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!

Finding normalized eigenfunctions of a linear operator in Matrix QM

  1. Oct 18, 2013 #1
    1. The problem statement, all variables and given/known data
    Hey everyone!
    The question is this:
    Consider a two-state system with normalized energy eigenstates [itex]\psi_{1}(x)[/itex] and [itex]\psi_{2}(x)[/itex], and corresponding energy eigenvalues [itex]E_{1}[/itex] and [itex]E_{2} = E_{1}+\Delta E; \Delta E>0[/itex]

    (a) There is another linear operator [itex]\hat{S}[/itex] that acts by swapping the two energy eigenstates: [itex]\hat{S}\psi_{1}=\psi_{2}[/itex] and [itex]\hat{S}\psi_{2}=\psi_{1}[/itex]. Show that the corresponding normalized eigenfunctions of [itex]\hat{S}[/itex] are [itex]\phi_{1}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)+\psi_{2}(x))[/itex] and [itex]\phi_{2}(x)=\frac{1}{\sqrt{2}}(\psi_{1}(x)-\psi_{2}(x))[/itex], with eigenvalues [itex]\lambda_{1}=+1[/itex] and [itex]\lambda_{2}=-1[/itex]

    2. Relevant equations
    - The regular eigenvalue/eigenvector stuff.
    - In matrix mechanics, the wavefunction [itex]\psi_{n}(x)=C_{1}(1,0...n)^{T}+C_{2}(0,1,0...n)^{T}+...C_{n}(0....n)^{T}[/itex]

    3. The attempt at a solution
    So I've found the matrix associated with the linear operator [itex]\hat{S}[/itex] in the natural basis [itex]U_{1}=(1,0)^{T}, U_{2}=(0,1)^{T}[/itex]; which is:
    http://imageshack.com/a/img266/5713/bfo6.jpg [Broken]
    This gives the correct eigenvalues of +1 and -1. But I dont know how to find the eigenfunctions that the question is talking about. I'm not even sure if the matrix is right...
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 19, 2013 #2

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    I think the hint I can give is to use that S is linear and consider a linear combination of the 2 vectors psi1 and psi2.
     
  4. Oct 19, 2013 #3
    But is my matrix correct for S..?
     
  5. Oct 20, 2013 #4

    dextercioby

    User Avatar
    Science Advisor
    Homework Helper

    Who cares, you needn't use any matrix to solve the problem.
     
  6. Oct 20, 2013 #5
    what other ways are there that I can solve this? is it the bra-kets? And yea that matrix is wrong, it should be 0 along the diagonal and 1 otherwise. That's the only way it can be linear always and hermitian, which is the requirement.
     
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: Finding normalized eigenfunctions of a linear operator in Matrix QM
Loading...