A particle is known with certainty to be located between the positions x=-1 and

Question

A particle is known with certainty to be located between the positions x=-1 and x=+1. But within that region, it could be at any position with equal probability. Write down a normalized wave function that describes this particle. Then, determine the probability distribution of its momentum values.

Explanation / Answer

Within the box, the Schrodinger equation for the particle is precisely the same as that for a particle in free space ( V = 0), which was solved on this page. We may thus immediately write out the wavefunction for the particle within the box . However, it will prove more convenient to use the basic relation ei? = cos? + isin? to write the solution in this form: where C and D are two more arbitrary constants, and 0 £ x £ L . Now, because of the presence of the potential walls, we shall see that further restrictions upon the permitted wavefunctions arise. For x > L or x < 0 , the wavefunction may be simply be stated to be ? = 0 , since the particle may not be found in these regions. We have established that for the total wavefunction of the particle to be acceptable it must be continuous. The wavefunction for the particle within the box must be zero at the walls for it to be continuous with the wavefunction for the particle within the walls. This imposes two boundary conditions upon the wavefunction ?k:

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