please help me solve these questions, the teacher included the final answer in the screenshots but i want the steps for it so i can understand how they got the final answer

1) Consider a particle of mass m in the finite well, given by the potential energy: Vo for x a II The particle is trapped in a stationary bound V state of this potential with energy 0 a a. Write the general form of the time-independent wavefunction y (x) of the particle for each of the three regions: I (x a). Wherever possible write your answers in terms of the constants: 2m (Vo - E) and k = 1 2m E . 12 Answer: W, = Ae" + Be", y, = Fcos kx + G sin(kx), ", = Ce-"* + De"* b. What is the result of applying the convergence conditions of the wavefunction for x - - co in region I and for x - + co in region III ? Answer: R = 0 and D = 0 c. Find all the boundary conditions satisfied by the wavefunction at x = 0 and at x = a. (Use the fact that y and - must be continuous at the boundaries.) dx A =F aA = kG Answer: \\ Ce= = F cos(ka) + G sin(ka) -aCe- = - kF sin(ka) + kG cos(ka) d. Given the symmetry of V(x) around x = a/2, the eigenfunctions will be either symmetric or antisymmetric around r = a/2. Show that a symmetric wavefunction will satisfy at the boundaries the condition A = Ce-do2) Consider a particle of mass m in the finite well, given by the potential energy: co for x a The particle is trapped in a stationary bound state of this potential with energy 0 a ? Explain why in detail. Answer: Zero - in Classical Mechanics the Kinetic Energy cannot be negative d. Write the equations satisfied by the wavefunction y(x) of the particle in each of the regions: I (x a). Write the general form of the time- independent wavefunction for each of these three regions. 4,(x) =0 Answer: { Wy(x) = A coskx + B sin(kx) where a = 2m (Vo - E) and k = 2m E 4In(x) = Ce-ax e. Find all the boundary conditions satisfied by the wavefunction at x = 0 and at x = a. A =0 Answer: Ce= B sin(ka) -aCea = k B cos(ka)3) Consider a particle of mass m in the the potential energy: co for x 5 - b (region 1) Vo for - b b (region V) where Vo, a and b are positive real constants and b > a. The particle is trapped in a stationary bound state of this potential with energy 0 b O