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One dimensional infinite square well potential

a) Show that the classical probability distribution function for a particle in a one dimensional infinite square well potential of length L is given by P(x) = 1/L

b) Use the result from part (a) to find the expectation value for X and the expectation value for X^2 for a classical particle in such a well.

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The probability distribution function is the square of the absolute value of the wavefunction. As you approach the classical limit the wavefunction will oscillate faster and faster. To get a well defined classical probability distribution, you define a coarse-grained average by integrating over a small length interval epsilon before you take the classical limit. This is not just a mathematical trick. If you want to measure the position of the particle you must use some device that has a finite length. So, you will always have to deal with the average of the probability distribution over some finite length interval.

If you take the classical limit then this coarse grained average will be smooth function because you've averaged out the infinite fast oscillations. You can then take the limit epsilon --> 0 to zero without problems. That function will still be smooth and this is the classical probability distribution P(x).

Let's start with deriving the energy eigenfunctions.
Outside the square well the wavefunction must be zero. Because the wavefunction is continuous the ...

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