What does it mean to "quantise" a system?

Question

Suppose we have a physical system, let's say a ring of $N$ atoms held together by elastic force. (This is just an example, we could have picked any physical system)

Classically we can easily find the Lagrangian of this system, and then, using the Euler-Lagrange equations, we can find the equations of motion $q_i(t)$ for the atoms. Once we have the canonical coordinates $q_i(t)$ we can also find the conjugated momenta $p_i(t)$ and then, via Legendre transform, finally find the Hamiltonian of the system.

Point is: looking at the physical system classically we can easily derive anything we want about it (Lagrangian, canonical coordinates, ecc.). However this is clearly not a quantum mechanical description of our system.

In my lecture notes this problem is fixed by a procedure denominated quantisation, it works as follows:
We take our classical description of the system, classically derived, and we perform two steps onto it:

• Step 1: we promote the coordinates $q_i,p_i$ to operators $\hat{q}_i,\hat{p}_i$
• Step 2: we impose the following commutations relation: $$[\hat{q}_i,\hat{p}_j]=i\hbar \delta _{i,j} \ \ , \ \ [\hat{q}_i, \hat{q}_j]=0 \ \ , \ \ [\hat{p}_i, \hat{p}_j]=0 \tag{1}$$

And we should get a perfectly good quantum description of our physical system (in this case of the atom chain).

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This procedure does not convince me, I have several questions about it:

• Coordinates in classical mechanics and observables in QM (represented via hermitian operators) are two vastly different things. What does it mean to just promote one into the other?
• Why should impose the canonical commutation relation (1) automatically transport the system from classically described to quantistically described?
• At last: we have performed our derivations on the system (about the Lagrangian, canonical coordinates, momenta, ecc.) classically at first, and only after we tried to make the description of it quantum mechanical. How can we rely on our classical description? What ensures us that this procedure (quantisation), starting from a classical description, will produce an accurate quantum description of the system?

Answer 1

The goal of quantization is to produce a quantum mechanical system that reproduces a known classical limit. The rules you are describing are a (very successful) way to guess what quantum mechanical equations we should use to represent a system.

Strictly logically speaking, we should simply write down some operators describing the observables of our system, define the commutation relations between them, and propose a Hamiltonian that describes the dynamics. Then, we can check the predictions of this model against experiments. If we want to know how the system behaves classically, there are various methods to take a classical limit of a quantum system.

However, often in practice, we build our intuition from classical mechanics. For example, we are very familiar with positions and momenta and the Kepler problem in classical mechanics, and we use this intuition when devising the observables for the Hamiltonian of an electron in the Hydrogen atom. Quantization is a method to let us use this intuition to guide us in defining the quantum system.

Answer 2

It is assumed you have appreciated Todorov's 2012 classic overview "Quantization is a Mystery" ( “but second quantization is a functor”).

• "Promote" indicates the "first quantization" step of one oscillator to the one-to-many map from classical entities to operators. This is the "mystery part" of the Nelson quote. There is quite a bit of freedom in this promotion, as you introduce ℏ-dependent information by sheer guesswork. But after 100 years of QM, people have agreed on the recipes that work, at least for the oscillator. The nontrivial commutation relations lead to the discrete spectrum and generic quantum behavior of the oscillator, as covered in elementary QM texts.

• Your second step, however, is completely predictable, one-to-one: a functor. That is, once you have understood that a classical continuous system is but an infinity of repackaged formal classical oscillators (normal modes), then you don't do anything drastic in quantizing all of these oscillators exactly the way you quantized one, in the mystery part. You just keep track of indices, symmetries, etc, in putting together the pieces in the repackaging. The "second quantization" map is banal (unambiguous), and all the complication is resolvable, even though it confuses novices--probably because they complacently gave short shrift to the breathtakingly beautiful details of the repackaging.

Remember, the oscillators are formal objects, having little to do with positions and momenta for the oscillator you learn about in quantum mechanics: they live in a mathematical, notional space (this is the part most students shrug off, in my experience, with devastating conceptual consequences).

It appears to me you are mostly confused by the shorthand language in field quantization; but once you appreciate the repackaging, completely predictable and non-mysterious in the classical case, quantization eventuates in a blink...