Conservation of Probability

Recall Schrodinger’s Equation 

The complex conjugate of Eq. (1) is

Multiply Eq. (1) by Y*(r, t)

Multiply Eq. (2) by -Y(r, t)

Add Eq. (3) to Eq. (4)

Note that the term inside the brackets on the left hand side of Eq. (5) can be rewritten using the relation

With this equality Eq. (5) can be written as

Rearranging terms we can write Eq. (7) as

Inserting the probability density

Eq. (8) becomes

The probability current is defined as

Take the divergence of Eq. (11)

We can now right Eq. 10 in terms of the probability current

Eq. (13) is known as the local conservation of probability.