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Quantum mechanics-hamiltonian matrix and stationary states

  1. Jul 16, 2014 #1
    1. The problem statement, all variables and given/known data
    let [1> and [2> mutually orthogonal states (eigenstates of some Hermitian operator).
    the Hamiltonian operator is given by H=c[1><2]+c[2><1], where c is a real number.

    (a) calculate the eigenstates and corresponding eigenvalues of H
    (b) if the initial state of the system is [1>, how does the state evolve with time?


    2. Relevant equations
    H[ψ>=ih[itex]\frac{∂[ψ>}{∂t}[/itex]

    3. The attempt at a solution
    for (a), I first earned hamiltonian matrix as below
    0 c
    c 0

    from this, I earned two eigenvalues +c and -c.
    and earned [itex]\sqrt{1/2}[/itex]( 1 1 ) and [itex]\sqrt{1/2}[/itex]( 1 -1 ) as two eigenstates of hamiltonian matrix.

    however what I don't know is the answer of (b)
    the problem I was given does not have any information of the state [1>. Is there anyway to solve this problem? if I assume [1> as (1 0) (and [2> as (0 1)) this problem can be solved. but I don't know more... can anyone help me?
     
  2. jcsd
  3. Jul 16, 2014 #2

    BruceW

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    Homework Helper

    nice work for 'part a'. I get the same answer. And for part b, it says you need to work out the time dependence. I think there are a couple of slightly different ways you could go about this. The way I prefer to do it is to think about [1(t)>, and write it out in the basis of [1(0)> and [2(0)>. So, you are right that in this basis, [1(t)> starts off as (1 0) But at some time later it will evolve to other values, for example (0 -1). So, the problem is what kind of function of time should these coefficients be? To work that out, you can use your 'relevant equation', and you can also make use of the energy eigenstates if you want, since you have already calculated them. but in the end, I think you should write [1(t)> in terms of [1(0)> and [2(0)>
     
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