Editing Uncertainty principle (section)

===Limitations===
The derivation of the Robertson inequality for operators <math>\hat{A}</math> and <math>\hat{B}</math> requires <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> to be defined. There are quantum systems where these conditions are not valid.<ref>{{Cite journal |last=Davidson |first=Ernest R. |date=1965-02-15 |title=On Derivations of the Uncertainty Principle |url=https://pubs.aip.org/jcp/article/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle |journal=The Journal of Chemical Physics |language=en |volume=42 |issue=4 |pages=1461–1462 |doi=10.1063/1.1696139 |bibcode=1965JChPh..42.1461D |issn=0021-9606 |access-date=2024-01-20 |archive-date=2024-02-23 |archive-url=https://web.archive.org/web/20240223160247/https://pubs.aip.org/aip/jcp/article-abstract/42/4/1461/208937/On-Derivations-of-the-Uncertainty-Principle?redirectedFrom=fulltext |url-status=live }}</ref>
One example is a quantum [[particle in a ring|particle on a ring]], where the wave function depends on an angular variable <math>\theta</math> in the interval <math>[0,2\pi]</math>. Define "position" and "momentum" operators <math>\hat{A}</math> and <math>\hat{B}</math> by
<math display="block">\hat{A}\psi(\theta)=\theta\psi(\theta),\quad \theta\in [0,2\pi],</math>
and
<math display="block">\hat{B}\psi=-i\hbar\frac{d\psi}{d\theta},</math>
with periodic boundary conditions on <math>\hat{B}</math>. The definition of <math>\hat{A}</math> depends the <math>\theta</math> range from 0 to <math>2\pi</math>. These operators satisfy the usual commutation relations for position and momentum operators, <math>[\hat{A},\hat{B}]=i\hbar</math>. More precisely, <math>\hat{A}\hat{B}\psi-\hat{B}\hat{A}\psi=i\hbar\psi</math> whenever both <math>\hat{A}\hat{B}\psi</math> and <math>\hat{B}\hat{A}\psi</math> are defined, and the space of such <math>\psi</math> is a dense subspace of the quantum Hilbert space.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | page = 245 | bibcode = 2013qtm..book.....H }}</ref>

Now let <math>\psi</math> be any of the eigenstates of <math>\hat{B}</math>, which are given by <math>\psi(\theta)=e^{2\pi in\theta}</math>. These states are normalizable, unlike the eigenstates of the momentum operator on the line. Also the operator <math>\hat{A}</math> is bounded, since <math>\theta</math> ranges over a bounded interval. Thus, in the state <math>\psi</math>, the uncertainty of <math>B</math> is zero and the uncertainty of <math>A</math> is finite, so that
<math display="block">\sigma_A\sigma_B=0.</math>
The Robertson uncertainty principle does not apply in this case: <math>\psi</math> is not in the domain of the operator <math>\hat{B}\hat{A}</math>, since multiplication by <math>\theta</math> disrupts the periodic boundary conditions imposed on <math>\hat{B}</math>.<ref name="Hall2013">{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 245 | bibcode = 2013qtm..book.....H }}</ref>

For the usual position and momentum operators <math>\hat{X}</math> and <math>\hat{P}</math> on the real line, no such counterexamples can occur. As long as <math>\sigma_x</math> and <math>\sigma_p</math> are defined in the state <math>\psi</math>, the Heisenberg uncertainty principle holds, even if <math>\psi</math> fails to be in the domain of <math>\hat{X}\hat{P}</math> or of <math>\hat{P}\hat{X}</math>.<ref>{{Citation | last = Hall | first = B. C. | title = Quantum Theory for Mathematicians | publisher = Springer | year = 2013 | pages = 246 | bibcode = 2013qtm..book.....H }}</ref>