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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?

  • 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 $q$. – Ryan Unger Feb 21 '17 at 02:43
  • @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 http://physics.stackexchange.com/questions/223633/derivation-of-canonical-position-momentum-commutator-relation/223682#223682. Regarding Hilbert space construction this answer may help, although it concerns a slightly different question: http://physics.stackexchange.com/questions/204094/wavefunctions-in-different-hilbert-spaces/205157#205157 – udrv Feb 21 '17 at 06:49
  • See the Stone-von Neumann theorem. – Qmechanic Feb 21 '17 at 09:35

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