Show that it is a energy eigenstate and find the corresponding energy

[Denver Dang]

Homework Statement

Hi.

I'm looking at a hydrogen atom, which normalized stationary states is defined as |nlm>

The hydrogen atom is described by the normalized wavefunction:
$$\left| \psi \right\rangle =\frac{1}{\sqrt{2}}\left( \left| 210 \right\rangle +\left| 211 \right\rangle \right)$$
Now, show that $\left| \psi \right\rangle$ is an energy eigenstate, and find the corresponding energy.

Homework Equations

I'm told that:

$${{L}^{2}}\left| nlm \right\rangle =l\left( l+1 \right){{\hbar }^{2}}\left| nlm \right\rangle$$
$${{L}_{z}}\left| nlm \right\rangle =m\hbar \left| nlm \right\rangle$$
$${{L}_{+}}={{L}_{x}}+i{{L}_{y}}$$
$${{L}_{-}}={{L}_{x}}-i{{L}_{y}}$$

The Attempt at a Solution

In my mind, it seems so easy, but I don't have my book at my side, so I can't even check how it is done. Does it has something to do with:
$$H\left| \psi \right\rangle =E\left| \psi \right\rangle$$

If so, what hamiltonian am I suppose to use ?

Well, I'm kinda lost right now, so I was hoping to get a push in the right direction.

Thanks in advance.


Regards

 

[Simon Bridge]

Yes - it is an energy eigenstate if it is an eigenvector of the Hamiltonian.
You will need the Hamiltonian for the hydrogen potential.

Note: out of the quantum numbers shown in the state vectors, which refer to the energy eigenstates?

 

[Denver Dang]

Ahh yes, thank you very much :D