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Time dependent eigenvalues?

  1. Nov 15, 2007 #1
    Given an operator [tex]\hat{Q}[/tex] (in the Schrodinger picture) in non-relativistic quantum mechanics and a state [tex]|\psi(t)\rangle[/tex] such that

    [tex]\hat{Q} |\psi(t)\rangle=q(t)|\psi(t)\rangle[/tex]

    where q(t) is explicitly time-dependent, can we properly say that [tex]|\psi(t)\rangle[/tex] is an eigenstate of Q with a time-dependent eigenvalue. That is, [tex]|\psi(t)\rangle[/tex] remains a eigenstate of Q for all times but its eigenvalue is different depending on when you measure it?

    For example, suppose we had a wave function of the form

    [tex]\psi(x,t)\propto e^{ixg(t)+h(t))}[/tex]

    then applying the momentum operator we find

    [tex]-i\frac{\partial}{\partial x}\psi(x,t)=g(t)\psi(x,t)[/tex].

    Would you say that [tex]\psi(x,t)[/tex] is momentum eigenstate with momentum = g(t)?
    Last edited: Nov 15, 2007
  2. jcsd
  3. Nov 15, 2007 #2
    This wave function is a eigenfunction of the Hamiltonian, and because the momentum operator commutes with the hamiltonian, it is also a eigenfunction of the momentum. You can write this wave function as

    \psi(x,t)\propto e^{i(p x-E t))}

    Where p corresponds to the eigenvalue of the momentum operator and E corresponds to the eigenvalue of the energy operator.
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