# Calculating eigenvalues and eigenstates

1. Apr 22, 2007

### fkliment2000

Hi!

i want to calculate the eigenvalues and the eigenstates of the momentum operator and the Hamilton operator of a free particle.

How do i do this?

2. Apr 23, 2007

### neu

3. Apr 23, 2007

### fkliment2000

I know what an eigenvalue equation is, but i don't know how to use this knowledge in this case.

For the Hamilton Operator I think I have to use the Schrödinger equation

H * phi(x) = E * phi(x) with H = -h^2/(2m) * nabla^2

But what is then the answer? Is it "E" is the Eigenvalue and phi(x) is the eigenfunction of H?
Which equation do I have to use for the momentum operator?

4. Apr 23, 2007

### Staff: Mentor

The Hamiltonian operator doesn't enter into this problem at all. On that page, the generic eigenvalue equation is

$\hat A f(x) = a f(x)$

where $\hat A$ is the operator and $a$ is an eigenvalue. In your situation, $\hat A$ is the momentum operator, and $f(x)$ is $\psi(x)$. Have you seen yet, what the momentum operator looks like?

5. Apr 23, 2007

### Repetit

For a free particle the wave function is given by (one dimensional) $$\Psi(x)=e^{i k x}$$. If you operate the hamiltonian on this eigenfunction you see that this particular wave function is a solution to the Schrödinger equation with eigenvalue $$E=\hbar^2 k^2 / 2m$$.

The momentum operator is $$-i \hbar d/dx$$. Operating on the free particle wave function you get

$$-i \hbar \frac{d \Psi}{dx} = p \Psi$$

where p is the eigenvalue of the momentum operator. So from this you see that the momentum eigenvalue of a free particle is just $$\hbar k$$.

6. Apr 23, 2007

### masudr

Indeed. OP: your momentum eigenvalue equation for a single free particle,

$$-i \hbar \frac{d \psi}{dx} = p \psi,$$

is a standard differential equation, assuming the wavefunction is static in time (you can worry about ones that change later). So, if you can solve differential equations, solve this one! See what you get for solutions to the D.E.

7. Apr 24, 2007

### mavyn

Hi!
I'm working on the same exercise as fkliment2000. I already got the eigenstates like Repetit said.
But I have some more questions to answer, but i dont know how..

1) Is cos(k*x) an eigenstate of both of these operators? (k is the wave number of this state, x is the position coordinate)
2) search all the eigenstates, which are common for both of these operators (Hamilton, momentum).
3) If a particle is in the eigenstate of the momentum operator, what is the probability to find it within the interval 0<x<1?

Thanks!

8. Apr 24, 2007

### cristo

Staff Emeritus
That said;
2. If two operators commute, what can be said about their sets of eigenstates?
3. How do you find the probability in QM of a particle being found in an interval, given the eigenstate of the particle?

9. Apr 24, 2007

### mavyn

2. Then they must have they same set of eigenstates.

3. I calculate the Integral from this interval over the probability density.