# QM Spin Function Question

• bmb2009
In summary, we are given a spin wave function (ψ1, ψ2)τ and need to determine a possible normalized spin wave function. By expressing the norm of the wave function in terms of ψ1 and ψ2, we can derive a normalized wave function (Ψ) that is a superposition of states. Using inner products, we can then determine the probability of finding the particles in the +z and -z directions.

## 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
1, ψ2)τ

## 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!

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.