# Help solving Eigenvalue problem

1. Sep 23, 2008

### Felicity

1. The problem statement, all variables and given/known data

solve the eigenvalue problem

(-∞)x dx' (ψ(x' ) x' )=λψ(x)

what values of the eigenvalue λ lead to square-integrable eigenfunctions?

3. The attempt at a solution

(-∞)xdx' (ψ(x' ) x' )=λψ(x)

differentiate both sides to get

ψ(x)x=λ d/dx ψ(x)

ψ(x)x/λ= d/dx ψ(x)

2xe x^2 =d/dx e x^2

so ψ(x) = e x^2 and λ = 2

but this is not square integrable so either this is incorrect or there are other solutions I am not seeing

Can anyone help me find what I am missing?

Thank you

2. Sep 23, 2008

### Dick

You want to solve the differential equation, ψ(x)x/λ= d/dx ψ(x) Separate the variables. You should get a solution with a lambda in it. Figure out what values of lambda make it square integrable.

3. Sep 23, 2008

### yaychemistry

Hello,
I think there are other solutions.
$$x\psi\left(x\right) = \lambda\frac{d}{dx}\psi\left(x\right)$$
and substitute $$\psi\left(x\right) = \exp\left(f\left(x\right)\right)$$ and see if you don't get an equation for $$f\left(x\right)$$ which has a solution that depends on $$\lambda$$.

Also, don't forget your solution has to be finite at the lower endpoint of the integral ($$-\infty$$) in the original problem statement.

4. Sep 23, 2008

### Felicity

ok i separated variables, integrated and got

ψ(x)=e1/(2λ) x2

however I don't see how this can be square integrable since for any value of λ I can think of the integral of the square will equal infinity. I feel like I am missing something obvious here but I don't know what it is

5. Sep 23, 2008

### Dick

Will it equal infinity even if lambda is negative?

6. Sep 23, 2008

### Felicity

of course! im so embarrassed, thank you