Expectation values

1. Jan 28, 2007

thenewbosco

1. The problem statement, all variables and given/known data
Calculate the expectation values of x, $$x^2$$ for a particle in a one dimensional box in state $$\Psi_n$$

2. Relevant equations
$$\Psi_n = \sqrt{\frac{2}{a}}sin(\frac{n\pi x}{a})$$

3. The attempt at a solution
i formed the integral
$$\int_{-\infty}^{+\infty}\Psi ^2 x dx$$ as the expectation value of x. (Psi squared simply because this psi is not complex)
this gives $$<x>=\frac{2}{a} \int_{-\infty}^{+\infty}x sin^2(\frac{n\pi x}{a})dx$$. The problem is i do not know a way to simplify this integral, similarly i have the problem when there is an x^2 in the integral.
any help would be appreciated

2. Jan 28, 2007

G01

First, the limits of integration are the ends of the box, not infinity...(since psi is zero outside the box anyway)

To solve, try integration by parts. (BTW I just did the exact same problem!)

3. Oct 22, 2007

gardman007

I am working on this same problem for x^2. Since x is just a multiplicative operator, shouldn't you just be able to put in x^2 in front of the sin term? It doesn't seem to give me the correct result.
Thanks

4. Oct 22, 2007

G01

Can I see your work? You may be making a computational error. You should be able to do that integral by integration by parts as well, just like the previous one. It just takes one more step.

5. Oct 22, 2007

gardman007

I found your solution on another page G01, I'm still not sure what I was doing wrong though. It seems like my method should work. Any ideas?

6. Oct 22, 2007

G01

What was your method? If you mean putting the x^2 in front of the sine term that should be fine. How did the solution you find solve it?

Last edited: Oct 22, 2007
7. Oct 22, 2007

gardman007

I am just calculating the actual values, so I'm letting my calculator due the integration. Could you explain the process to get to the correct integration?

8. Oct 22, 2007

gardman007

So, for <x^2>, I should just be able to replace x with x^2 in front of the sin term in the last equation of the first post?