Order of product of $x$ and $p$ while deriving Hamiltonian from a Lagrangian in Quantum Mechanics

Question

Everyone who has taken a course in Quantum Mechanics has at some point derived a quantum Hamiltonian from a Lagrangian. However, I can't seem to find any reference on the topic. My question is regarding the product $p_i x_i$ in the Legendre transform: $$H = \sum_i p_i \dot x_i - KE+ PE.\tag{1}$$ Is this the correct order or does the order not matter? At what point does one make the upgrade from classical variables to quantum variables and introduce the commutation relations?

Answer 1

Normally, one quantizes a theory in the following order:

$$\begin{align} \text{classical}& \text{ Lagrangian} \cr \downarrow&\text{ Legendre transformation} \cr \text{classical}& \text{ Hamiltonian}\cr \downarrow&\text{ quantization} \cr \text{quantum}& \text{ Hamiltonian}\end{align}\tag{I}$$

see e.g. this, this & this Phys.SE posts.

However, OP apparently wants to quantize in the opposite order:

$$\begin{align} \text{classical}& \text{ Lagrangian} \cr \downarrow&\text{ quantization ??} \cr \text{quantum}& \text{ Lagrangian ??}\cr \downarrow&\text{ quantum Legendre transformation ??} \cr \text{quantum}& \text{ Hamiltonian}\end{align}\tag{II}$$

There are several issues with OP's proposal (II) before one could address OP's question about a quantum Legendre transformation: E.g. how do we define CCR using only Lagrangian variables $q$ and $v$? Of course, one could in principle work backwards using the conventional method (I).

OP's eq. (1) has related issues, as OP already seems to be aware of: How should one order the $p\dot{q}$ term to ensure that the quantum Hamiltonian is Hermitian? It seems easier to just use the conventional quantization scheme (I).