1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quantum Mechanics: States

  1. Mar 3, 2015 #1
    1. The problem statement, all variables and given/known data

    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?##


    2. Relevant equations

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




    3. 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?

     
  2. jcsd
  3. Mar 3, 2015 #2
    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...
     
  4. Mar 3, 2015 #3
    This is the rigorous way, yes. You get zero because the states ##\vert +z\rangle## and ##\vert -z\rangle## are orthogonal.
     
  5. Mar 3, 2015 #4
    Thank you very much!
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Quantum Mechanics: States
Loading...