- #1

slimjim

- 11

- 0

## Homework Statement

Consider as an unperturbed system H0 a simple harmonic oscillator with mass m,

spring constant k and natural frequency w = sqrt(k/m), and a perturbation H1 = k′x =

k′sqrt(hbar/2m)(a+ + a−)

Determine the exact ground state energy and wave function of the perturbed system

here a+ and a- are the ladder operators: a+/-=1/sqrt( 2hbar*m*ω)(-/+p^2 +(mωx) )

and H0=1/(2m)*(p^2 +(mωx)^2) = hbar*ω*[(a+)*(a-)+1/2]

## The Attempt at a Solution

I just need a nudge in the right direction for this one. The questoin confuses me because I didnt realize exact solutions could be found for perturbed systems.

we want to solve Hψ=Eψ where H= H0 + λH1

(the unperturbed hamiltonian plus the perturbation)

I am confused about the factor of λ on H1.

griffiths text says " for the moment we'll take lambda to be a small number, later we'll crank it up to one, and H will be the true hamiltonian"

so should I take λ=1?

either way, when I write out Hψ=Eψ, i get a very nasty diff. eq. which leads me to believe there is a better method.