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Simple quastion about superposition

  1. Nov 29, 2014 #1
    Is it possible to have a unitary evolution from ##|\psi_1\rangle\to a |\psi_1\rangle+b |\psi_2\rangle##, where ##|\psi_1\rangle##, ##|\psi_2\rangle## are two orthogonal states of a system?

    What if the state of the system is time dependent state of the form ##|\psi (t)\rangle = c_1(t) |\psi_1\rangle+c_2(t) |\psi_2\rangle##. Is it possible to have initial conditions ##c_1 (0)=1## and ##c_2 (0)=0## and at a later time t>0 get ##c_1 (t)\neq0## and ##c_2 (t)\neq0##?
     
    Last edited: Nov 29, 2014
  2. jcsd
  3. Nov 29, 2014 #2

    mfb

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    2016 Award

    Staff: Mentor

    Let
    $$|\psi_1\rangle = \frac{1}{\sqrt{2}} ( |1\rangle + |2\rangle ) \\
    |\psi_2\rangle = \frac{1}{\sqrt{2}} ( |1\rangle - |2\rangle )$$
    Where ##|1\rangle## and ##|2\rangle## are energy eigenstates with different energy. Then your evolution happens.

    Neutral meson mixing is an example from particle physics, but I'm sure there are similar systems in atomic transitions.
     
  4. Nov 29, 2014 #3
    Are you saying that ##|\psi_1\rangle \to a|\psi_1\rangle+b|\psi_2\rangle## is possible ?
    This means that there is some unitary operator ##U## such that
    $$U|\psi_1\rangle = a|\psi_1\rangle+b|\psi_2\rangle$$
    And because it's unitary, there is ##U'=U^{\dagger}## such that
    $$U' (a|\psi_1\rangle+b|\psi_2\rangle) =|\psi_1\rangle $$
    This seems like a "unitary wave-function collapse"
     
    Last edited: Nov 29, 2014
  5. Nov 29, 2014 #4
    Obviously |w1> to begin with isn't the correct state for the description of a system. Even though we see definite reality, the system is still described by QM which declares it in superposition.
     
  6. Nov 29, 2014 #5
    Well, what I am asking is if the evolutions like ##U |\psi_1\rangle = a|\psi_1\rangle+b|\psi_2\rangle## are possible.
    For me it seems totally legit since U is simply a rotation in Hilbert space.
    It's like in the ordinary vector space where, for example, a rotation matrix ##R(45^{o})## acts on the vector ##\vec{x}=(0,1)## and rotates it to the vector ##\frac{1}{2}\vec{x}+\frac{1}{2}\vec{y}##.
    $$R\vec{x}=\frac{1}{2}\vec{x}+\frac{1}{2}\vec{y}$$
    $$R^{-1}\left(\frac{1}{2}\vec{x}+\frac{1}{2}\vec{y}\right)=\vec{x}$$
    Am I right?

    My problem then is that a unitary evolution of the form ##U^{\dagger}(a|\psi_1\rangle+b|\psi_2\rangle)=|\psi_1\rangle ## seems like a "unitary wave-function collapse" and would solve the measurement problem.
     
    Last edited: Nov 29, 2014
  7. Nov 29, 2014 #6
    If it did, I'm sure we'd know about it now. Again, the state |w1> isn't alone.
     
  8. Nov 29, 2014 #7
    What do you mean that |w1> isn't alone (and why it isn't the correct state for the description of a system)?
     
  9. Nov 30, 2014 #8

    Nugatory

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    Staff: Mentor

    Except that it only works at one particular time ##t##.

    Say ##|\psi_1\rangle## and ##|\psi_2\rangle## are eigenfunctions of the observable A (which necessarily does not commute with the Hamiltonian) with eigenvalues ##a_1## and ##a_2##. There is a moment when a measurement of A will yield ##a_1## with certainty, but at any time arbitrarily close to that moment, there is some probability of getting ##a_2## instead, and the collapse into ##|\psi_1\rangle## or ##|\psi_2\rangle## will be non-unitary and the measurement problem is still there.

    As far as the measurement problem is concerned, this situation is no different from when I prepare a system in an eigenstate of some observable that does commute with the Hamiltonian, then make a measurement of that observable. The post-collapse state is the same as the pre-collapse state, but only because I made a very particular type of measurement on a system specifically prepared to produce that outcome.
     
    Last edited: Nov 30, 2014
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