Problem

Consider the one-dimensional, time-dependent probability density

Hydrogen Density Plots.png

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;

Advanced Remark

The conservation of probability in 3D space can be compactly expressed by the vector equation

where and are the wavefunction and its complex conjugate, is the probability density, and is the current density.

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