Let Pab(t) be the probability of finding a particle in the range (a < x < b), at tim...

Let Pab(t) be the probability of finding a particle in the range (a < x < b), at time t.

(a) Show that

where

What are the units of J(x, t)? Comment: J is called the probability current, because it tells you the rate at which probability is “flowing” past the point x. If Pab(t) is increasing, then more probability is flowing into the region at one end than flows out at the other.

(b) Find the probability current for the wave function in Problem 1. (This is not a very pithy example, I’m afraid; we’ll encounter more substantial ones in due course.)

Problem 1

A particle of mass m is in the state

where A and a are positive real constants.

(a) Find A.

(b) For what potential energy function V(x) does Ψ satisfy the Schrödinger equation?

(c) Calculate the expectation values of x, x2, p, and p2.

(d) Find σx and σp. Is their product consistent with the uncertainty principle?