Using Perturbation Theory to Calculate Probability in Quantum Mechanics

[andyfreesty1e]

Quantum Physics Help!

Homework Statement

The spring constant for a particle in the ground state of a simple harmonic oscillator changes by a factor a^4 instantaneously, what is the probability that the particle is observed in the ground state of the new potential immediately following the change

can anyone give me a step in the right direction please, i think i have to use perturbation theory, but haven't really been taught how to

Homework Equations

v(x)=(a^4)(kx^2)/2 is the new potential

The Attempt at a Solution

 

[gabbagabbahey]

There's no need for perturbation theory here. Just ask yourself two questions:

(1)What is the ground state of the harmonic oscillator both before and after the change in spring constant?

(2)If I'm told that a particle is in the state [itex]|\psi_1\rangle[/itex] and I want to find the probability that it is measured in the state [itex]|\psi_2\rangle[/itex], what equation would I use?

The answer to number (1) is derived in most introductory QM texts, and the answer to number (2) is often taken to be one of the postulates of QM.

 

[andyfreesty1e]

so the ground state before is (1/a(sqrt Pi))^1/2 exp[-x^2/2a^2] where a=(h_bar/mw)^1/2
and w=(k/m)^1/2 so the ground state after is the same but w=(a^4k/m)^1/2 ?

so then calculate c=integral (psi*Psi(x,0))dx
and then c^2 is the probability?

does this seem about right?

 

[gabbagabbahey]

Seems about right to me

 

[andyfreesty1e]

sorry to keep asking questions, i think I am being a bit stupid, just to clarify something, in my above post would Psi(x,0) be my initial wavefunction, the one where the potential has not changed, and psi* would be the wavefunction when the potential has changed?
thanks