How can we compute expectation values for spin states using Pauli matrices?

[Robben]

Homework Statement

When calculating expectation values for spin states I encountered $\langle \hat{\mathbb{S}}_+\rangle = \langle+z|\hat{\mathbb{S}}_+|+z\rangle = \frac12\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle.$

How do we compute $\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle?$

Also, similary $\langle+z|\left(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-\right)^2|+z\rangle?$

Homework Equations

$\hat{\mathbb{S}}_x=\frac12(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-)$

The Attempt at a Solution

I know that $\langle+z|\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle = 0$ but I am not sure how to compute it to get zero.

Do I compute $\left(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle\right)$ first, which gives $\hbar|-z\rangle$ and then use the bra $\langle +z|$ to get zero?

 

[matteo137]

One approach, which I think is very pedagogic, is to write everything in terms of vectors and matrices. This can be done in this case because a spin has a finite length, while it would be more difficult for the case of a harmonic oscillator.

For example you can define $\vert +z \rangle = (1,0)^{T}$, $\vert -z \rangle = (0,1)^{T}$.
Then you have to express the spin operator in terms of Pauli matrices, and multiply everything.

e.g. $\langle +z \vert S_z \vert +z \rangle = (1,0) \dfrac{\hbar}{2}\sigma_z (1,0)^{T} = \dfrac{\hbar}{2}$

P.S. note that if your spin is not $1/2$ then it might be tough...

 

[matteo137]

Homework Statement

Do I compute $\left(\hat{\mathbb{S}}_++\hat{\mathbb{S}}_-|+z\rangle\right)$ first, which gives $\hbar|-z\rangle$ and then use the bra $\langle +z|$ to get zero?

This is the rigorous way, yes. You get zero because the states $\vert +z\rangle$ and $\vert -z\rangle$ are orthogonal.

 

[Robben]

One approach, which I think is very pedagogic, is to write everything in terms of vectors and matrices. This can be done in this case because a spin has a finite length, while it would be more difficult for the case of a harmonic oscillator.

For example you can define $\vert +z \rangle = (1,0)^{T}$, $\vert -z \rangle = (0,1)^{T}$.
Then you have to express the spin operator in terms of Pauli matrices, and multiply everything.

e.g. $\langle +z \vert S_z \vert +z \rangle = (1,0) \dfrac{\hbar}{2}\sigma_z (1,0)^{T} = \dfrac{\hbar}{2}$

P.S. note that if your spin is not $1/2$ then it might be tough...

Thank you very much!