Perturbation in Infinite Square well

[ed2288]

Homework Statement

Calculate the 1st order probability an electron in the ground state of an infinite sqaure well (width 1) will be found in the first excited state t seconds after the pertubation H=sin(PI*x) is switched on.

Homework Equations

Transition frequency is omega_12

The Attempt at a Solution

I know to begin this question I need to calculate the integral between 0 and 1 of the product of the ground state wavefunction, pertubation, and 1st excitied wavefunction. The product comes to sin^2(PI*x)sin(2PI*x), which when integrated comes to zero!
Any suggestions??
Thanks
(ps I know that this integral will not give the final answer, but I know exactly what to do after this step)

 

[Jhero]

Hello,

Thank you for your post. It seems like you are on the right track with your approach. However, there are a few things to consider in order to get the correct answer.

First, when calculating the probability of finding an electron in a particular energy state, we use the wavefunction squared, rather than the product of two wavefunctions. This is because the wavefunction squared gives us the probability density, which tells us the likelihood of finding the electron in a particular position.

Second, in order to calculate the probability at a specific time t, we need to use the time-dependent Schrodinger equation. This equation takes into account the time evolution of the wavefunction and allows us to calculate the probability at any given time.

In this case, we can use the time-dependent perturbation theory to solve for the probability at t seconds after the perturbation is switched on. This involves expanding the wavefunction in terms of the unperturbed states and using the transition frequency to determine the probability at the desired time.

I hope this helps guide you in the right direction. If you have any further questions, please let me know.