# Finding spin up/spin down eigenstates along some arbitrary direction

So let's say I have some particle in some arbitrary state where its ket vector is given as a linear combination of the spin up eigenstate and the spin down eigenstate. The magnitude square of a, which is the factor in front of the spin up eigenstate, gives the probability that a measurement on the particle yields spin up, in which case the spin angular momentum (SAM for short) points in the same direction as our z-axis. The same applies for b, which is the factor in front of the spin down eigenstate, except that it gives the probability of getting spin down, in which case the SAM points antiparallel to the z-axis.

But now, let's say that we want to measure the spin of the particle in some arbitrary direction, perhaps along some vector x. Given that the particle is in the same state as before, what is now the probability of getting spin up (in which case the SAM points along x), and the probability of getting spin down (the SAM points antiparallel to x)?

That is to say, how can I express the spin up/spin down eigenstates along x in terms of our previous spin up/spin down states (which points along the z-axis)? Using this, I can easily calculate the probabilities stated above by calculating the magnitude square of the inner products between the new eigenstates and the wave function.

You can use the projection operators $$P_\pm = \frac 12 (1+{\bf n}\cdot {\boldsymbol \sigma}).$$ Applied to any starting state they give you the eigenstates of $${\bf n}\cdot {\boldsymbol \sigma}$$ with spin $$\pm$$ along the direction specified by the unt vector $${\bf n}$$.

For small matrices projection operators are usually the fastest route the eigenvectors.

• Wow, that looks really cool. I did manage to find the general form of the two orthonormalized eigenvectors (in terms of the previous eigenstates) of that operator, but I had to check that they worked through "brute force", but this seems like a very easy way to find them in practise. At least it's much easier than the method presented by MannyC. Thank you for this! Jun 26, 2020 at 14:22

To compute the spin components along $$\hat{n}$$ consider the matrix $$\sigma_{n} \equiv \hat{n} \cdot \vec{\sigma}\,.$$ Diagonalize $$\sigma_n$$ and find the list of eigenvectors $$|i\rangle_n$$. Then express your ket in terms of that basis. So find the coefficients $$c_i$$ in $$|\mathrm{your\;state}\rangle = \sum_i c_i \,|i\rangle_n\,.$$ The probability of having spin $$i$$ along $$\hat{n}$$ is $$|c_i|^2$$.

As a sanity check, if you wish to measure the spin along $$\hat{z}$$, then $$\sigma_n = \sigma^3$$, which is already diagonal. So the coefficients $$c_i$$ are just the components of your ket.

By the way, this is the procedure to follow for all operators, not just for the spin.