Solving Probability of Particle in 2nd Excited State of Box

[eck]

Here's the problem I'm trying to solve:
Suppose a particle of mass m is sitting in the ground state of a 1-dimensional box of length L. The length of the box is suddenly doubled, and the energy of the particle is measured. What is the probability that the particle will be found in the second excited state of the box?
I know the particle initially has some energy corresponding to the first energy level in the initial box, and I can calculate the energy corresponding to the third energy level (second excited state) in the new box. To solve the problem I need to the new wavefunction for the particle. So I'm going to have something (I think) like $$A \sin frac{n \pi x}{L}$$. The only problem is I don't know how to get A... I can't just normalize it, because I'd get a probability of one. Any pointers?

 

[eck]

the LaTeX got sort of meseed up... it should be A sin (n*pi*x/L)

 

[jtbell]

To solve the problem I need to the new wavefunction for the particle.

Actually, the wavefunction of the particle immediately after the box expands is exactly the same as the wavefunction immediately before the box expands. So, start by writing the complete wave function (for all values of x) for the ground state of the unexpanded box, substituting the appropriate value of n.

Now, you need to think of this function as a linear combination of the energy eigenfunctions of the expanded box, and find the coefficient that goes with the new n=2 eigenfunction. There should be something in your textbook about this linear combination business, and how to find the coefficients. If you can't find it, tell us which book you have and someone might have it and be able to point you to the right place.

 

[George Jones]

the LaTeX got sort of meseed up... it should be A sin (n*pi*x/L)

As always, jtbel has given good physical advice. Let me add a few words about using Latex when posting to Physics Forums.

You can edit your own posts by clicking the edit button, which only appears when you're logged in. Also, you can view the Latex source for expressions in any post (not just your own) by left-clicking on the expression. Doing this to your post reveals the missing "\".

There used to to be a preview button that allowed Latex expressions to be previewed before posting. Now that this feature is no longer available, I, when posting stuff that includes Latex expressions, use the method of successive approximations. Post; edit; post; edit; post;... . I do this until I get to within $$\epsilon$$ of what I want to say.

Regards,
George