PERIODIC WAVEFUNCTIONS HAVE QUANTIZED EIGENVALUES OF MOMENTA AND ANGULAR MOMENTA

Summary

The wavefunction $$\psi(L,t)$$ is confined to a circle whenever the eigenvalues L of a particle are only nonzero on the points along a circle. When the wavefunction $$\psi(L,t)$$ associated with a particle has non-zero values only on points along a circle of radius $$r$$, the eigenvalues $$p$$ (of the momentum operator $$\hat{P}$$) are quantized—they come in discrete multiples of $$n\frac{ℏ}{r}$$ where $$n=1,2,…$$ Since the eigenvalues for angular momentum are $$L=pr=nℏ$$, it follows that angular momentum is also quantized.

Proof

The wavefunction $$\psi$$ is a tensor field defined at each point along the circle: at a particular point the value of this field reads “$$\psi$$” independent of the coordinate value used to label that point. Notice that the coordinate values $$x$$ and $$x+(n)(2πr)$$ label the same point. It follows that

$$\psi(x,t)=\psi(x+2πrn,t)$$

since the value of a scalar field must be the same at a particular point. If the wavefunction is confined to a circle, then this condition that the wavefunction must “come back to itself” applies to any wavefunction corresponding to any state. The eigenfunction $$\psi_p$$ (associated with momentum) must satisfy

$$\psi_p(x,t)=\psi_p(x+2πr,t).$$

It is fairly straightforward to show that $$\psi_p$$ is always given by $$\psi_p(x,t)=\psi_p(x)=Ae^{ipx/ℏ}$$. The momentum eigenvectors $$|\psi_p⟩$$ are those special vectors which satisfy the equation

$$\hat{P}|\psi_p⟩=p|\psi_p⟩.$$

We can rewrite this equation in terms of the wavefunction as

$$-iℏ\frac{∂}{∂x}\psi_p(x,t)=p\psi_p(x,t).$$
Let’s multiply both sides of Equation # by $$i/ℏ$$ to obtain
$$\frac{∂}{∂x}\psi_p(x,t)=\frac{ip}{ℏ}\psi_p(x,t)⇒\frac{d}{dx}\psi_p(x)=\frac{ip}{ℏ}\psi_p(x).$$
The solution to Equation # is given by
$$\psi_p (x)=Ae^{ipx/ℏ}.$$
If we substitute this result into Equation # we get
$$Ae^{ipx/ℏ}=Ae^{ip(x+2πr)/ℏ}=Ae^{ipx/ℏ}e^{2πrip/ℏ)}.$$
We can now use algebra to determine what values of $$p$$ satisfy Equation #:
$$e^{2πrnip/ℏ}=1⇒2πrip/ℏ=(n)2π\text{ (n=1,2,…)}$$
$$p=n\frac{ℏ}{r}$$
$$L=nℏ$$