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 ( that's all screenshot so there is nothing missing that I didn't submit here )

1) An electron moves in a three-dimensional box whose sides have a length of 0.2 nm. The mass of the electron is me = 9.11 x 10 31 kg. The electron is in the third excited level corresponding to n? = 11 . a. Calculate the energy of the electron in eV . Answer: E = 103.41 eV b. What is the degeneracy of this state? Find all the combinations of the quantum numbers nr, ny and na that would yield this energy. Answer: Degeneracy = 3; (nx, ny, na) = (1,1,3), (1,3,1), (3,1,1) c. Write the wave functions for these different states in terms of the 1-dim Infinite Well eigenfunctions. Answer: y (x)y (y)y;(z); y(x);(y)y(z); 4(xu()y(z) 2) Assume that the nucleus of an atom can be regarded as a three-dimensional cubic box with width 2 x 10-14 m. Considering a proton (m = 1.67 x 10-27 kg ) in this three- dimensional box, a. Calculate the ground-state energy of the proton in MeV. Answer: 1.54 MeV b. Calculate the energies of the first and second excited states Answer: 3.07 MeV and 4.61 MeV c. What are the degeneracies of the first and second excited states? Answer: Both states have a degeneracy of 33) Consider an electron (m. = 9.11 x 10-31 kg) confined to a 2-dimensional infinite well of sides L, = Ly = L. a. Obtain a general expression for the energy levels of this particle. Answer: En,my = 2mlz (n2 + n; ) 25 hx2 b. Consider the electron in a state with energy 2,/2 -. Find the degeneracy of this state and write the wavefunctions for each degenerate state. Answer: Degeneracy = 2; y;(x).(y), WA(x)y;(y) c. You are given that the lateral dimension of the 2-dim square box is L = 1.0 x 10 'm, i. Find the lowest (ground state) energy of the electron in this 2-dim box in Joules and in ev. Answer: F. = 1.21 x 10-17 1 = 75.3 eV ii. What would be in Classical Mechanics the speed of an electron with a kinetic energy equal to the value you found in (c.i)? Answer: v = 5.15 x 105 m/s = 0.017 c 4) Consider the 3-dim Harmonic Oscillator for which the time-independent Schrodinger Equation is h2 2m dx2 y ( x. y. z )+ k (x2 + yz + 2? ) w ( x, y.z) = Ey (x, y,z) Using separation of variables, one obtains the following eigenfunctions and Energy eigenvalues: (You might enjoy trying to obtain this result) Why.my.n; (x, y. z) = Wn, (x)4, (y)ym, (2) where non,, n. = 0, 1, 2, 3. ... and y,, (x), 4,, (y) and y, (z) are one-dimensional Harmonic Oscillator eigenfunctions. a. Find the values of the four lowest energy states in terms of han. Answer: -hoo, zhao, hoo hoo. b. Find the degeneracy of each of these four levels. Answer: 1, 3, 6 and 105) Consider a particle of mass m in a three-dimensional cubic infinite well of side a, with potential energy: V(x, y, z ) = 0 for O S x, y.z Sa o otherwise The wavefunction of this particle at / = 0 is given by the following superposition of the eigenfunctions Un n n, (x, y, z): y (x, y, z) = A (641.1.1 - 34/2.1.1 + 94/1.2.3) Recall that for this particle, the energy eigenvalues are Engmy n. = (n2+ +nz)e where e = 2ma2 Answer the following questions without performing any actual integrations but using the orthonormality property of the eigenfunctions: -m -m -m where or " is the Kronecker delta function. a. Find the normalization constant A. Answer: A = 3V/14 b. i. What is the probability of finding the system in the state with energy 3c ? Answer: 2/7 ii. What is the probability of finding the system in the state with energy be? Answer: 1/14 iii. What is the probability of finding the system in the state with energy 12c? Answer: 0 iv. What is the probability of finding the system in the state with energy 14c? Answer: 9/14 c. What is the expectation value of the energy? Answer: - x3+ 14 X6+- 14 x 14 6 = 10.286e6) Consider a particle of mass m in a three-dimensional cubic infinite well of side a. We use the notation of Problem 5. In this case the normalized wavefunction of the particle is given by: 2 y ( x, y, z ) = 42.1.1 V21 41.2.1 21 V21 a. Show that y (x, y, z) is normalized. b. i. What is the probability of finding the system in the state with energy 3c ? Answer: 16/21 ii. What is the probability of finding the system in the state with energy be? Answer: 5/21 c. What is the expectation value of the energy? Answer: 16 -X3+ 21 X6+7X6) e = Te=3.71e 7) Consider for this problem the eigenfunctions and energy eigenvalues of a 3-dimensional Harmonic Oscillator given in Problem 4. A particle of mass m is in a three-dimensional harmonic oscillator potential energy. The system has angular frequency @, and a wavefunction given by 5 3 2 y (x, y, z ) = -42.1.0 yo.2,1 - 40.0.3 39 V39 39 39 a. Show that w(x, y, z) is normalized. b. i. What is the probability of finding the system in the state with energy -how? Answer: 0 ii. What is the probability of finding the system in the state with energy -ho ? Answer: 1 c. What is the expectation value of the energy? Answer: -hoo