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I have some books on quantum mechanics. These explains that states of a system are unit vectors and observables are represented by self-adjoint linear operator on a Hilbert space, etc ... but don't explain how to construct these Hilbert spaces and operators which satisfies expected axioms such as $[\hat{q}, \hat{p}]=i\hbar.$

How do you construct them?

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  • $\begingroup$ The usual Hilbert space is $L^2(\Bbb R)$, with $q$ the operator that sends $\psi(x)$ to $x\psi(x)$, and $p$ is the operator $-i\hbar d/dx$. It's then easy to verify that $[q,p]=i\hbar$ on the common domain of $qp$ and $pq$, but hard to check that it works on the common domain of $p$ and $q$. $\endgroup$
    – Ryan Unger
    Commented Feb 21, 2017 at 2:43
  • $\begingroup$ @user53216 For $[{\hat q}, {\hat p}] = i\hbar$ domain issues may be bypassed once momentum eigenfunctions in position representation are known (or postulated) to be plane waves, see physics.stackexchange.com/questions/223633/…. Regarding Hilbert space construction this answer may help, although it concerns a slightly different question: physics.stackexchange.com/questions/204094/… $\endgroup$
    – udrv
    Commented Feb 21, 2017 at 6:49
  • $\begingroup$ See the Stone-von Neumann theorem. $\endgroup$
    – Qmechanic ♦
    Commented Feb 21, 2017 at 9:35

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