A particle of mass moves in a plane in the isotropic oscillator

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Quantum Mechanics · Symmetry, angular momentum and spin

The Structure of Quantum Mechanics, Ch Central potentials as a rotation problem

Problem

A particle of mass mm moves in a plane in the isotropic oscillator H^=ℏω(a^x†a^x+a^y†a^y+1)\hat H=\hbar\omega(\hat a_x^\dagger\hat a_x+\hat a_y^\dagger\hat a_y+1), two copies of Chapter 8's ladder algebra. (a) Show that the level with N=nx+nyN=n_x+n_y holds N+1N+1 states. (b) The only rotations in the plane are about zz, generated by L^z=x^p^y−y^p^x\hat L_z=\hat x\hat p_y-\hat y\hat p_x. Show that L^z=iℏ(a^y†a^x−a^x†a^y)\hat L_z=i\hbar(\hat a_y^\dagger\hat a_x-\hat a_x^\dagger\hat a_y), and explain why rotation symmetry by itself cannot account for an (N+1)(N+1)-fold degeneracy. (c) Define K^+=a^x†a^y\hat K_+=\hat a_x^\dagger\hat a_y, K^−=a^y†a^x\hat K_-=\hat a_y^\dagger\hat a_x and K^3=12(a^x†a^x−a^y†a^y)\hat K_3=\tfrac12(\hat a_x^\dagger\hat a_x-\hat a_y^\dagger\hat a_y). Show that they commute with H^\hat H and obey [K^3,K^±]=±K^±[\hat K_3,\hat K_\pm]=\pm\hat K_\pm and [K^+,K^−]=2K^3[\hat K_+,\hat K_-]=2\hat K_3. Identify the multiplet that level NN forms and the group behind it, and deduce the eigenvalues of L^z\hat L_z on level NN.

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