(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Starting with [tex]\sigma_{y}[/tex], calculate the momentum eigenstates of spin in the y direction.

[tex]\sigma_{y} = \left[\stackrel{0}{i} \stackrel{-i}{0}\right][/tex] (Pauli spin matrix in the y direction)

[tex]S_{y} = \frac{\hbar}{2}\sigma_{y} [/tex] (spin angular momentum operator for the y direction)

2. Relevant equations

[tex]A\left|\psi\right\rangle = a\left|\psi\right\rangle [/tex] where A is some linear operator and a is the corresponding eigenvalue

3. The attempt at a solution

The solution I tried was determining the eigenvalues for the matrix, [tex] det (A - \lambda I) = 0[/tex], where [tex] A \equiv S_{y} [/tex], [tex]\lambda[/tex]

are the eigenvalues and I is the 2x2 identity matrix.

After working through the determinant expression, I obtain eigenvalues of [tex]\lambda = \pm \frac{\hbar}{2}[/tex]

Then for momentum eigenstates, since the eigenstates aren't given I just used an arbitrary eigenstate, defined as [tex]\left|\psi\right\rangle[/tex]

Therefore, the momentum eigenstates I obtain are just

[tex]S_{y}\left|\psi\right\rangle = \pm \frac{\hbar}{2} \left|\psi\right\rangle [/tex]

I'm just wondering if my logic is correct as I step through my calculations. First I tried operator the spin angular momentum (y-direction) operator in the known matrices for spin-up, spin-down states. But, I realised that these were states in the z-direction. So, for momentum eigenstates in the y-direction the only way I could think of was the eigenvalue equation method.

Thanks.

p.s. Does anyone know how to write matrices in latex? Sorry, about my dodgy matrix up above for sigma y

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# Homework Help: Momentum eigenstates of spin

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