# Homework Help: Energy eigen value

1. Feb 23, 2006

### Reshma

The ground state of a one-dimensional Harmonic oscillator described by the Hamiltonian $H = \frac{p^2}{2m} + \frac{kx^2}{2}$ is of the form, $\psi = Ae^{-ax^2}$. Determine 'A' and 'a' so that the wavefunction $\psi$ is a normalized eigenstate of the Hamiltonian. What is the energy eigenvalue of the wavefunction?

Well, I was able to normalize the wavefunction and obtained the value of 'A'.
$$\int_{-\infty}^{\infty}\psi \psi^* dx =1$$

$$A^2\int_{-\infty}^{\infty}e^{-2ax^2}dx =1$$

$$A^2\sqrt{\frac{\pi}{2a}} =1$$

$$A = (\frac{2a}{\pi})^{1/4}$$

How do I determine 'a'? Any clues to obtain energy eigen value?

Last edited: Feb 24, 2006
2. Feb 24, 2006

### Reshma

Sorry, I could not correct my errors yesterday. I have rectified the LaTeX typos. Now can someone help me....?

3. Feb 24, 2006

### Hurkyl

Staff Emeritus
I would say use the definitions of "eigenstate" and "eigenvalue".

4. Feb 24, 2006

### Reshma

You mean use the eigenfunction and obtain the eigenvalue?
$$i \hbar \frac{\partial}{\partial t} \psi = \mathcall H \psi$$

5. Feb 24, 2006

### inha

Just apply the harmonic oscillators hamiltonian to the eigenfunction. And I don't think you can determine a but you can set some constraints on it. a just tells you how wide the gaussian is.

6. Feb 24, 2006

### Hurkyl

Staff Emeritus
The definitions are that $\psi$ is an eigenfunction of H with eigenvalue $\lambda$ if and only if $H \psi = \lambda \psi$.

7. Feb 24, 2006

### qtp

yep you should just be able to operate on the wavefunction with the hamiltonian to obtain the eigenvalues which are the energy values

8. Feb 25, 2006

### Reshma

$$H = {p^2 \over 2m} + {1\over 2} m \omega^2 x^2$$
$$p = -i \hbar \partial / \partial x$$
$${-\hbar^2\over 2m}{\partial^2 \psi \over \partial x^2} + {1\over 2} m \omega^2 x^2 \psi = E_n \psi$$
$$E_n = \hbar \omega \left(n + {1\over 2}\right)$$