# Relation Between the Wave Function and Probability Density

### From FSUPhysicsWiki

The quantity can be interpreted as probability density. In order for us to do so, two conditions must be met. First, the probability amplitude must be positive semi-definite (equal to or greater than zero). This condition is trivial because is always a non-negative function. Second, the probability density, integrated over all space, must be equal to one:

We will show that, if this relation is satisfied for a specific time, then it is satisfied for all times shortly.

Because of the fact that we may interpret as a probability density, we may calculate expectation values of observables, such as position and momentum, in terms of it. In general, the expectation value of an observable is given by

In particular, the expectation value of a position coordinate *x*_{i} is

and that for a component of momentum *p*_{i} is

## Conservation of Probability

We will now show that the solution to the Schrödinger equation conserves probability, i.e. the probability to find the particle somewhere in the space does not change with time. To see that it does, consider

Now multiply both sides by the complex conjugate of

Now, take the complex conjugate of this entire expression:

Taking the difference of the above equations, we finally find

Note that this is in the form of a continuity equation,

where

is the probability density, and

is the probability current.

Once we know that the densities and currents constructed from the solution of the Schrödinger equation satisfy the continuity equation, it is easy to show that the probability is conserved.

To see this, note that

Here, we used the fact that the wave function is assumed to vanish outside of the boundary, and thus the current vanishes as well. Therefore, we see that the normalization of the wave function does not change over time, so that we only need to normalize it at one instant in time, as asserted earlier.

## Problems

**(1)** Consider a particle moving in a potential field

**(a)** Prove that the average energy is where *W* is energy density.

**(b)** Prove the energy conservation equation, where is the energy flux density.

**(2)** Assume that the Hamiltonian for a system of *N* particles is , and is the wave fuction. Defining

and similarly for the other ρ_{i} and , prove the following relation: