Eigenvalue for harmonic oscillator

[Habeebe]

Homework Statement

The Hamiltonian for a particle in a harmonic potential is given by
$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{Kx^2}{2}$, where K is the spring constant. Start with the trial wave function $\psi(x)=exp(\frac{-x^2}{2a^2})$
and solve the energy eigenvalue equation $\hat{H}\psi(x)=E\psi(x)$. You must find the value of the constant, a, which will make applying the Hamiltonian to the function return a constant time[sic, I assume he meant "times"] the function. Then find the energy eigenvalue.

Homework Equations

$\hat{H}=\frac{\hat{p}^2}{2m}+\frac{Kx^2}{2}$
$\psi(x)=exp(\frac{-x^2}{2a^2})$
$\hat{H}\psi(x)=E\psi(x)$

The Attempt at a Solution

Application of the Hamiltonian gave me:
$\hat{H}\psi(x)=[\frac{h^2}{2m}(\frac{1}{a^2}+\frac{x^2}{a^2})+\frac{1}{2}Kx^2]\psi(x)=E\psi(x)$

If I understand the problem correctly, since E and a must be constant, I must come up with an a, devoid of any x's or functions of x's so taking the Hamiltonian will yield something equally devoid of x's and functions of x's.

I mean, I get that I should have $[\frac{h^2}{2m}(\frac{1}{a^2}+\frac{x^2}{a^2})+\frac{1}{2}Kx^2]=E$ for some constants E, a. The problem is, I have no clue how to go about solving that and getting a to not involve any x's, nor am I convinced that it's possible. Using the quadratic formula on it gives a big mess (involving x), so I'm pretty sure that's the wrong route.

Thanks for the help.

 

[ehild]

You have a sign error: How is the operator $\hat{p}$ defined?

To remove x from the equation, collect the terms with x². You have to choose the parameter a properly, so as the two terms cancel each other.


ehild

 

[TSny]

Also check to see if you have enough factors of $a$ in the denominator of your $x^2/a^2$ term.

 

[Habeebe]

Thanks, I found the sign error and got it worked out. As for the factors of a, you're right, I actually typed it in wrong.

 

[paranormal]

I would first commend the student for their attempt at solving the problem and for recognizing the need for a constant value of a. I would then suggest that they try to simplify the equation by factoring out a common term, like x^2, and see if that leads to any solutions. Additionally, they could try substituting the given wave function into the energy eigenvalue equation and see if they can solve for a that way. If they are still having trouble, I would suggest consulting with a professor or fellow classmates for guidance and further clarification on the problem. It is important to remember that not all problems have simple solutions and it is okay to seek help and collaborate with others in the scientific community.