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: Expectation value via trace

  1. Jun 11, 2012 #1
    1. The problem statement, all variables and given/known data
    [itex]\mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle )[/itex]
    where [itex]\mid1\rangle, \mid2\rangle[/itex] are orthonormal
    i)density operator
    ii) [itex]\langle A \rangle[/itex] where A is an observable
    2. Relevant equations

    3. The attempt at a solution
    i) [itex]\rho = \frac{1}{2} (\mid1\rangle\langle1\mid + \mid2\rangle\langle2\mid)[/itex]

    ii) [itex]\langle A \rangle = \frac{1}{2}\langle1\mid A\mid1\rangle + \frac{1}{2}\langle2\mid A\mid2\rangle + \frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle[/itex]
    i guess [itex]\frac{1}{2}\langle2\mid A\mid1\rangle + \frac{1}{2}\langle1\mid A\mid2\rangle[/itex] = 0 (because first 2 terms are each half the expected value but surely depends on operator therefore not necessarily zero??)

    also I would like to use the trace to solve this:
    [itex]\langle A \rangle = tr[\rho A] = tr[\frac{1}{2}\mid1\rangle \langle1\mid A + \frac{1}{2}\mid2\rangle \langle2\mid A][/itex] but what does this mean?
    Last edited: Jun 11, 2012
  2. jcsd
  3. Jun 11, 2012 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Education Advisor

    This is the density matrix for a mixed state, where half the particles are in state ##\vert 1 \rangle## and half are in state ##\vert 2 \rangle##. The state you were given is a pure state.

    You can't really say anything about those matrix elements since you don't know anything about ##\hat{A}## except that it's an observable.

    Say you have some basis and you find the matrix representing ##\rho\hat{A}##. The trace is simply the sum of the diagonal elements. How would you write that in Dirac notation?
  4. Jun 12, 2012 #3
    thanks, I have since worked out that the density matrix I gave is not complete (I assumed that the outer product of two orthogonal vectors would be zero!!)
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook