Quantizing Velocity: Exploring the Celerity Operator in Lagrangian Mechanics

[eljose]

Let be the Lagrangian of a particle:

$$L(q,\dot q,t)$$ my question is if we can get its quantizied version in the form:

$$p\dot q-L(q,\dot q,t)| \Psi>=E_{n}|\Psi >$$

of course we know how to quantizy the Momentum operator the question is..¿how do we quantizy the "celerity" $$\dot q$$ operator acting over an state?..thanks.

 

[masudr]

You can't write the Hamiltonian like that and still use the $\dot{q}$. Just as in classical Hamiltonian mechanics, you have to eliminate $\dot{q}$ in favour of $p = \partial L / \partial \dot{q},$ and so you shouldn't have any "generalized velocity" (which is what you have called celerity) quantities appearing in the expression.

Just to be picky as well, I would ask that you put brackets around your operator acting on $|\psi\rangle$ on the left-hand-side.

Alternatively, you might want to eliminate the $p$ that you have put on the left-hand-side with the relation between $p$ and $\dot{q}$ given by $p = \partial L / \partial \dot{q}.$ In this case, I guess the operator $\dot{q}$ would be given by $\hat{\dot{q}} =d \hat{x}/dt.$

 

[selfAdjoint]

I was just reading the perspective on Julian Schwinger in the History of Physics section of the arxiv. It mentioned his papers on the quantum action principle in fully relativistic form. Apparently he was concerned to have a way to define relativistic QM without starting with a classical theory, quantizing it, and then throwing it away.

Has anybody looked at these papers? Can anybody comment on the methodology?

 

[eljose]

The problem with Hamiltonian Quantization is that in several theories you have $$H\Psi= 0$$ so there wouldn,t be any time evolution of the system..

"self-adjoint" could you give the references to the arxiv papers about Julian Schwinger?..i would like to take a look at them.

 

[masudr]

Well if H = 0, then by definition

$$\sum_{i=1}^N p_i\dot{q}_i - L(\vec{q}, \dot{\vec{q}}, t) = 0$$

so you will get the same problem, or have I missed something?

 

[selfAdjoint]

"self-adjoint" could you give the references to the arxiv papers about Julian Schwinger?..i would like to take a look at them.

The biography paper is at http://xxx.lanl.gov/PS_cache/physics/pdf/0606/0606153.pdf. Schwinger's own papers aren't on the arxiv; they were published in journals of the American Physical Society or Proceedings of the National Academy of Science, and those organizations are still charging for the online versions. Almost worth joining the APS ($109) to get to those great old works.