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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]