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Probability and Current Densities:

The probability density for an infinitesimal region is equal to the square modulus of the wave function, as proposed by the physicist Max Born.

eq.1

One can utilize a form of Gauss' divergence theorm to obtain more information concerning this, which is:

eq.2

Rewriting the time derivative fo the probability density function:

eq.3

Expanding:

eq.4

The time-dependent Schrödinger Equation for the free particle is of the form:

eq.5

Rewriting:

eq.6

Applying these two equations in eq.4:

eq.7

Collecting:

eq.8

Canceling the negative sign and the operator-vector del:

eq.9