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4. Solve the 1D Schrodinger equation for a particle of mass, m, in a box. The particle is confined in the
box, modeled as an infinite potential well that is O inside the region from -<x< and an infinite
amount of potential energy outside this region. Notice this problem is similar to the problem fully
worked out in the lecture. Here the length of this box is a.
(a) Using this new coordinate system, solve the Schrodinger equation using the same procedure
shown in the lecture. Remark: You need to follow the setup exactly the way it is defined in this
problem. You cannot use a change in coordinate system. Solving means you will find the energy
spectrum and all corresponding wave functions that solve the Schrodinger equation.
(b) Compare the allowed energy levels that you derive in this problem to those derived in the
lecture: Are the two energy spectrums the same or different? Make sense of the answer based
on physical intuition.
(c) Compare the solutions for the allowed wave functions. These functions will be mathematically
different. However, show that these solutions give the same physical results. Hint: Do this using
trigonometric identities and show that one set of solutions map to the other set once a shift in
coordinates is made.

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