Question

3. Particle in a cylindrically symmetrical potential: Let pw. be the cylindrical coordinates of a spinless particle (z = pcos y, y psiny: P 20, OS <2m). Assume that the potential energy of this particle depends only one, and not on yor: Vin-V ). Recall that & P R 1 18 dr2 + dy? - apa pap + 2 day? (a) Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian in the representation. Show that H commutes with L, and P. Show from this that the wave functions of eigenstates of the Hamiltonian can be chosen to be of the form Vn.m. .. :) = ...(pemeka where you are to specify that possible values that m and k can take (b) Write, in cylindrical coordinates, the eigenvalue equation for the Hamiltonian H. Derive from it the differential equation which yields fnm(p). (c) Let , be the operator whose action, in the 7) representation, is to change y to -y (it reflects the system in the plane). Does , commute with H? Show that E, anticommutes with L ( EL, + L2, 0), and show from this that E .m.) is an eigenvector of L. What is the corresponding eigenvalue? What can be concluded about the degeneracy of the energy levels of the particle? Could this result be predicted directly from the differential equation derived in part (b)?

particle in a cylindrically symmetric potential: do only C please

Answer 1

C lefs : mhY(x,y,z) Lz 4(z,y,z) say H Y( *>y, z) E P(2,y, 2) has oylindrical agminehry The System p(x,-y, 2) -> YCz,y, z) HY = E4 E Plx, - y, z) = E 4C2,4, z) # ZHY(x,y, z) : Eɛ Y(x,y, 2) = -0 EgH (xj),z). Ycx,-y, z) H4 (2,-y,2) = Ep(23y, z) [,, 4]=0 Lz# xPy -ypc Eylz ih EybY: - ih [* -]vcy,2) -lz 4(2sy,z) -lz P(2)-y,z) Eylz4 - -lz Zy (3, y, z) Zylz t LzEy=0 {=4,} =0 Zy 4(2,y, z) = Y(2,-4,2) = 4(2,y,2) € eigenuector of Lz Coresponding eigen value is 1 s { Frg} due ** Symmety ,any State wim same -the will be same , ie there will be no energy cost to aake 4(x2, Y2sz) Ciren aitg - a3+ ys] state from 4(Z,g4i,Z) to degerevacy = oo %3D