Write down the equations and the associated boundary conditions for solving particle in a 1-D box of dimension L with a finite potential well, i.e., the potential energy U is zero inside the box, but finite outside the box. Specifically, U = U, for x < 0, U = 0 for 0 ≤ x ≤ L, and U = U, for x > L. Assuming that particle's energy E is less than U, what form do the solutions take? Without solving the problem (feel free to give it a try though), qualitatively compare with the case with infinitely hard walls by sketching the differences in wave functions and probability densities and describing the changes in particle momenta and energy levels (e.g., increasing or decreasing and why), for a given quantum number.

Write down the equations and the associated boundary conditions for solving particle in a 1-D box of dimension L with a finite potential well, i.e., the potential energy U is zero inside the box, but finite outside the box. Specifically, U = U, for x < 0, U = 0 for 0 ≤ x ≤ L, and U = U, for x > L. Assuming that particle's energy E is less than U, what form do the solutions take? Without solving the problem (feel free to give it a try though), qualitatively compare with the case with infinitely hard walls by sketching the differences in wave functions and probability densities and describing the changes in particle momenta and energy levels (e.g., increasing or decreasing and why), for a given quantum number.