Lie symmetries of Schrodinger equation on a sphere

Abstract

‎We are going to discuss an object with a mass attached to a spring and vibrating on the surface of a sphere (see Figure‎ 1). ‎To do this‎, ‎we first reconstruct the Schrodinger equation on a sphere‎. ‎In fact‎, ‎the paper considers the question of a quantum system obeying the Schrodinger equation on a sphere‎. ‎After a brief introduction we set up the Hamiltonian of the system and the corresponding Schrodinger equation‎. ‎We provide the Lie algebra of symmetries and build‎ ‎the optimal system of Lie subalgebras and its adjoint presentation‎. ‎Reductions of similarities related to subalgebras are obtained which are used in our study of the‎ 3‎D quantum harmonic oscillator on a sphere as a special case of the new equation‎, ‎and possible solutions are proposed‎.