Consider a particle of mass m that is free to move in a one-dimensional region of length L that closes on itself (for instance, a bead that slides frictionlessly on a circular wire of circumference L, as in Problem 2.46).
(a) Show that the stationary states can be written in the form
ψn(x)=L1e2πinx/L,(−L/2<x<L/2),
where n=0,±1,±2,…, and the allowed energies are
En=m2(Lnπℏ)2.
Notice that—with the exception of the ground state (n=0)—these are all doubly degenerate.
(b) Now suppose we introduce the perturbation
H′=−V0e−x2/a2,
where a≪L. (This puts a little "dimple" in the potential at x=0, as though we bent the wire slightly to make a "trap".) Find the first-order correction to En, using Equation 7.33. *Hint:* To evaluate the integrals, exploit the fact that a≪L to extend the limits from ±L/2 to ±∞; after all, H′ is essentially zero outside −a<x<a.
(c) What are the "good" linear combinations of ψn and ψ−n, for this problem? (*Hint:* use Eq. 7.27.) Show that with these states you get the first-order correction using Equation 7.9.
(d) Find a hermitian operator A that fits the requirements of the theorem, and show that the simultaneous eigenstates of H0 and A are precisely the ones you used in (c).