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.

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# Homework Help: Eigenvalues and Eigenstates of Spin Operator

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