Wave function

Wave function generally obeys the Schrodinger Equation
and is normalizable i.e.
The normalized wave function is then .

Note that doesn’t depend on time (just differentiate under integral).

We call the probability density function,
where is the probability of finding the particle in .
Furthermore, we define the probability current:

It follows that

Superposition

Suppose and are normalizable wave functions.
Then so is .

Inner product

For and define

Also write using Dirac Notation.
Not in Part IB though.