Question

Particle in a cylindrically symmetrical potential Let p, o, z be the cylindrical coordinates of a spinless 1. (x = ? coso, y = ? sin ?, p 0, 0 <p < 2?). Assume that the potential en of this particle depends only on , and not on ? and z. Recall that: a. Write, in c ylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with L, and P. Show fr the wave functions chosen in the form: associated with the stationary states of the particle can be where the values that can be taken on by the indices m and k are to be specified. b. Write, in cylindrical coordinates, the eigenvalue equation of the Hamiltonian H of the particle. Derive from it the differential equation which yields c. Let 2, be the operator whose action, in the r representation, is to change y to-y (reflection with respect to the xOz plane). Does ?, commute with H? Show that ?y anticommutes with Lz, and show from this that ?! ??.mk>isan ue? What can be concluded ld this result eigenvector of L,. What is the corresponding eigenvalue? Wha concerning the degeneracy of the energy levels of the partice be predicted directly from the differential equation established in (b)?

Answer 1

Let p, phi, z be the cylindrical coordinates of a spinless particle (x = rho cos phi, y = rho sin phi: p greaterthanorequalto 0, 0 lessthanorequalto phi < 2Pi Recall that: partial differential^2/partial differential x^2 + partial differential^2/partial differential y^2 = partial differential^2/partial differential rho^2 + 1/rho partial differential/partial differential rho + 1/rho^2 partial differential^2/partial differential phi^2 (a) The Hamiltonian in cartesian system can be written as H = -h^2/2m [partial differential/partial differential x^2 + partial differential/partial differential y^2 + partial differential/partial differential z^2] + v(x, y, z) In cylindrical coordinate system we know partial differential/partial differential x^2 + partial differential/partial differential y^2 = partial differential/partial differential rho^2 + 1/rho partial differential/partial differential rho + 1/rho^2 partial differential^2/partial differential phi^2 Therefore H = -h^2/2m [partial differential^2/partial differential rho^2 + 1/rho partial differential/partial differential rho + 1/rho^2 partial differential^2/partial differential phi^2 + partial differential^2/partial differential z^2] + v(p) It is said that v is a function of p. i.e. x f y and is of z state rho = x^2 + y^2 Now L_z = -ih partial differential/partial differential theta & rho_2 = -ik partial differential/partial differential z Therefore [H, iz] = H L_z - L_z H operating this on a function F(rho, phi, z) we get \r\n (H L_z - L_z H) f = H(L_z f) - L_z(H F)