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: Canonical perturbation theory

  1. Oct 27, 2007 #1
    1. The problem statement, all variables and given/known data

    An electron is inside a magnetic field oriented in the z-direction. No measurement of the electron has been made. A magnetic field in the x-direction is now switched on. Calculate the first-order change in the energy levels as a result of this perturbation.

    3. The attempt at a solution

    I got this wrong, claiming that the first order perturbation is zero. I said this because if you evaluate the expectation of the [itex]S_x[/itex] matrix in the states up/down individually, you obviously get zero since [itex]S_x[/itex] is anti-diagonal.

    Apparently you are supposed to evaluate the expectation of [itex]S_x[/itex] in the superposition of states [itex]\psi = 1/\sqrt{2}(1,0)^T + 1/\sqrt{2}(0,1)^T[/itex]. I have trouble understanding this because [itex]\psi[/itex] is not an eigenstate of the orginial Hamiltonian, which it should be for canonical perturbation theory.

    Any help is greatly appreciated.
  2. jcsd
  3. Oct 27, 2007 #2

    Physics Monkey

    User Avatar
    Science Advisor
    Homework Helper

    Hi noospace,

    Something is very strange here. Assuming the z-direction magnetic field is large and the x-direction magnetic field is small, your calculation is correct. There is no first order change in the energy levels in such a situation. (One could just solve the complete problem trivially and see this fact.) Using eigenstates of the perturbation to calculate expectation values is certainly wrong. I suspect there is some kind of confusion here. Any ideas?
  4. Nov 2, 2007 #3
    Hi Physics Monkey,

    Thanks for your response. My lecturer's approach to this question is to evaluate the expectation of the operator [itex]S_x[/itex] in the superposition of states [itex]\psi = \frac{1}{\sqrt{2}} | 1/2 1/2 \rangle + \frac{1}{\sqrt{2}}}|1/2 \pm 1/2\rangle[/itex]. The answer given is

    [itex] \pm\frac{B_x\hbar}{2m_e}[/itex]

    My lecturer claims that since [itex]| 1/2 1/2 \rangle, | 1/2 -1/2 \rangle[/itex] are separately eigenstates of the Hamiltonian then so should their superposition. I claim this is false since the sum of two eigenvectors need not be an eigenvector.

    Who is correct?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook