A Modern Approach to Quantum Mechanics · Problem 11.9

Identical Particles — John S. Townsend

Physics — Quantum Mechanics · three-dimensional-box-perturbation

Problem

A particle of mass

m

is confined in the three-dimensional potential energy box

V(x,y,z) = \begin{cases} 0 & 0 < x < L,\ 0 < y < L,\ 0 < z < L \\ \infty & \text{elsewhere} \end{cases}

Determine the first-order shift in the energies of the ground state and the first excited states due to the perturbation

\hat{H}_1 = \begin{cases} V_1 & 0 < x < L/2,\ 0 < y < L/2,\ 0 < z < L \\ 0 & \text{elsewhere} \end{cases}

which raises the potential energy by an amount

V_1

in one quarter of the box.