Stark effect energy shifts in hydrogen n=2 states

[PHamnett]

Homework Statement

Considering the n = 2 states of hydrogen: In the absense of an external field, the four j = 1/2 states are degenerate. Using degenerate pertubation theory, I am supposed to show that for a very weak field the Stark effect shifts the energy levels by

$$\mp \sqrt{3}a_0 e \epsilon$$

where $$a_0$$ is the bohr radius, e is the electric charge and $$\epsilon$$ is the electric field


The solutions state that we should get the following equations:
<1 1/2 1/2 | z | 0 1/2 1/2> = $$-\sqrt{1/3}$$<1 0 | z | 0 0>

and

<1 1/2 -1/2 | z | 0 1/2 -1/2> = $$\sqrt{1/3}$$<1 0 | z | 0 0>

and then from there a pertubed hamiltonian is constructed.

But to be honest, none of it seems to make sense and me and my friend do not know where even the first line of the solutions comes from. Yet alone the rest.
Any light that could be shed on the situation would be most appreciated.
Thanks.

 

[vela]

What term is being added to the unperturbed Hamiltonian? Start by figuring that out. Then it should be clear why you need to find the matrix elements of z.