As a concrete illustration of these ideas, we study the
particle in a box
(in one dimension).
This is just a particle (of mass
which is free to move inside the walls of a box
but which cannot penetrate the walls.
We represent that by a potential which is zero inside the box and infinite outside.
We solve the Schrödinger equation inside the box and realize that the probability
for the particle to be outside the box, and hence the wavefunction there, must be zero.
Since there is no potential inside, the Schrödinger equation is
where we have anticipated that there will be many solutions indexed by
We know four (only 2 linearly independent) functions
which have a second derivative which is a constant times the same function:
The wave function must be continuous though, so we require the boundary conditions
The sine function is always zero at
and none of the others are.
To make the sine function zero at
So the energy eigenfunctions are given by
where we allow the overall constant
because it satisfies the differential equation.
back into the Schrödinger equation,
we find that
Only quantized energies are allowed when we solve this bound state problem.
We have one remaining task.
The eigenstates should be normalized to represent one particle.
So the wave function will be normalized if we choose
We can always multiply by any complex number of magnitude 1, but, it doesn't change the physics.
This example shows many of the features we will see for other bound state problems.
The one difference is that, because of an infinite change in the potential at the walls
of the box, we did not need to keep the first derivative of the wavefunction continuous.
In all other problems, we will have to pay more attention to this.