I'm slightly confused as to whether quantum (hermitian) operators, which we get by promoting observables to operators, are dimensionless or not?
Clearly the Hamiltonian of the system, say of the harmonic oscillator, has units of energy. Does the hamiltonian of the corresponding quantum mechanical system also have units of energy?
I suspect this is the case because only then would the procedure of making the Hamiltonian dimensionless by introducing creation and annihilation operators make sense, as done on: http://en.wikipedia.org/wiki/Creation_and_annihilation_operators, starting from the position representation of the SE, though. But the idea of operators having dimensions doesn't strike me intuitively.
It'd be great if someone could explain this.