What is a Possible Normalized Spin Wave Function?

[bmb2009]

Homework Statement

A number of spin 1/2 particles are run through a Stern-Gerlach apparatus and when the emerge they all have the same spin wave function (ψ₁, ψ₂)^τ and 9/25 are in the +z direction and 16/25 are in the -z direction with the normal basis column vectors for +z and -z.

Determine a possible normalized spin wave function
(ψ₁, ψ₂)^τ

Homework Equations

The Attempt at a Solution

I am not sure where to start, all i can think of is to have 9/25(+z) = (9/25,0)^τ and 16/25(-z) = (0, 16/25)^τ to yield a total wave function of (9/25,16/25)^τ
but I don't know what to do for the normalization or if my initial thought was even on the right track? Thanks for the guidance!

 

[fzero]

You are on the right track, but have confused the relationship between the wavefunction and the probability to find a particle in a given quantum state.

First, since you are worried about the normalization, if we are given a spin wavefunction $( \psi_1 ~~\psi_2)^T$, can you express the norm of this vector in terms of $\psi_{1,2}$?

Can you use the previous result to derive a normalized wavefunction (call it $\Psi$ for the same superposition of states?

Now, given $\Psi$ can you express the probability to find the particle in the +z direction in terms of $\psi_{1,2}$? Think in terms of inner products.

Do the same for the -z direction.