# Dosen't energy eigenvalue depend on x?

1. Sep 24, 2013

### rar0308

there is potential V(x).
If at some point x=a wavefunction have some energy eigenvalue,
then Is it guaranteed that It has same energy throughout whole region?

2. Sep 24, 2013

### Staff: Mentor

You can't talk about the energy at a point. You get the energy as an expectation value (not eigenvalue, unless you are in an eigenstate of the Hamiltonian). So, if you want to know the total energy of the system described by the wave function $\psi(x)$, you calculate
$$E = \int_{-\infty}^{\infty} \psi^*(x) \hat{H} \psi(x) dx.$$
If you want only the potential energy, you use instead
$$E_\mathrm{pot} = \int_{-\infty}^{\infty} \psi^*(x) V(x) \psi(x) dx.$$
(I'm assuming that the wave function is normalized.)

3. Sep 24, 2013

### rar0308

Let quantum state be given as one of energy eigenfunctions.
V(x) is unknown except some point x=a.
Calculated energy eigenvalue using Hamiltonian operator at x=a.
Is it guaranteed that energy eigenvalue is the same throughout whole region other than x=a?

4. Sep 24, 2013

### dauto

Yes, otherwise the state wouldn't be an energy eigenfunction would it?

5. Sep 24, 2013

### rar0308

dauto is Right. How can i delete this thread?