# Gross Structure of Hydrogen

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1. Feb 17, 2015

### Robsta

1. The problem statement, all variables and given/known data
Show that for hydrogen the matrix element <2 0 0|z|2 1 0> = -3a0 where a0 is the Bohr Radius.

On account of the non-zero value of this matrix element, when an electric field is applied to a hydrogen
atom in its first excited state, the atom's energy is linear in the field strength.

2. Relevant equations

Energy of electron: -ħ2/2a02μn2

3. The attempt at a solution
<2 0 0| and |2 1 0> are bra and ket states of Hydrogen |n l m> where n is the principle quantum number, l is the orbital number and m is the magnetic number. I think I'm just struggling to work out what the operator z does (does it just point out the z coordinate of the electron?) Any advice on how I can approach this, specifically what matrix is being referred to, would be great.

2. Feb 17, 2015

### lightgrav

the "z" between the bra and the ket is the z function , which is anti-symmetric along the z coordinate.
It is needed so that the L=1 ket state, after multiplied by z, has non-zero overlap with the (symmetric) L-0 bra state.
(so that, any operator that is non-symmetric in z (I wonder what that might be?) might initiate a transition).

3. Feb 18, 2015

### Robsta

I'm not really sure what operator would be non-symmetric in z since the hydrogen atom is spherically symmetrical?

4. Feb 18, 2015

### Robsta

And I'm still not really sure what matrix is being referred to in the question

5. Feb 18, 2015

### Staff: Mentor

Read the second part of the question.

When you have a basis of states $|\phi_i\rangle$, you can construct a matrix representation of any operator $\hat{A}$, where the elements are
$$A_ij = \langle \phi_i | \hat{A} | \phi_j \rangle$$
This is why these bracket "sandwiches" are often referred to as matrix elements. Note that the wave function can then be written as a vector.