Transition: Classical Mechanics to Quantum Mechanics

[go quantum!]

Imagine that I have a system that is described classically by a given Hamiltonian which is a function of a given set of parameters $$q$$ and their canonical conjugate momenta $$p=\frac{\partial L}{\partial \dot{q}}$$.
Then, I will say that the quantum description of the same system is guided by setting the commutator [ tex ] [q_a,p_a]=i [ /tex ] because the Poisson bracket is [tex ]{q_a,p_a}=1[ /tex ].

This step is crucial and it is the cornerstone of the process of quantizing. I would like to ask if you know some motivations for this step. Do you understand it?

Thanks for you help!

 

[Bill_K]

The motivation for replacing Poisson brackets by commutators is that it works for simple one-particle systems in Cartesian coordinates - {x, p} → [x, p].

Note, that this replacement is not the end of the story. Now that the Hamiltonian is given in terms of operators, you have to decide a consistent ordering for those operators.

 

[go quantum!]

Is that the only motivation? It can't be...