Consider a particle of mass mu (do not use m, otherwise will confuse with a quan

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Consider a particle of mass mu (do not use m, otherwise will confuse with a quantum number in this problem) moving in the xy plane under a two-dimensional central potential V(r). The coordinates of the particle is expressed as (r, theta) as shown. The Hamiltonian of the particle is H = h^2/2 mu (partial differential^2/partial differential r^2 + 1/r partial differential/partial differential r + 1/r^2 partial differential^2/partial differential theta^2) + V(r). Show that the wavefunction psi/(r, theta) can be written as psi(r, theta) = R(r)e^plusminus im theta, where R(r) is independent of theta. Explain why m cannot be an imaginery number. In the above notation, m can be chosen to be non-negative. Explain what values of m are allowed. The two-dimensional angular momentum operator is defined as L = -ih partial differential/partial differential theta. Do H and L have simultaneous eigenfunctions? If yes, what are they? If no, explain why. Suppose the particle is in a two-dimensional infinite circular well, V(r) = {0, r a. Show that the radial equation is -h^2/2 mu [(d^2/dr^2 + 1/r d/dr) - m^2/r^2] R = ER. You are given that eigenvalues and eigenfunctions of the radial equation (21) are E_nm = h^2 x_nm^2/2 mu a^2 and R_nm (r), n = 0,1, 2, 3,.... The values of x_nm are shown below: Show that each energy level E_nm is two-fold degenerate except when m = 0. Note that you have to find two orthogonal wavefunctions having the same energy. What is the degeneracy of the m = 0 energy level? We use the notation s, p, d, f, ... to denote states with angular momentum quantum number m = 0, 1, 2, 3, ... respectively. What are the five lowest energy states in this notation (for example, a state with n = 0 and m = 0 should be denoted as 0s)? List them in increasing order of energy. Suppose we put two identical non-interacting spinless (S = 0) particles in this circular well, what are the energies and normalized total wavefunctions Psi(r_1, theta_1, r_2, theta_2) in the ground state (lowest energy) and first excited state (next lowest energy) respectively. You should express the total wavefunctions in terms of R_nm. Repeat part h) if the two particles are electrons, again assuming that they are non-interacting, i.e., ignore electron-electron repulsion. Remember that since the spin is nonzero, wavefunctions have both orbit and spin parts. Discuss how electron-electron repulsion splits the first excited state in part i). Give a qualitative reason why one has higher energy than the other. No mathematical expression or proof is necessary. Now consider a particle in a three-dimensional infinite cylindrical well, i.e., it is inside a cylinder which is infinitely long along the z direction and has radius a in the x-y plane. Find its energies and wavefunctions Psi{r, theta, z) using your knowledge of the two-dimensional infinite circular well above. Give conditions and allowed values for any variable or quantum number you introduce. Are the energy levels discrete or continuous? Are the wavefunctions normalizable?

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