4. a) Consider a square potential well which has an infinite barrier at x = 0 and a barrier of height U at x = L, as shown in the figure. For the case E < U, obtain the solutions of the Schodinger equation inside the well (0 s x s L), as well as for outside the well (x > L) that satisfy the appropriate boundary conditions at x = 0 and x = 00. Put the appropriate conditions on x =L to find the allowed energies of the system. Are there conditions for which the solution is not possible? explain. E

4. a) Consider a square potential well which has an infinite barrier at x = 0 and a barrier of height U at x = L, as shown in the figure. For the case E < U, obtain the solutions of the Schodinger equation inside the well (0 s x s L), as well as for outside the well (x > L) that satisfy the appropriate boundary conditions at x = 0 and x = 00. Put the appropriate conditions on x =L to find the allowed energies of the system. Are there conditions for which the solution is not possible? explain. E

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The essence of the statement of the uniqueness theorem is that if we know the conditions the limit that needs to be met by the potential of the system, then we find the solution of the system , then that solution is the only solution that exists and is not other solutions may be found. If we know potential solutions of a system, can we determine the type of system that generate this potential? If so, prove the statement! If no, give an example of a case that breaks the statement!
Question A2 Consider an infinite square well of width L, with V = 0 in the region -L/2 < x < L/2 and V → ∞ everywhere else. For this system: a) Write down and solve the time-independent Schrödinger equation for & inside the well, where -L/2< x
Problem 15. In the case of a particle tunneling through the barrier, we made equal both functions and derivatives at the boundary between region I and II, and between II and III, see eq. (9a – 9d). But in the case of the particle in the box, we made both functions equal at x = 0 and at x = L. Explain why we did not consider derivatives in this case.