Total time derivatives of operators in elementary quantum mechanics

Abstract

The use of a total time derivative of operators, that depends on the time evolution of the wave function as well as on any intrinsic time dependence in the operators, simplifies the formal development of quantum mechanics and allows its development to more closely follow the corresponding development of classical mechanics. We illustrate the use of the total time derivative for a free particle, the linear potential, the harmonic oscillator, and the repulsive inverse square potential. In these cases, operators whose total time derivative is zero can be found and yield general properties of wave packets and several useful time-dependent solutions of Schrödinger’s equation, including the propagator. Show more