Perturbative Methods for Modelling (M2)

Section 7.3,  p. 330
Example 2, p. 334


We consider the Schrödinger equation $\fs2 -y'' + \left({x^2\over 4} + \epsilon\,x\right) y = E \,y$  subject to the boundary condition $\fs2 \lim_{|x|\to \infty} y(x)=0$. In the harmonic potential case $\fs2 \epsilon=0$, show that $\fs2 E_0=1/2$ is an eigenvalue and $\fs2 y_0(x)=e^{-x^2/4}$ an associated eigenfunction.

Using perturbation theory, show that $\fs2 E=1/2-\epsilon^2 +O(\epsilon^3)$ for $\fs2 \epsilon\to 0$. Recover that result through a translation for $\fs2 x$.