Quantum Mechanics: Concepts and Applications · Problem 6.1

Three-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · three-dimensional-potential-well-harmonic-oscillator

Problem

Consider a spinless particle of mass mm which is moving in a three-dimensional potential V(x,y,z)={12mω2z2,0<x<a, 0<y<a,∞,elsewhere.V(x, y, z) = \begin{cases} \frac{1}{2}m\omega^2 z^2, & 0 < x < a,\ 0 < y < a, \\ \infty, & \text{elsewhere.} \end{cases} (a) Write down the total energy and the total wave function of this particle. (b) Assuming that ℏω>3π2ℏ2/(2ma2)\hbar\omega > 3\pi^2\hbar^2/(2ma^2), find the energies and the corresponding degeneracies for the ground state and first excited state. (c) Assume now that, in addition to the potential V(x,y,z)V(x, y, z), this particle also has a negative electric charge −q-q and that it is subjected to a constant electric field ϵ\epsilon directed along the zz-axis. The Hamiltonian along the zz-axis is thus given by H^z=−ℏ22m∂2∂z2+12mω2z2−qϵz.\hat{H}_z = -\frac{\hbar^2}{2m}\frac{\partial^2}{\partial z^2} + \frac{1}{2}m\omega^2 z^2 - q\epsilon z. Derive the energy expression EnzE_{n_z} for this particle and also its total energy EnxnynzE_{n_x n_y n_z}. Then find the energies and the corresponding degeneracies for the ground state and first excited state.

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