Let p, p, z be the cylindrical coordinates of a spinless particle (x = p cos...

 

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Let p, p, z be the cylindrical coordinates of a spinless particle (x = p cos , y = p sin ; p≥ 0, 0≤ < 27). Assume that the potential energy of this particle depends only on p, and not on and z. Recall that: 8² 8² + 8² 1 ə 1 8² + + əx² Əy² др2 рдр p² 26² a. Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with Lz and Pz. Show that this allows writing the wave functions associated with the stationary states of the particle as: Pn,m,k (P, 4, z) = fnm (p) imç ikz 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 that fnm (p) obeys. c. Let Ey be the operator whose action, in the {|r)} representation, is to change y to -y (reflection with respect to the xOz plane). Does Ey commute with H? Show that Σy anticommutes with Lz, and show that, as a result, Ey|n,m,k) is an eigenvector of L. What is the corresponding eigenvalue? What can be concluded concerning the degeneracy of the energy levels of the particle? Could this result be predicted directly from the differential equation established in (b)? Let p, p, z be the cylindrical coordinates of a spinless particle (x = p cos , y = p sin ; p≥ 0, 0≤ < 27). Assume that the potential energy of this particle depends only on p, and not on and z. Recall that: 8² 8² + 8² 1 ə 1 8² + + əx² Əy² др2 рдр p² 26² a. Write, in cylindrical coordinates, the differential operator associated with the Hamiltonian. Show that H commutes with Lz and Pz. Show that this allows writing the wave functions associated with the stationary states of the particle as: Pn,m,k (P, 4, z) = fnm (p) imç ikz 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 that fnm (p) obeys. c. Let Ey be the operator whose action, in the {|r)} representation, is to change y to -y (reflection with respect to the xOz plane). Does Ey commute with H? Show that Σy anticommutes with Lz, and show that, as a result, Ey|n,m,k) is an eigenvector of L. What is the corresponding eigenvalue? What can be concluded concerning the degeneracy of the energy levels of the particle? Could this result be predicted directly from the differential equation established in (b)?