefunctions satisfying a time-dependent Schrödinger equation with the same Hamiltonian H, and 𝒪 is an arbitrary operator. For a Hamiltonian H0= (2m)−1 p2+V (r) the equations of change of angular momentum density and kinetic energy density are derived. The equations of change are developed for systems having Breit–Pauli type Hamiltonians which involve spin as well as external static electromagnetic fields and relativistic corrections. The velocity operator ṙ is shown to contain a spin-dependent term. The relativistic correction to the kinetic energy (p2)2 term is not consistent with the equation of continuity. An expression is derived for the rate of ’’spin flipping’’ of a hydrogen atom in an external electromagnetic field. Methods are derived for projecting the equations of change onto real three-dimensional space. Natural spin orbitals are useful for describing the equation of continuity in three dimensions; however, they cannot describe the equation of motion or the other equations of change.
The Journal of Chemical Physics•Jun 1, 1978•65 citations