Introduction to Quantum Mechanics · Problem 6.33

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · discrete_rotational_symmetry

Problem

Consider a particle of mass mm in a two-dimensional infinite square well with sides of length LL. With the origin placed at the center of the well, the stationary states can be written as ψnxny(x,y)=2Lsin⁡ ⁣[nxπL(x−L2)]sin⁡ ⁣[nyπL(y−L2)],\psi_{n_x n_y}(x,y) = \frac{2}{L}\sin\!\left[\frac{n_x\pi}{L}\left(x - \frac{L}{2}\right)\right]\sin\!\left[\frac{n_y\pi}{L}\left(y - \frac{L}{2}\right)\right], with energies Enxny=π2ℏ22mL2(nx2+ny2),E_{n_x n_y} = \frac{\pi^2\hbar^2}{2mL^2}\left(n_x^2 + n_y^2\right), for positive integers nxn_x and nyn_y. (a) The two states ψab\psi_{ab} and ψba\psi_{ba} for a≠ba \neq b are clearly degenerate. Show that a rotation by 90∘90^\circ counterclockwise about the center of the square carries one into the other, R^ ψab∝ψba,\hat{R}\,\psi_{ab} \propto \psi_{ba}, and determine the constant of proportionality. *Hint*: write ψab\psi_{ab} in polar coordinates. (b) Suppose that instead of ψab\psi_{ab} and ψba\psi_{ba} we choose the basis ψ+\psi_+ and ψ−\psi_- for our two degenerate states: ψ±=ψab±ψba2.\psi_{\pm} = \frac{\psi_{ab} \pm \psi_{ba}}{\sqrt{2}}. Show that if aa and bb are both even or both odd, then ψ+\psi_+ and ψ−\psi_- are eigenstates of the rotation operator. (c) Make a contour plot of the state ψ−\psi_- for a=5a = 5 and b=7b = 7 and verify (visually) that it is an eigenstate of every symmetry operation of the square (rotation by an integer multiple of π/2\pi/2, reflection across a diagonal, or reflection along a line bisecting two sides). The fact that ψ+\psi_+ and ψ−\psi_- are not connected to each other by any symmetry of the square means that there must be additional symmetry explaining the degeneracy of these two states.

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