I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,[itex]\phi[/itex]) using the Pauli Matrices

2. Relevant equations

Spin operator in arbitrary direction:

n.[itex]\sigma[/itex] = [itex]\hbar[/itex]/2(cos[itex]\phi[/itex]sin[itex]\theta[/itex][itex]\sigma_x[/itex] + sin[itex]\phi[/itex]sin[itex]\theta[/itex][itex]\sigma_y[/itex]+cos[itex]\theta\sigma_z[/itex])

[itex]\sigma_x[/itex],[itex]\sigma_y[/itex],[itex]\sigma_z/[itex] are the Pauli spin matrices.

3. The attempt at a solution

The way I did it was to express the pauli matrices in their matrix form, sum up the expression to get one matrix, then solve the eigenvalue equation

n.[itex]\sigma[/itex][itex]\Psi[/itex] = [itex]\lambda[/itex][itex]\Psi[/itex].

This gives me the answer [itex]\pm[/itex][itex]\hbar[/itex]/2

My question is: Is there a better/quicker way to do this (and problems similar to this) without having to solve the eigenvalue equation directly? I have other similar questions where solving the eigenvalue equation becomes long and tedious.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Eigenvalues and Eigenstates of Spin Operator

**Physics Forums | Science Articles, Homework Help, Discussion**