Quantum mechanics: one thousand neutrons in an infinite well

[Broseidon]

I apologize in advance for not being familiar with LaTex.

1. Homework Statement
One thousand neutrons are in an infinite square well, with walls x=0 and x=L. The state of the particle at t=0 is :
ψ(x,0)=Ax(x-L)

How many particles are in the interval (0,L/2) at t=3?

How many particles have energy E₅ at t=3s? (That is the <E> at t=3s)

Homework Equations

Wavefunction of the infinite square well: ψ(x)=Sqrt[2/L]Sin[n*Pi*x/L]

Time dependence of wavefunctions: ψ(x,t)=∑c_nψ_n(x,t)

c_n coefficients: c_n=∫ψ_n(x)*f(x)dx

The Attempt at a Solution

This problem is quite similar to this one: https://www.physicsforums.com/threads/neutrons-in-a-one-dimensional-box.242003/, except that now, one must calculate the quantities at a later time, forcing us to construct the full time-dependent solution (I suspect).

I begin by normalizing the initial wavefunction, where I get: A=Sqrt[30/L²].

Then, in principle, we can get ψ(x,t) by calculating those c_n coefficients, and plug it in the general solution (Griffiths example 2.2 does this).

Well, this is all very nice and beautiful, but let's not forget what the question asks: calculating a probability at a later time. When putting this in Born's postulate to find the probability, the time dependence (a complex exponential) cancels out, so we don't even have to opportunity to 'plug in 3s! Not to mention, the infinite summation would also give me trouble when trying to perform the integral. (I hope I made sense...)

So, does anyone have any advice on how else to attack this problem?
Thank you!

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[DrClaude]

The problem can be simplified by consider the symmetry of the situation. Also, look out for red herrings.

 

[DrClaude]

I begin by normalizing the initial wavefunction, where I get: A=Sqrt[30/L²].

Check that result also.

 

[Broseidon]

Right, probability is not supposed to change with time for stationary states, and it doesn't matter if it's 3 seconds or 0 seconds.
So to calculate the probability, I can use the sin wavefunction, and I suppose the initial wavefunction must then be used in the second part, since the expectation value of energy formula involves finding the c_n.
Am I missing anything else?

 

[DrClaude]

Right, probability is not supposed to change with time for stationary states, and it doesn't matter if it's 3 seconds or 0 seconds.

The probability of being in a given state doesn't change, but different states can interfere with each other. You need to see if this affects the answer here.