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We have:

$$i\hbar\frac{d |\Psi(t)\rangle}{d t} = H|\Psi(t)\rangle$$

$|\Psi\rangle$ is an element of the Hilbert space.

However, the Hilbert space is unspecified.

As an analogy, in classical mechanics, the configuration space will be different for a system of 2 particles versus a system of 3 particles.

So, what are minimal things we need to do to specify the background space of a system in QM? From e.g. Ballentine this is what I came up with:

  1. Make a list of dynamical variables (this is where the physics comes in) and replace those variables by Hermitian operators and their actions on the Hilbert space. Do we need an irreducible set? Specify commutation relations.

  2. Appeal to symmetry principles? Do we need this? It helps to specify the commutation relations and the form of the operators. This is where you can make use of unitary operators corresponding to displacements, rotations etc.

  3. Write down the Hamiltonian using the operators listed in step 1.

Now given boundary conditions you can solve for the state vector.

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    $\begingroup$ In (1) I think you mean Hermitian operators, not unitary. $\endgroup$
    – jacob1729
    Jun 2, 2019 at 17:50
  • $\begingroup$ Right. Unitary is another requirement for symmetry transformations (displacement, time translation, rotations). Unitary operators can be used to determine hermitian operators such as momentum and position. That would step 2) $\endgroup$
    – yalis
    Jun 2, 2019 at 18:05
  • $\begingroup$ Although it does seem as if the hermitian constraint is not a strict requirement: en.wikipedia.org/wiki/Non-Hermitian_quantum_mechanics $\endgroup$
    – yalis
    Jun 2, 2019 at 18:20
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    $\begingroup$ Defining algebraic relations between observables (and a technical detail such as "the spectrum of the Hamiltonian is bounded from below") is enough to fully specify the theory. Think of the algebraic approach to the 1D harmonic oscillator. $\endgroup$
    – DanielC
    Jun 2, 2019 at 19:40

1 Answer 1

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First, a clarification. Based on the text in the OP, the question isn't really about the Hilbert space. It's really about the observables. I'll start by explaining what I mean by this.

All complex infinite-dimensional separable Hilbert spaces are the same (isomorphic) as far as their Hilbert-space structure is concerned. For example, when doing single-particle QM in $1$-dimensional space, we typically represent elements of the Hilbert space using using complex-valued functions $\psi(x)$ of a single real variable $x$. When doing single-particle QM in $3$-dimensional space, we typically represent elements of the Hilbert space using using complex-valued functions $\psi(x,y,z)$ of three real variables $x,y,z$. These are the same Hilbert space, just presented differently. They both have a countable orthonormal basis, and we can map from one such basis to another using a unitary transformation. The two Hilbert spaces are unitarily equivalent as abstract Hilbert spaces, even though they are described using functions of different numbers of variables.

The difference between these two models is in the observables. The same Hilbert space can be presented in different ways to simplify the implementation of the observables in different models, but the distinctions are in the observables, not in the Hilbert space. The unitary transformation described above preserves the abstract Hilbert-space structure, but it completely scrambles the observables.

With that clarification in mind, we can separate this into two sub-questions:

  • Sub-question A: What physical observables are needed in order to describe the given system?

  • Sub-question B: What mathematical operators should be used to represent those physical observables?

Let's assume that sub-question A has already been answered, at least loosely. In other words, let's assume that you already have a pretty good idea of which physical system you want the model to describe.

The most widely used approach to answering sub-question B is quantization. This is what points $1$ and $3$ in the OP are about. In this approach, the loose initial answer to sub-question A is based on a "classical" model. This "classical" model isn't necessarily a good approximation to the system's dynamics (although sometimes it is, especially in non-relativistic few-particle systems). The word "classical" here simply means "involving only mutually commuting dynamical variables." (To accommodate fermions, this can be generalized to allow anticommuting variables. It's still "classical.") This is where point $2$ in the OP comes into play: symmetry, such as Lorentz symmetry, is often one of the most important guiding principles we have in choosing a good initial "classical" model.

The goal of quantization is to promote these "classical" dynamical variables to non-commuting operators in such a way that retains, as much as possible, the same equations of motion (in the Heisenberg picture). The dynamical variables may or may not be observables themselves, but observables are expressed in terms of them. The trick is to choose commutation relations that are consistent with the equations of motion.

Do we need an irreducible set?

Actually, using a redundant set of dynamical variables is okay, and in fact this is often easier than trying to eliminate all of the redundancies. How to make quantization work properly in the presence of redundant dynamical variables is a huge topic about which entire books have been written, like [1] and much of [2]. With or without redundant variables, the goal to end up with a clear mapping from physical observables to non-commuting operators on a Hilbert space, a mapping that is at least mostly consistent with the "classical" equations of motion that we started with. Perfect consistency is sometimes impossible: this is the subject of anomalies, which is a huge subset of the subject of quantization.


References to illustrate the depth of the subject called "quantization":

[1] Henneaux and Teitelboim (1992), Quantization of Gauge Systems, Princeton University Press

[2] DeWitt (2003), The Global Approach to Quantum Field Theory (two volumes), Oxford University Press

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