14 votes
Accepted

Can the wavefunction be inferred from the expectation values of operators?

It actually suffices to know the expectation values of all projection operators of the form $P_\psi:=|\psi\rangle\langle \psi|$ for $\psi \in H$ (which are of course observables). Indeed, suppose we ...
8 votes
Accepted

Separable Hilbert space in quantum mechanics

Separability is not a necessary physical requirement, at least in modern approaches. First of all, all mathematical technology, as the spectral theory, is valid both for separable or non-separable ...
5 votes

Can the wavefunction be inferred from the expectation values of operators?

Intuitively, if you know the expectation values of all possible observables, that should be enough to fix the state of the system. This almost sounds tautological, since the state is just a ...
  • 6,111
4 votes
Accepted

Operators and periodic boundary conditions

Your intuition is right. The salient detail can be understood by studying a single particle on a ring, with Hilbert space $L^2(\mathrm S^1)$. The first thing to do is consider how we write down a ...
  • 55.5k
3 votes

Can the wavefunction be inferred from the expectation values of operators?

Just a generalization to Meng Cheng's answer and theoretically how you can construct $O_{mn}$ given known operators like $x$ and $p$. Given some Hermitian operator $H$ which we have full knowledge ...
  • 861
2 votes

Can the wavefunction be inferred from the expectation values of operators?

Can the wavefunction be inferred from expectation value of operators? The answer is yes. If we consider "the wavefunction" to be the ground state wavefunction then the first Hohenberg-Kohn ...
  • 9,375
2 votes

Non-distributivity of quantum logic according to C. Piron

The point is that if $\land$ and $\vee$ corresponded exactly to classical and and or, then they would be distributive simply because the classical operators are distributive. It does not follow from ...
  • 111k
2 votes

Square root of number operator for quantum harmonic oscillator

The square root exists and it is defined by standard functional calculus for every selfadjoint operator $A: D(A) \to H$ with non negative spectrum (which is equivalent to $\langle x, Ax\rangle \geq 0$...
2 votes
Accepted

Question on ordered exponential explanation in Wikipedia

OP is presumably missing that Wikipedia mentions that $\gamma$ is an infinitesimal rectangle. Each of the 4 contour integrals of the 4 sides of $OE[-J]$ are replaced to with the initial value of $J$ ...
  • 174k
1 vote

Zero frequency quantum oscillator rigorously

The spectrum of $p^2$ is fixed by the one of $p$ which, in turn, is fixed by the CCR and by the Stone-Von Neumann theorem. $p^2$ has continuous spectrum with no (proper) eigenvectors. There is no hope ...
1 vote

Non-distributivity of quantum logic according to C. Piron

First of all, nowadays, one learns it logic courses that "truth" is a complex notion, and in particular that "truth" per se has no real meaning; only "truth within a certain ...
  • 379
1 vote

Square root of number operator for quantum harmonic oscillator

Asking for such references may belong to the MSE, instead of this site. Most sensible, well-meaning, physicists work out the requisite relations in Fock space themselves, extending the humdrum ...

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