Canonical Commutation Relations

Question

Is it logically sound to accept the canonical commutation relation (CCR)

$$[x,p]~=~i\hbar$$

as a postulate of quantum mechanics? Or is it more correct to derive it given some form for $p$ in the position basis?

I understand QM formalism works, it's just that I sometimes end up thinking in circles when I try to see where the postulates are.

Could someone give me a clear and logical account of what should be taken as a postulate in this regard, and an explanation as to why their viewpoint is the most right, in some sense!

Answer 1

Your running into circles will stop once you commit yourself to a choice.

What to regard as postulate is always a matter of choice (by you or by whoever writes an exposition of the basics). One starts from a point where the development is in some sense simplest. And one may motivate the postulates by analogies or whatever. The CCR are a simple coordinate-independent starting point.

However it is more sensible to introduce the momentum as the infinitesimal generator of a translation in position space. This is its fundamental meaning and essential for Noether's theorem, and has the CCR as a simple corollary.

Answer 2

You can either accept it as a postulate (in which case it is often more convenient to postulate the CCR and CAR for creation and annihilation operators) or you can derive the relation in the position basis with

$$ \hat x = x \wedge \hat p = -i \hbar \nabla \Rightarrow [ \hat x , \hat p ] = - i \hbar x \nabla + i \hbar + i \hbar x \nabla $$

as you have to take the product rule when you apply $\nabla x$ to a function $f$.

You could also get these by the equivalence principle with classical mechanics, which says that $\{ q , p \} = 1$ for the Poisson brackets $\{\cdot,\cdot\}$ which are related to the commutator by a factor of $i \hbar$. That this equivalence principle holds is visible for example in the Ehrenfest theorem.

Answer 3

The choice of postulates is somewhat arbitrary in the sense that given a set of postulates you almost always can find an alternative set. The choice is guided by subjective criteria such as simplicity, closeness to experiment, or theoretical elegance.

However there are situations where some postulates/theorems do not make sense. For instance, $[\hat{x},\hat{p}] = i\hbar$ makes no sense in the Wigner & Moyal formulation of quantum mechanics, neither as postulate nor as theorem, because this formulation of quantum mechanics does not use operators:

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space.

Although the phase space formulation of quantum mechanics does not use commutation relations, them can be still obtained as a theorem when one makes the transition from the general phase space state to the configuration space wavefunction: $W(p,x;t) \rightarrow \Psi(x;t)$. Precisely, an explicit derivation of the $[\hat{x},\hat{p}] = i\hbar$ is given in my paper Positive definite phase space quantum mechanics