Crandalls Puzzle. Stationary states of the one-dimensional Schrdinger equation ordinarily respect three rules of thumb: (1) The

Crandall’s Puzzle. Stationary states of the one-dimensional Schrödinger equation ordinarily respect three “rules of thumb”: 

(1) The energies are nondegenerate, 

(2) The ground state has no nodes, the first excited state has one node, the second has two, and so on, and 

(3) If the potential is an even function of x, the ground state is even, the first excited state is odd, the second is even, and so on. We have already seen that the “bead-on-a-ring” (Problem 2.46) violates the first of these; now suppose we introduce a “nick” in at the origin:

(If you don’t like the delta function, make it a gaussian, as in Problem 7.9.) This lifts the degeneracy, but what is the sequence of even and odd wave functions, and what is the sequence of node numbers?

Transcribed Image Text:

H' = -α8(x).