The chapter shows that total probability is conserved because the potential is real

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Wave mechanics in one dimension · Quantum dynamics and its formulations

Quantum Mechanics: From the Photon to the Hydrogen Atom, Ch The Schrödinger equation and the Born rule

Problem

The chapter shows that total probability is conserved because the potential

V(x)

is real. Suppose instead a particle moves under the potential

V=-i\Gamma

, with

\Gamma>0

a real constant (imaginary, not real). ((a)) Redo the steps used earlier for a real potential, now with this

V

, to find

\dfrac{\partial}{\partial t}|\Psi|^2

, and show that it no longer has the pure continuity form

\partial\rho/\partial t + \partial j/\partial x = 0

. ((b)) Show that

\dfrac{d}{dt}\displaystyle\int_{-\infty}^{\infty}|\Psi|^2\,dx = -\dfrac{2\Gamma}{\hbar}\displaystyle\int_{-\infty}^{\infty}|\Psi|^2\,dx

, and hence that the total probability decays as

e^{-2\Gamma t/\hbar}