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Homework Help: Stupid question about superposition of quantum states

  1. May 29, 2015 #1
    1. The problem statement, all variables and given/known data
    A quantum-mechanical harmonic oscillator with frequency ω has Hamiltonian eigenstates |n with eigenvalues En = (n + 1/2) ħω. Initially, the oscillator is in the state (|0> + |1>)/√2. Write down how the state of the oscillator evolves as a function of time t. Calculate the first time for which the time-evolved state is orthogonal to the initial one.

    3. The attempt at a solution
    I know how to evolve the states in time using the exponential solution to the TDSE.

    |ψ(t)> = (e-iωt/2|0> + e-3iωt/2|1>)/√2.

    This is fine. now I want the probability of finding the system in state |0> at time t. I bra through with <0|
    <0|ψ(t)> = (e-iωt/2<0|0> + e-3iωt/2<0|1>)/√2.

    <0|1> are orthogonal states so their inner product dissapears. <0|0> = 1

    <0|ψ(t)> = e-iωt/2/√2

    The probability of finding the state is |<0|ψ(t)>|2

    P(|0>) = 1/2

    Now here's the problem. When I take the mod square, the time dependence disappears, implying that the probability is constant for all time. This isn't true, at least I don't think it's true. Isn't the probability supposed to oscillate? I've forgotten how this oscillation comes about in bra-ket notation.
  2. jcsd
  3. May 29, 2015 #2


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    There is no problem with your calculation.

    However, this is not what they are asking: they are asking the first time at which the state is orthogonal to the initial one so you need [itex] \Bigl| \langle \psi(0) | \psi(t) \rangle \Bigr|^2 [/itex]
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