Probability of staying in same state after time-dep perturbation?

[vincebs]

Let's say that you've got a time-dependent perturbation to your potential (say, the particle-in-a-box to make things simple). Say you start in energy eigenstate #3. What's the probability that the particle will stay in eigenstate 3 after time T?

This is not a homework problem. I'm not understanding the equation for the probability which seems to give a number larger than 1.

 

[genneth]

I take you're staring at Fermi's Golden Rule. That rule is a first order time dependent perturbation result, and so will not give the correct answer at large times or for large perturbations.

 

[denian]

I would approach this question by first understanding the concept of time-dependent perturbation and its effect on the system. Time-dependent perturbation refers to a change in the potential energy of a system over time. In this case, the potential energy of the particle-in-a-box is changing over time.

When a system experiences a perturbation, it can cause the energy eigenstates to mix and evolve over time. This means that the particle's initial state, in this case, eigenstate #3, may change over time. Therefore, the probability of the particle staying in eigenstate #3 after time T is not a fixed value and will depend on the specific details of the perturbation.

To calculate the probability, we can use the time-dependent Schrödinger equation, which describes how the wave function of a system evolves over time. We can solve this equation to determine the evolution of the particle's state over time and then use the resulting wave function to calculate the probability of the particle being in eigenstate #3 at time T.

It is important to note that the probability calculated using this method may be greater than 1. This is because the time-dependent perturbation can cause the eigenstates to mix, resulting in a probability amplitude that is greater than 1. However, the probability of the particle being in a specific eigenstate cannot exceed 1, as this would violate the laws of quantum mechanics.

In summary, the probability of the particle staying in eigenstate #3 after time T will depend on the specific details of the time-dependent perturbation. It is important to carefully consider the effects of the perturbation on the system and use the appropriate mathematical tools, such as the time-dependent Schrödinger equation, to calculate the probability accurately.