Dosen't energy eigenvalue depend on x?

[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?
Where can I find explanation about this?

 

[DrClaude]

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?
Where can I find explanation about this?

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.)

 

[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?

 

[dauto]

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

 

[rar0308]

dauto is Right. How can i delete this thread?