Introduction to Quantum Mechanics · Problem 8.18

The Variational Principle — David J. Griffiths

Physics — Quantum Mechanics · variational_harmonic_oscillator

Problem

(a) Generalize Problem 8.2, using the trial wave function

\psi(x) = \frac{A}{(x^2 + b^2)^n},

for arbitrary

n

. *Partial answer:* The best value of

b

is given by

b^2 = \frac{\hbar}{m\omega}\left[\frac{n(4n-1)(4n-3)}{2(2n+1)}\right]^{1/2}.

(b) Find the least upper bound on the first excited state of the harmonic oscillator using a trial function of the form

\psi(x) = \frac{Bx}{(x^2 + b^2)^n}.

*Partial answer:* The best value of

b

is given by

b^2 = \frac{\hbar}{m\omega}\left[\frac{n(4n-5)(4n-3)}{2(2n+1)}\right]^{1/2}.

n \to \infty

. Why is that? *Hint:* Plot the trial wave functions for

n = 2

n = 3

, and

n = 4

, and compare them with the true wave functions (Equations 2.60 and 2.63). To do it analytically, start with the identity

e^z = \lim_{n\to\infty}\left(1 + \frac{z}{n}\right)^n.