Can a Quantum State Have Time-Dependent Eigenvalues?

[pellman]

Given an operator $$\hat{Q}$$ (in the Schrodinger picture) in non-relativistic quantum mechanics and a state $$|\psi(t)\rangle$$ such that

$$\hat{Q} |\psi(t)\rangle=q(t)|\psi(t)\rangle$$

where q(t) is explicitly time-dependent, can we properly say that $$|\psi(t)\rangle$$ is an eigenstate of Q with a time-dependent eigenvalue. That is, $$|\psi(t)\rangle$$ 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

$$\psi(x,t)\propto e^{ixg(t)+h(t))}$$

then applying the momentum operator we find

$$-i\frac{\partial}{\partial x}\psi(x,t)=g(t)\psi(x,t)$$.

Would you say that $$\psi(x,t)$$ is momentum eigenstate with momentum = g(t)?

 

[Magister]

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.

 

[denian]

I would say that the concept of time-dependent eigenvalues is an interesting and relevant topic in quantum mechanics. In this scenario, it is possible for the state |\psi(t)\rangle to be an eigenstate of the operator \hat{Q} at different times, with varying eigenvalues. This is because the operator \hat{Q} and the state |\psi(t)\rangle are both time-dependent.

In the example given, it is correct to say that \psi(x,t) is a momentum eigenstate with a time-dependent eigenvalue of g(t). This means that the state has a well-defined momentum at any given time, but the value of that momentum may change over time due to the time-dependent function g(t). This concept is important in understanding how quantum systems evolve over time and how their properties, such as momentum, can change.

It is important to note that time-dependent eigenvalues are a result of the time-dependence of the system and do not violate any fundamental principles of quantum mechanics. In fact, they can provide valuable insights into the behavior of quantum systems and can be used to make predictions about their future states.

In summary, the concept of time-dependent eigenvalues is a valid and useful concept in quantum mechanics, allowing us to describe the behavior of quantum systems at different points in time and understand how their properties may change over time.