Required math: calculus
Required physics: Schrödinger equation
References: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.46.
Suppose we have a particle that slides on a frictionless circular loop of circumference . Since there is no friction, , but unlike the free particle, we have a periodic constraint that .
Since , the solution is the same as that for the free particle, with the added condition that :
Normalizing this by integrating the square modulus over gives .
The constraint here is that , where is the circumference of the wire. This leads to , so the condition on becomes . Since the solution above assumes that can be positive or negative, takes on all positive and negative integer values. From this we can get the allowed energies:
Note that the energy is the same for , so in this case there are two wave functions for each energy meaning that we have a one-dimensional system with degenerate states. However, in the proof that such states cannot exist, we assumed that the wave function was defined over all , not just a restricted range. Thus the condition that the wave function goes to zero at infinity is not true here, so the conditions for non-degeneracy specified in the proof do not hold here.