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## Problem

Consider the one-dimensional, time-dependent probability density

and the current density

where and .

Show by calculation that

.

Hint: Use the Schrödinger equation and its complex conjugate.

## Solution

Calculate the partial derivative of the probability density with respect to .

The Schrödinger equation and its complex conjugate can be written as

Thus

Calculate the partial derivative of the current density with respect to .

It is clear from the above calculations that , therefore;