What Happens When Propagator Formalism Is Applied to an Eigenstate?

[ehrenfest]

If you use propagator formalixm to calculate the future time dependence of a state that starts in an eigenstate, what happens?

The equation for the propagator is

$$K(x, x';t,0) = \sum_n \psi_n^*(x')\psi_n(x)e^{-iE_nt/\hbar$$

So if we start in an eigenstate does that mean that the summation

$$\sum_n \psi_n^*(x')\psi_n(x)e^{-iE_nt/\hbar$$

can be rewritten

$$\sum_n \psi_n^*(x')\psi_9(x)e^{-iE_nt/\hbar$$

say if we started in the eigenstate corresponding to n = 9?

 

[Jhero]

it is important to clarify the terminology used in this forum post. "Propagator formalism" refers to a mathematical framework used in quantum mechanics to calculate the time evolution of a quantum state. It involves the use of a propagator, which is a mathematical operator that describes the transition of a state from one point in time to another.

In the context of this forum post, the question is asking about using the propagator formalism to calculate the time evolution of a state that starts in an eigenstate. An eigenstate is a special type of quantum state that represents a particular energy level of a system. In this case, the question is asking what happens when we use the propagator formalism to calculate the time evolution of a state that starts in a specific energy level (eigenstate).

To answer this question, we need to understand the equation for the propagator, which is correctly stated in the forum post as:

K(x, x';t,0) = \sum_n \psi_n^*(x')\psi_n(x)e^{-iE_nt/\hbar

This equation represents the transition amplitude from an initial state at time 0 to a final state at time t. The summation over n includes all possible energy levels of the system. So, if we start in an eigenstate, say the one corresponding to n = 9, then the summation will only include the term for n = 9. This means that the propagator will only involve the eigenstate corresponding to n = 9, and the equation can be rewritten as:

K(x, x';t,0) = \psi_9^*(x')\psi_9(x)e^{-iE_9t/\hbar

In other words, the propagator formalism will give us the time dependence of the state in terms of the eigenstate that it started in. This is a fundamental concept in quantum mechanics, known as the time evolution of states.

In conclusion, if we use the propagator formalism to calculate the time dependence of a state that starts in an eigenstate, the equation will involve only that particular eigenstate and its corresponding energy level. This is a powerful tool in understanding the behavior of quantum systems and is essential in many areas of physics and chemistry.