Suppose a particle of mass m moves in a box of length a, with boundaries at x =

Question

Suppose a particle of mass m moves in a box of length a, with boundaries at x = 0 and x = a.
Instead of the potential being zero inside the box, suppose the potential is at a constant value of
1 eV across the box and is infinite outside of the box. Physically, you would expect that all of the
energies would be simply shift up by 1 eV and the wavefunctions would not be altered. Show
this by setting up the Schrodinger Equation for this problem and solving for the normalized
eigenstates and the eigenvalues.

Explanation / Answer

Suppose you have a particle of mass, "m," in a box. The walls of the box are separated by a distance, "L." The walls are infinitely high. This system can be described using a wavefunction, "?(x)," as a possible solution to the Schrodinger equation. The Schrodinger time-independent equation is: (-h^2 /8p^2 m) * (d^2 ?(x) / dx^2) + V(x) ?(x) = E ?(x). "E" is the total energy, "V(x)" is the potential energy, "?(x)" (hereafter just "?") is the wavefunction, and "h" is Planck's constant -- a constant you'll meet often in quantum mechanics. 2 Note that outside the box, the particle will have infinite potential energy. The Schrodinger equation then becomes: (-h^2 /8p^2 m) * (d^2 ?(x) / dx^2) + ?(x) * infinity = E ?(x). The only way this relationship can now hold true is if ?(x) = 0, because then both sides will be equal. Therefore, the wavefunction is 0 everywhere outside the box. Since the probability of finding the particle in a given location is just the square of the wavefunction, there is zero probability the particle can be outside the box.

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