Consider a three-dimensional (3d) isotropic harmonic oscillator described by the hamil- tonian (no hats on operators) h = p+ mw'ra, (p = p. + pă+ p2, p2 = x2 + y2 + 22). 2m 20 2 as implied by our discussion in class, its energy spectrum is a simple equidistant ladder: the eigenvalues of the hamiltonian (1) can be labeled as en = (n+3) hiw, n= 0, 1, hw, n= 0, 1, (2)
(a) explore the degeneracy of an arbitrary nth level in ladder (2) and find the analytical dependence dan), that is, how many linearly independent states d correspond to the same level n.
(b) when we were discussing 1d systems with symmetric potentials v(x) = v(-x), we realized that the stationary states y(2) can be conveniently described as either even (+(-x) = x(x)) or odd ) = - in 3d systems with spherically-symmetric potentials v(r) = v(r), we generalize this idea to the symmetry with respect to the inversion transformation: r → -r. we define parity p = +1 (even/odd) of state y(r) whenever y(-r) = py(r). find the parity of stationary states belonging to ladder (2). is each energy level n "composed" of states with the same parity p?