Infinite potential well energy question

[phys2]

Homework Statement

A particle of mass m is confined (in one dimension) to the region 0 ≤ x ≤ a by a potential which is zero inside the region and infinitely large outside.

If the wavefunction at time t = 0 is of the form

ψ (x,0) = Ax(a - x) inside the region
ψ (x, 0) = 0 outside the region

(a) Find the value of A to normalise the wavefunction
(b) The probability of measuring the ground state energy of the particle.

Homework Equations

P = integral of ψ times its complex conjugate = 1

The Attempt at a Solution

So for (a), I used the formula above and integrating with respect to x from 0 to a, I got A = square root of 30/a⁵

(b) Since this is an infinite potential well, the energy values would be E = ħ²n²π² / 2ma² so the ground state energy would be ħ²π² / 2ma²

So am I supposed to find the probability of getting the above ground state energy function?

I was thinking that this might have something to do with energy expectation values but then that has a dψ/dt under the integral so I would end up getting zero which wouldn't make any sense.

 

[phys2]

Does anyone have an answer?

 

[Steely Dan]

When thinking about the probability to get a certain energy level, another way to ask the question is what is the probability that the particle will be in the state corresponding to that energy level? In other words, you know the actual wavefunction at t = 0 and you want to know what the probability is that it will be found in the state with the wavefunction corresponding to the ground state energy level. What is the process for evaluating that likelihood?