# One-dimensional linear harmonic oscillator perturbation

1. Jan 29, 2014

### Firben

1. The problem statement, all variables and given/known data

Consider a one-dimensional linear harmonic oscillator perturbed by a Gaussian perturbation H' = λe-ax2. Calculate the first-order correction to the groundstate energy and to the energy of the first excited state

2. Relevant equations

ψn(x) = $\frac{α}{√π*2n*n!}$1/2 * e2x2$\frac{1}{2}$

E1n = <ψ0n|H'|ψ0n>

3. The attempt at a solution

E10 = <ψ00|H'|ψ00> =

∫$\frac{α}{√π*2n*n!}$1/2 * e2x2$\frac{1}{2}$*$\frac{α}{√π*2n*n!}$1/2 * e2x2$\frac{1}{2}$*H'* dx

Is this right ? What is α in this case ?

2. Jan 29, 2014

### vela

Staff Emeritus
Your expression for the n-th wavefunction is missing the Hermite polynomial Hn(x). It should be
$$\psi_n(x) = \left(\frac{a}{\sqrt{\pi}2^n n!}\right)^{1/2} e^{-a^2x^2/2} H_n(ax).$$ Your expression for the first-order energy correction for the ground state turns out to be fine because H0(x)=1. The quantity $a$ should be defined in your notes or textbook. It's the characteristic length scale for the harmonic oscillator.