Principles of Quantum Mechanics · Problem 5.3.1

Simple Problems in One Dimension — R. Shankar

Physics — Quantum Mechanics · complex-potential-continuity-equation

Problem

Consider the case where V=Vr−iViV = V_r - iV_i, where the imaginary part ViV_i is constant. Is the Hamiltonian Hermitian? Go through the derivation of the continuity equation and show that the total probability for finding the particle decreases exponentially like e−2Vit/ℏe^{-2V_i t/\hbar}. Such complex potentials are used to describe processes in which particles are absorbed by a sink.

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