Particle in a box

Boundary value problem where one needs to find the solution for the problem given that the potential is ∞ at the boundaries. Thus the particle is completely confined to a region and the wave function that describes this state must reflect that condition.

Method divide the spce up into regions and solve the problem in each region:

Region 1: x<0

Region 3: x>0

Region 2: a>x>0 Solutions that are energy eigenstates.

Examine the Hamitonian and find the eigenstates

To get the BC one has B=0 and

We find that :

· Energy eigenstates are not eigenstates of either momentum or position.

· We can find the average location and average momentum for any eigenstates.

· There are many acceptable states that have a definite energy.

· The time evolution for a specific state is a solution to the Schrodinger equation

· A general solution, a legitimate possibility for the particle located in a box is

A solution at some time that will now need to evolve based on the Schrödinger equation.

Be clear:

· is a solution or a prepared state at a fixed time that must evolve

· is a very specific solution it usually has the property that the wavefunction is a solution to some operator eigenequation and therefore there are an associated set of eginevalues labeled by n.

· implies that that time dependence of this state is included. This can be very general. So far we have only examined the particularly simple time evolution of an energy eigenstates.