Electron Clebsch-Gordon coefficients

[Mastern00b]

Homework Statement
The state of an electron is,

|Psi> =a|l =2, m=0> ⊗ |up> + Psi =a|l =2, m=1> ⊗ |down>,

a and b are constants with |a|² + |b|² = 1
choose a and b such that |Psi> is an eigenstate of the following operators: L², S², J² and J_z.
The attempt at a solution

I am really not sure where to start on this, i have tried reading through previous threads on the coefficients and i am still confused.

For a start does the statement above mean that the wavefunction describes both a spin up and spin down arrangement, both of which are possible at any time? And do i need to treat this as two particles, one spin up and one spin down?

 

[Mastern00b]

Anyone got any ideas that could help?

 

[Orodruin]

It simply means that your electron is in a superposition between two states, one which has orbital angular momentum 2 with orbital angular momentum z-component 0 and spin up and another which has orbital angular momentum 2, orbital angular momentum z-component 1, and spin down. Your task is to adjust the linear combination such that the resulting state is an eigenstate of the list of operators.

As a first step, do you know how these operators act on this state?