Infinite potential well question with wave function $\psi (x) = (x -a/2)^2$

Question

In an infinite potential well with width $a$, a particle in this potential well is at state with wave function is $\psi (x) = (x -a/2)^2$ (not normalized).

1. If you measure the energy of the system just once, what is the energy value you get?

2. If you measure the energy 10000 times, roughly how many times do you get the ground state energy?

For the first question. I assume that the wave function here is the superposition of plenty eigenfunctions. So here comes the problem.... If this is true, then when I measure the energy of the system just once... which energy will I get?

For the second question. I though that when a wave had been measured, then the energy you get will be the same as you did the measurement second time or more....... Is that true? Thus, for here if we measure 10000 times. The probability of getting the group state is very low right? For that the $\psi (x)$ can be superposition by $\infty$ of state...

Answer 1

A general wavefunction for a system can be expanded as a sum of the eigenfunctions, $\psi_i$:

$$ \Psi = a_0\psi_0 + a_1\psi_1 + a_2\psi_2 + \,... $$

The coefficients $a_i$ are calculated using:

$$ a_i = \langle\psi_i | \Psi\rangle $$

If you do a measurement just once you will get one of the eigenvalues $E_i$, and the probability of getting each eigenvalue $E_i$ is $a_i^2$.

For the second question I would guess the question means 10,000 independent measurements, otherwise as you say after the first measurement you'll just keep getting the same result. In that case the question is asking for the expectation value of the energy $\langle E \rangle$ given by:

$$ \langle E \rangle = \langle\Psi | H | \Psi\rangle $$

which you can calculate using the expansion of $\Psi$. So feed in the expression for $\Psi$ that you're given in the question and away you go.