Normalization of the momentum eigenstates:

There are two normalization conditions. One is the periodic boundary condition (box normalization). The other is to normalize so that over an infinite range you find a delta function.

First let us consider the periodic boundary conditions. The goal is to have the appropriate boundary conditions for a particle in a box with rigid walls (V=∞) and have states of definite momentum. Schiff explains the problems associated with confining a particle in terms of the reflections off of the rigid walls of a finite box. Basically this problem does not conserve momentum. Twice the balls momentum is absorbed by the wall. Thus the way you require the ball to act at the boundary can lead to complications when trying to find momentum eigenstates. However, by assuming periodic boundary conditions the ball passes through the wall and its entry into a new box in some sense puts the ball at the opposite side of the original box with momentum unchanged. The confined particle problem localizes the quantum states while in free space the momentum wave functions are smoothly spread out over an infinite range. For momentum eigenstates the wave must extend over all space. It is important to realize that in any confined space momentum cannot be determined to arbitrary precision. We know this from the uncertainty principle. Putting the particle in a box gives you some knowledge of its location thereby making it impossible to specify the momentum exactly. So we impose boundary conditions that are periodic. This allows us to force the wavefunction to be zero at the walls but still be defined outside the box and therefore be a legitimate momentum eigenstates. The limit can be taken as to approach the free particle solution.

Problem 4.6 examines these states over a finite interval a [-a/2èa/2] with a focus on the normalization as the range goes to infinity. The detail associated with whether you choose the function to be zero outside the box or periodic is not discussed.

This shows that the integral of the proposed wave function is properly normalized as the range goes to infinity.

Notice, however, for a finite range that for any arbitrary values of k and k’ with and for a finite interval a, the integral is not zero. If one chooses the k values such that then these k’s have an integral equal to 0 and are orthogonal.

It is also interesting to note that for a particle in a box the energy can have eigenvalues and it appears to be made up of eigenstates of right and left going momentum but the state

Choose range a=10 and plot an example. The plot shows a cosine function or the real part of the state above. The plot emphasizes that wavefunctions go to zero outside the box.

Are these momentum eigenstates? From the above discussion the answer is clearly no. Also note that at the boundary the derivative of the function is discontinuous. Thus the momentum operator in position space is not defined at this point.

A second approach is to examine the states normalized over an infinite range

the normalization leads to a delta function.

Using this state as a representation of a momentum eigenstate wrt the position basis we find:

So the question becomes how are the two methods of normalization equivalent?

To clarify this relationship I have added a subscript to the momentum label in the first set of states. This is because the limit as will require a transition of the momentum eigenvalues from discrete to continuous.

The integral is over a finite range and again only values of p allowed are .

When the sums in the above relationships go over to integrals the integer