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Help me with the Time Dependent Schrodinger Equation please

  1. Dec 1, 2015 #1

    RJLiberator

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    Gold Member

    1. The problem statement, all variables and given/known data
    Let |v(t)> ∈ℂ^2 by the time-evolving state of a qubit.
    [tex]If |v(0)> =\begin{pmatrix}
    0 \\
    1
    \end{pmatrix}[/tex]
    , and the Hamiltonian of the system is [tex]H =
    \begin{pmatrix}
    0 & -iλ \\
    iλ & 0
    \end{pmatrix} (where λ∈ℝ)[/tex]
    what is |v(t)>?


    2. Relevant equations

    Time dependent schrodinger Equation:
    iħ*d/dt(|v(t)>=H*|v(t)>

    iħ*d/dt(α_j (t)) = α_j(t)*λ_j
    3. The attempt at a solution

    We just learned this material at the end of last lecture and I need to apply it to a couple final homework problems.

    Overall, this seems like a straightforward computation but I'm severely struggling to decipher what I need to do with the giving material.

    I first calculated the eigenvalues of the Hamiltonian as the teacher stated. This came out to be +/- λ.
    Next, the teacher suggested expanding |v(t)> to something, I'm not really sure what he means by this.
    What do the alphas represent? I have no idea... All I have is:
    α_j (t) = α_j(0) e^(-iλ_i*t/ħ)

    Can some please help me decipher this / lead me to start?

    Thank you.
     
  2. jcsd
  3. Dec 2, 2015 #2

    blue_leaf77

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    Science Advisor
    Homework Helper

    What does ##If## mean in the left hand side?
    Anyway, apart from calculating the eigenvalues, you should also calculate the corresponding eigenstates of the Hamiltonian. Having found them, determine how the initial state ##v(0)## expands in term of these eigenstates, that is, try to find the expansion coefficients. The steps beyond this should be easy for you, to express the time evolution you simply need to apply the time evolution operator ##e^{iHt/\hbar}## to the initial state.
     
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