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 the time evolution of a state

  1. Mar 10, 2013 #1
    1. The problem statement, all variables and given/known data

    1) Using energies and eigenstates that I've worked out, find time evolution ψ(t) of a state that has an initial condition ψ(0) = [tex]
    \begin{pmatrix}
    1 \\
    0\\
    \end{pmatrix}
    [/tex]

    2) Find the expectation values < Sy> and <Sz> as a function of time.


    2. Relevant equations

    The Hamiltonian is [tex] H = \alpha (B_x S_x + B_y S_y + B_z S_z) [/tex]


    The energies that I worked out were the eigenvalues:

    [tex]\lambda_1= \frac{ \alpha \hbar B_x }{2}[/tex]

    [tex]\lambda_2= - \frac{ \alpha \hbar B_x }{2}[/tex]


    The eigenstates were the eigenvectors
    [tex]
    \begin{pmatrix}
    1 \\
    1\\
    \end{pmatrix}
    [/tex]

    and

    [tex]
    \begin{pmatrix}
    1 \\
    -1\\
    \end{pmatrix}
    [/tex]

    3. The attempt at a solution

    I tried using the time evolution operator
    [tex] U(t)= exp( \frac{ -i H t}{\hbar} ) [/tex]

    I ended up with something that looks like this:

    [tex]\psi(t) = A exp( \frac{ -i \alpha B_x t}{2} ) (1, 1) + B exp( \frac{ i \alpha B_x t}{2} ) (1, -1) [/tex]


    But I'm really unsure of where to go from here, or whether this is even right.
     
    Last edited: Mar 10, 2013
  2. jcsd
  3. Mar 10, 2013 #2
    Just check if your answer satisfies the inital condition and determine the constants A and B from normalization.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted