Spin-1/2 with expectation values other than $\hbar/2$

Question

I came across a problem that is related to the expectation value of spin-1/2. Assuming I want to find a single (or possibly a set of) spin state(s) $$\lvert\psi\rangle$$ that gives me an expectation value of say,

$$ \langle\psi\rvert Sx\lvert\psi\rangle = \pm\hbar/4 $$ $$ \langle\psi\rvert Sy\lvert\psi\rangle = \pm\hbar/4 $$ $$ \langle\psi\rvert Sz\lvert\psi\rangle = \pm\hbar/4 $$

How should I approach this problem. Is it still a simple superposition of eigenspinors or does it require spin coupling?

Answer 1

Representing the spin state as a spinor $$\begin{pmatrix}a\\b\end{pmatrix}.$$ $a$ and $b$ are in general complex numbers and obey the normalization condition $$|a|^2 + |b|^2=1,$$ so that leaves three independent parameters to define your spin state.

Writing the spin operators in their matrix form each of your above written equations becomes an equation for $a$ and $b$. Hence you have three equations with three unknowns and can solve for the spin state.