You have carefully prepared the wave function of an electron to be in the first energy eigenstate of an infinite potential well spanning a length a. Now you decide to release the electron from the prison by turning off the infinite potential barrier. The electron thus becomes free to move with zero potential everywhere. Your aim now is to make an immediate measurement of the energy of the electron and verify that the probabilities work according to the laws of quantum mechanics.
a) To do this you first need to calculate the fourier component of the initial wave function. [Hint: It is useful to express the sinusoidal function as a linear combination of exponentials using Euler's formulas.]
b) Simplify your answer by rewriting the exponentials in terms of sines and cosines. What is the energy associated with a fourier component k? Now, you can find the probability density of finding the particle with energy E.
c) For very large and very small values of k (or E), what happens to this probability density?
d) Intuitively, at what values of energy do you expect the probability density to peak?