?A function of the form \(\mathrm{e}^{-g x^{2}}\) is a solution of the Schrödinger

Chapter 7, Problem P7E.9

(choose chapter or problem)

A function of the form \(\mathrm{e}^{-g x^{2}}\) is a solution of the Schrödinger equation for the harmonic oscillator (eqn 7E.2), provided that g is chosen correctly. In this problem you will find the correct form of g. (a) Start by substituting \(\psi=\mathrm{e}^{-g x^{2}}\) into the left-hand side of eqn 7E.2 and evaluating the second derivative. (b) You will find that in general the resulting expression is not of the form constant \(\times \psi\), implying that \(\psi\) is not a solution to the equation. However, by choosing the value of g such that the terms in \(x^{2}\) cancel one another, a solution is obtained. Find the required form of g and hence the corresponding energy. (c) Confirm that the function so obtained is indeed the ground state of the harmonic oscillator, as quoted in eqn 7E.7, and that it has the energy expected from eqn 7E.3.

Text Transcription:

e^-gx^2

psi=e^-gx^2

times psi

psi

x^2