Why Does the Matrix Element <2 0 0|z|2 1 0> Equal -3a0 in Hydrogen?

[Robsta]

Homework Statement

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.

Homework Equations

Energy of electron: -ħ²/2a₀²μn²

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.

 

[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).

 

[Robsta]

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

 

[Robsta]

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

 

[DrClaude]

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

Read the second part of the question.

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