Qm Uncertainty And Compatibility — Examples

Adopt the state and composition postulate P1, which assigns the ray of a unit vector as a pure preparation, the observable and probability postulate P2, and the massive-particle model postulate P5 read through The massive spin-s particle model of the framework: state space, operators, units and frame readings for a spinless particle ($s=0$, $r=1$) of mass parameter $m>0$. Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)), as required by the spectral probability and moment suppliers used below. Fix a width $\sigma >0$ and define ${\psi }_{\sigma }(x):=(\pi {\sigma }^2{)}^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(2{\sigma }^2)},x\in {\mathbb{R}}^3,$ with momentum width ${\sigma }_p:=\mathrm{\hslash }\mathrm{/}\sigma$ and momentum representative $U{\psi }_{\sigma }$ under the model's momentum transform $U$. Then:

(i) ${\psi }_{\sigma }\in \mathcal{S}({\mathbb{R}}^3;\mathbb{C})$ is a unit vector, so P1 assigns its ray $[{\psi }_{\sigma }]$ as a pure preparation; ${\psi }_{\sigma }\in D(Q_j)\cap D(P_k)$ for all $j,k$, and $P_j{\psi }_{\sigma }=i\mathrm{\hslash }{\sigma }^{-2}{x}_j{\psi }_{\sigma }$;

(ii) the position outcome law of $Q_j$ has density $(\pi {\sigma }^2{)}^{-1\mathrm{/}2}{e}^{-t^2\mathrm{/}{\sigma }^2}$ in the coordinate $t=x_j$, with mean $0$ and variance ${\sigma }^2\mathrm{/}2$, so ${\mathrm{\Delta }}_{{\psi }_{\sigma }}{Q}_j=\sigma \mathrm{/}\sqrt{2}$;

(iii) $U{\psi }_{\sigma }={\psi }_{{\sigma }_p}$, that is $(U{\psi }_{\sigma })(p)=(\pi {\sigma }_p^2{)}^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2{\sigma }_p^2)};$ the momentum outcome law of $P_k$ has density $(\pi {\sigma }_p^2{)}^{-1\mathrm{/}2}{e}^{-p_k^2\mathrm{/}{\sigma }_p^2}$, with mean $0$ and variance ${\sigma }_p^2\mathrm{/}2={\mathrm{\hslash }}^2\mathrm{/}(2{\sigma }^2)$, so ${\mathrm{\Delta }}_{{\psi }_{\sigma }}{P}_k=\mathrm{\hslash }\mathrm{/}(\sigma \sqrt{2})$;

(iv) for every $j$, ${\mathrm{\Delta }}_{{\psi }_{\sigma }}{Q}_j{\mathrm{\Delta }}_{{\psi }_{\sigma }}{P}_j=\frac{\mathrm{\hslash }}{2},$ and the centered vectors are linearly dependent with vanishing real part of their pairing: $Q_j{\psi }_{\sigma }=-\frac{i{\sigma }^2}{\mathrm{\hslash }}{P}_j{\psi }_{\sigma },\mathrm{Re}⟨Q_j{\psi }_{\sigma },P_j{\psi }_{\sigma }⟩=0,$ so the equality case of The position-momentum preparation relation of the massive spin-s model clause (c) is realized.

The computation is one family of preparations in the model; it is not a characterization of all preparations attaining $\mathrm{\hslash }\mathrm{/}2$, involves no dynamics, no detection model and no empirical value.

• No converse is claimed. The verification shows that the Gaussian family attains the bound and satisfies the equality conditions of The position-momentum preparation relation of the massive spin-s model; it does not claim that every preparation with $\mathrm{\Delta }{Q}_j\mathrm{\Delta }{P}_j=\mathrm{\hslash }\mathrm{/}2$ is Gaussian, and no classification of the equality set is recorded here.
• One family, three coordinates. The computation is uniform in the coordinate $j$ and in the width $\sigma$; different widths give different preparations, and the momentum width ${\sigma }_p=\mathrm{\hslash }\mathrm{/}\sigma$ is a property of this family in the model, not a measured resolution.

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment P6, read for the spin-$\frac{1}{2}$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings with mass parameter $m>0$, spin label $s=\frac{1}{2}$ and $r=2$ (State and composition postulate P1: Hilbert spaces, preparations and composite systems, Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations). Assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice), the hypotheses of the cited spectral, moment and Robertson suppliers. Write $S_i=\mathrm{\hslash }{J}_i$ for the model's internal-factor operators of The massive spin-s particle model of the framework: state space, operators, units and frame readings clause (c), so that $J_i=\frac{1}{2}{\sigma }_i$ in the Pauli matrices of Finite spin matrices clause (e), and let ${\mathrm{\Delta }}_{\psi }{S}_i$ be the preparation spread of Preparation variance and spread of an assigned sharp observable. Then, for every unit $\psi \in H_{1\mathrm{/}2}$:

