Lie Groups and Quantum Mechanics

Abstract:
Mathematical modeling should present a consistent description of physical phenomena. We illustrate an inconsistency with two Hamiltonians - the standard Hamiltonian and an example found in Goldstein - for the simple harmonic oscillator and its quantisation. Both descriptions are rich in Lie point symmetries and so one can calculate many Jacobi Last Multipliers. The Last Multiplier provides the route to the resolution of this problem and indicates that the great debate about the quantisation of dissipative systems should never have occurred.

[1] Nucci MC & Leach PGL (2007) Gauge Variant Symmetries for the Schrödinger Equation, Communications in Mathematical Physics (submitted)
[2] Nucci MC & Leach PGL (2007) Lagrangians Galore, Journal of Mathematical Physics (submitted)