# Expectation values

1. Homework Statement
Calculate the expectation values of x, $$x^2$$ for a particle in a one dimensional box in state $$\Psi_n$$

2. Homework 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

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G01
Homework Helper
Gold Member
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!)

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

G01
Homework Helper
Gold Member
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
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.

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?

G01
Homework Helper
Gold Member
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?
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?

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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?

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?