Time dependent eigenvalues?

  • Thread starter pellman
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  • #1
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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)?
 
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Answers and Replies

  • #2
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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

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

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

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