# Quantum Mechanics: States

## 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?

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...

## 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.

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!