Quantum Mechanics: Concepts and Applications · Problem 9.8

Approximation Methods for Stationary States — Nouredine Zettili

Physics — Quantum Mechanics · perturbation-theory-wkb-modified-infinite-well

Problem

Consider an infinite, one-dimensional potential well of length

L

, with walls at

x = 0

and

x = L

, that is modified at the bottom by a perturbation

V_p(x)

V(x) = \begin{cases} 0, & 0 < x < L \\ \infty, & \text{elsewhere} \end{cases}, \qquad V_p(x) = \begin{cases} V_0, & 0 \leq x \leq L/2 \\ 0, & \text{elsewhere} \end{cases},

where

V_0 \ll 1

. (a) Using first-order perturbation theory, calculate the energy

E_n

. (b) Calculate the energy

E_n

in the WKB approximation. Compare this energy with the expression obtained in (a).