Quantum Mechanics: PHAS0042, Autumn 2019

Problem Sheet 4

Use Dirac notation for all these questions, except where indicated.

1. Consider an infinitely deep square well lying between x = 0 and x = a (so that the potential is infinite for x < 0 or x > a and zero for 0 < x < a), with eigenfunctions φn(x) = An sin(knx), kn = nπ/a.

(a) Normalise the basis functions (i.e. find An) [1]

(b) For a perturbation H0 (x) = Bδ(x − a/2), calculate explicitly the pertubation matrix for the first four eigen-states.        [4]

(c) Hence, for B = 1, find the energy of the first four perturbed energies (to first order). [2]

(d) Which states change in energy, and why ? In general (i.e. going beyond just the first four states) which states will change in energy ? Justify your answer. [2]

(e) For this system, for what value of B will first-order perturbation theory start to fail (roughly) ? Justify your answer.          [1]

N.B. The Dirac delta function has the property:

2. A quantized system can rotate about its z-axis with energy levels Em (0) = Bm2 and wavefunctions | ± mi =  exp(±imφ), where m is an integer, B is a constant and φ varies between 0 and 2π. This system is perturbed by an term which has the form. λ (exp(i2φ) + exp(−i2φ)).

(a) Explain why to first order all energy levels with |m| 6= 1 are unaffected by this perturbation. [1]

(b) Show that for the case of |m| = 1, the total energy, E, of the levels perturbed to first order can be obtained as solutions of         [3]

(c) Hence obtain expressions for the two energy levels, E, of the system perturbed through to first order.        [3]

(d) Give the corresponding normalised eigenvectors of the perturbed state in terms of the unperturbed eigenvectors | + 1i and | − 1i .           [3]

The following integral may be assumed for n integer:

3. (a) The first-order term in the polarisability of a system perturbed by an electric field E is given by α = −2∆E/E2. By writing the perturbation as V = −qEx, use perturbation theory to find the first and second order energy changes in a quantum harmonic oscillator in a linear electric field, and calculate the polarisability to first and second order.               [4]

(b) Using the trial function:

ψ(x) = A(a+ | x |) −n

show that the energy of a particle in a delta-potential:

V (x) = −αδ(x)

can be written as:

where there are two parameters a and n. Minimise the energy with respect to a first (to obtain expressions for both a and E as functions of n). What is the minimum value of E ?        [6]

[Start by normalising ψ(x); use  also  and remember that we can write the kinetic energy as  dx]