Principles of Quantum Mechanics · Problem 5.2.2

Simple Problems in One Dimension — R. Shankar

Physics — Quantum Mechanics · attractive-potential-bound-state

Problem

(a) Show that for any normalized

|\psi\rangle

\langle \psi|H|\psi\rangle \geq E_0

, where

E_0

is the lowest-energy eigenvalue. (Hint: Expand

|\psi\rangle

in the eigenbasis of

H

.) (b) Prove the following theorem: Every attractive potential in one dimension has at least one bound state. Hint: Since

V

is attractive, if we define

V(\infty) = 0

, it follows that

V(x) = -|V(x)|

for all

x

. To show that there exists a bound state with

E < 0

, consider

\psi_a(x) = \left(\frac{a}{\pi}\right)^{1/4} e^{-ax^2/2}

and calculate

E(a) = \langle \psi_a | H | \psi_a \rangle, \quad H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} - |V(x)|

Show that

E(a)

can be made negative by a suitable choice of

a

. The desired result follows from the application of the theorem proved above.