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!

Homework Help: Variance of an eigenvector

  1. Aug 28, 2009 #1
    1. The problem statement, all variables and given/known data
    On page 5 of http://arcsecond.wordpress.com/2009...uality-and-heisenbergs-uncertainty-principle/ the author states (w/o proof) that if [tex]\psi[/tex] is an eigenvector (say with eigenvalue [tex]\lambda[/tex]) of an Hermitian operator A (I don't think the Hermitian-ness matters here), then its variance is 0; that is, [tex]\langle \psi| A^2\psi\rangle = \langle \psi| A\psi\rangle^2[/tex]. However, I've not been able to show this.

    2. Relevant equations

    3. The attempt at a solution
    I keep getting
    [tex]\langle \psi|A^2\psi\rangle = \lambda\langle \psi|A\psi\rangle = \lambda^2\langle\psi |\psi\rangle[/tex]​
    [tex]\langle \psi|A\psi\rangle^2 = \left(\lambda\langle \psi|A\psi\rangle\right)^2 = \lambda^2\langle\psi |\psi\rangle^2.[/tex]​
    Where am I going wrong?
  2. jcsd
  3. Aug 28, 2009 #2


    User Avatar
    Homework Helper
    Gold Member

    Eigenvectors are typically normalized, so [itex]\langle\psi\vert\psi\rangle=1[/itex]
  4. Aug 28, 2009 #3
    D'oh! Of course! Thanks!
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook