Introduction to Quantum Mechanics · Problem 8.16

The Variational Principle — David J. Griffiths

Physics — Quantum Mechanics · variational_infinite_well

Problem

(a) Use the function

\psi(x) = Ax(a - x)

(for

0 < x < a

, otherwise 0) to get an upper bound on the ground state of the infinite square well. (b) Generalize to a function of the form

\psi(x) = A[x(a - x)]^p

, for some real number

p

. What is the optimal value of

p

, and what is the best bound on the ground state energy? Compare the exact value. *Answer:*

\left(5 + 2\sqrt{6}\right)\hbar^2/2ma^2