Harmonic Oscillator (HO)

Consider a particle described by the Cartesian coordinates (x; y; z) _ r and their conjugate momenta (px; py; pz) ? p. The classical definition of the orbital angular momentum of such a particle about the origin is L = r × p , giving

Lx = y pz - z py (1)
Ly = z px - x pz (2)
Lz = x py - y px (3)


Let us assume that the operators (Lx; Ly; Lz) ? L which represent the components of orbital angular momentum in quantum mechanics can be defined in an analogous manner to the corresponding components of classical angular momentum. In other words, we are going to assume that the above equations specify the angular momentum operators in terms of the position and linear momentum operators. Note that Lx, Ly, and Lz are Hermitian, so they represent things which can, in principle, be measured. Note, also, that there is no ambiguity regarding the order in which operators appear in products on the right-hand sides Eqs. (1).(3), since all of the products consist of operators which commute.