We know that the canonical position-momentum commutator relation is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (in Dirac notation) from

1. Closure relations (both momentum and position bases).

2. Orthonormality relations for both bases.

3. The assumption that momentum eigenstates are plane waves in the position representation?

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (in Dirac notation) from

1. Closure relations $ \int|x\rangle \, \langle x| dx $ (both momentum and position bases)

2. $ \left\langle \left.x'\right|x\right\rangle = \delta \left(x-x'\right) $ Orthonormality relations for both bases

3. $ \left\langle \left.x\right|p\right\rangle = e^{\text{ipx}} $ the assumption that momentum eigenstates are plane waves in the position representation

We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (in Dirac notation) from 1) Closure relations (both momentum and position bases) 2) Orthonormality relations for both bases 3) the assumption that momentum eigenstates are plane waves in the position representation?

We know that the canonical position-momentum commutator relation is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (in Dirac notation) from

1. Closure relations (both momentum and position bases).

2. Orthonormality relations for both bases.

3. The assumption that momentum eigenstates are plane waves in the position representation?