QUANTUM THEORY OF MATTER Homework set #13: Degenerate and non-degenerate stationary perturbation theory Problem # 13.1 : Electric ﬁeld and harmonic oscillator Consider a charged particle (of charge q) in the one-dimensional harmonic oscillator poten- tial. Suppose we turn on a weak electric ﬁeld (E), so that the potential energy is shifted by an amount H = −qEx. (a) Show that there is no ﬁrst-order change in the energy levels. (b) Calculate the second order correction to the energy levels. o (c) The Schr¨dinger equation for H = H0 + H can be solved directly by a change of variables: x = x − (qE/mω 2 ). Find the exact energies, and show that they are consistent with the perturbation theory approximation. Problem # 13.2 : Bead on a ring Consider a particle of mass m that is free to move on a ring of length L (e.g. a circular wire of circumference L). (a) Show that the stationary states can be written in the form 1 0 ψn (x) = √ e2πinx/L , −L/2 < x < L/2 , L where n = 0, ±1, ±2, · · ·, and the allowed energies are 2 0 2 h nπ¯ En = . m L Note that, with the exception of the ground state (n = 0), these states are all doubly degenerate. (b) Now suppose we introduce the perturbation 2 /a2 H = −V0 e−x , where a << L. This perturbation puts a little dimple in the potential at x = 0, as though we bent the wire slightly to make a trap for the bead (particle). Find the ﬁrst order correction to En using degenerate perturbation theory when appropriate. Hint: To evaluate the integrals assume a << L and extend the limits of integration from ±L/2 to ±∞, since H is essentially zero outside −a < x < a. 0 0 (c) What are the ”good” linear combinations of ψn and ψ−n for this problem? What is the symmetry associated with these linear combinations. Which operator commutes with the Hamiltonian? Problem # 13.3 : Perturbed cubical well Consider the three-dimensional inﬁnite cubical well deﬁned by the potential 0 for 0 < x < a, 0 < y < a, 0 < z < a , V (x, y, z) = ∞ otherwise . Suppose we perturb this potential by placing a δ-function bump at the point (a/4, a/2, 3a/4): H = a3 V0 δ(x − a/4)δ(y − a/2)δ(z − 3a/4) . (a) Find the energy, wave function(s) and degeneracy of the ground state and ﬁrst excited state(s) for the unperturbed well. (b) Find the ﬁrst-order corrections due to H to the energy of the ground state and the ﬁrst excited state(s). Problem # 13.4 : Harmonic oscillator Consider the isotropic three-dimensional harmonic oscillator in Cartesian coordinates and discuss the eﬀects of the perturbation H = λx2 yz for some constant λ. (a) Find the energy, wave function(s) and degeneracy of the ground state and ﬁrst excited state(s) for the unperturbed oscillator. (b) Find the ﬁrst-order corrections due to H to the energy of the ground state and the ﬁrst excited state(s). Hint: The matrix elements of the displacement x for a simple harmonic oscillator are given by √ √ n|x|n = ¯ /2mω h n δn,n −1 + nδn ,n−1 .