How do you calculate the probabilities associated with eigenfunctions of a wave function? I'm watching Lecture 03-05 of the MIT 3.024 lecture series on Electronic, Optical and Magnetic Properties of Materials by Polina Anikeeva, specifically the discussion from the 23:30 mark onwards.
In her lecture Prof. Anikeeva states that if we do not know the probabilities of different eigenvalues of momentum then we can measure the momentum many many times and approximate the probability from that. However, before that, she mentions that any measurement causes the collapse of wavefunction into one of its eigenfunctions associated with that value. From that point onwards, the new wave function describing that particle is the previous eigenfunction.
My question is why don't repeated measurements of momentum change the probability distribution and will give us probability distribution due to the original wave function.
 A: Yes, the wavefunction does change when you make a measurement, and hence the probability of the system ending up in a given eigenstate.  You would need to reset the experiment each time as you made these repeated measurements.  If you just repeatedly measured the same system without resetting it, then you would always get the same answer (assuming no time evolution), because the wavefunction is now exactly one of the eigenstates, and so measuring it will return that eigenstate with 100% probability.
A: The first measurement will always collapse the wave function and if you quickly perform more measurements you will simply get the same outcome. You have two main options in order to deal with this collapse:
1) Returning the particle to its original state after each measurement.
2) Set up a whole ensemble of particles which are in the same state $\Psi$  and measure all of them. I like this analogy made by David Griffiths: 'Imagine a row of bottles on a shelf, each containing a particle in the state $\Psi$ relative to the center of the bottle.
Let's say you are measuring the energy. The average of all the outcomes of E you get would be $< E >$ (the expectation value of E).
