The probability density of a particle with wavefunction
– or statefunction
– is
The probability density function changes in space, but it may also change in time. If the probability density is a function of time, then the particle will be moving and the state function must include this. It does so through the concept of probability current
- quite literally the flow of probability from one point to another.
If
- the statefunction for a free particle - then
But
so
Since the wavefunction
must be normalized:
so
For a stationary state wavefnnction, where the wavefunction is an eigenstate of the Hamiltonian operator, a particle in a box for example,
For a article in an infinite square well between
so
if we ignore the time dependent factor, then
This is general. A particle moving in a potential, in an eigenstate of the Hamiltonian operator corresponding to that potential, exists in a stationary state. Though the probability density may vary inside the potential region, it does not vary with time. The wave nature of the particle becomes very apparent, with regions of maximum probability density corresponding to constructive interference of the wavefunction with itself and regions of minimum probability density corresponding to destructive interference of the wavefunction with itself, in the same way that standing waves form on a violin string.