(a) with $⟨\mathbf{S}{⟩}_{\psi }:=(⟨S_x\psi ,\psi ⟩,⟨S_y\psi ,\psi ⟩,⟨S_z\psi ,\psi ⟩)\in {\mathbb{R}}^3$, ${\sum }_{i\in \{x,y,z\}}({\mathrm{\Delta }}_{\psi }{S}_i{)}^2=\frac{3{\mathrm{\hslash }}^2}{4}-\mid ⟨\mathbf{S}{⟩}_{\psi }{\mid }^2\ge \frac{{\mathrm{\hslash }}^2}{2},$ so at least one component satisfies ${\mathrm{\Delta }}_{\psi }{S}_i\ge \mathrm{\hslash }\mathrm{/}\sqrt{6}$; in particular no unit state has all three component spreads equal to zero;

(b) for every pair of distinct components the spectral projection-valued measures fail to commute; explicitly, with $P_i^{\pm }:=\frac{1}{2}(I\pm {\sigma }_i)$ the spectral projections of $S_i$ at $\pm \mathrm{\hslash }\mathrm{/}2$, $P_x^{+}{P}_y^{+}-P_y^{+}{P}_x^{+}=\frac{i}{2}{\sigma }_z\mathrm{\ne }0,$ and cyclically; consequently no pair of the three components possesses a joint sharp observable in the sense of Compatible sharp observables, their joint law and the ideal joint instrument;

(c) in the state $\psi =f{e}_{1\mathrm{/}2}$ with any unit spatial factor $f\in L^2({\mathbb{R}}^3;\mathbb{C})$ and the basis vector $e_{1\mathrm{/}2}$ of the internal factor, $⟨S_z{⟩}_{\psi }=\frac{\mathrm{\hslash }}{2},⟨S_x{⟩}_{\psi }=⟨S_y{⟩}_{\psi }=0,{\mathrm{\Delta }}_{\psi }{S}_x={\mathrm{\Delta }}_{\psi }{S}_y=\frac{\mathrm{\hslash }}{2},{\mathrm{\Delta }}_{\psi }{S}_z=0,$ and ${\mathrm{\Delta }}_{\psi }{S}_x{\mathrm{\Delta }}_{\psi }{S}_y={\mathrm{\hslash }}^2\mathrm{/}4=\frac{1}{2}\mathrm{\mid }⟨[S_x,S_y]\psi ,\psi ⟩\mathrm{\mid }$, so the Robertson bound of The Robertson relation for preparation spreads of assigned sharp observables is attained in this state;

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P5, P6, the stated choice hypotheses and the cited mathematics of the finite internal matrices acting on $H_{1\mathrm{/}2}$. They concern the internal-factor coupling constants of the model; they assert no laboratory measurement of a spin component, no Stern-Gerlach apparatus model, no dynamics, no empirical value and no interpretation of $S_i$ as a rotating body or as orbital angular momentum.

Let $H\mathrm{\ne }\{0\}$ be a complex Hilbert space, let $A,B$ be self-adjoint operators on $H$, and let $D\subseteq D(A)\cap D(B)$ be a dense linear subspace with $AD\subseteq D(B)$ and $BD\subseteq D(A)$. If $ABu=BAu$ for every $u\in D$, then the spectral projection-valued measures of $A$ and $B$ strongly commute in the sense of Strong spectral commutation of self-adjoint operators clause (b).

• Where the domain identity loses information. Both operators are the differential expression $-iD$ on different self-adjoint domains; the dense invariant domain $C_c^{\mathrm{\infty }}(0,1)$ sees only the action, not the boundary conditions that determine the spectra. No graph-norm core property for either self-adjoint extension is asserted. The domain identity $[A,B]u=0$ is therefore not a sufficient condition for strong spectral commutation. Strong spectral commutation still implies equality of the products on every common invariant domain where both compositions are defined.
• Boundedness is the missing hypothesis. For bounded everywhere-defined self-adjoint operators, a commutator identity on a dense subspace extends by continuity to $AB=BA$ on all of $H$, which is equivalent to strong spectral commutation. This counterexample refutes only the converse implication for unbounded operators; it does not refute the implication from strong spectral commutation to equality on a common invariant domain. This is why Strong spectral commutation of self-adjoint operators is defined at the level of projection-valued measures.

Time is a parameter, not an observable. The framework's evolution postulate P4 assigns to an isolated system a self-adjoint energy operator and the flow $U(t)=\exp (-itH\mathrm{/}\mathrm{\hslash })$ with $t$ a real parameter of the curve (Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Assigned energy observable, unitary flow and the two pictures of autonomous evolution). The framework's assigned sharp observables are self-adjoint operators with spectral projection-valued measures and outcome laws (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes); time is not among them: there is no assigned self-adjoint time operator whose outcome law would enter the Robertson relation of The Robertson relation for preparation spreads of assigned sharp observables, and this page introduces none. Consequently the expression $\mathrm{\Delta }H\mathrm{\Delta }t$ is not an instance of the Robertson relation, and the symbol $\mathrm{\Delta }t$ has no definition until a timescale constructed from observables is specified.

What the framework does support. The graph-domain Ehrenfest theorem Graph-domain Ehrenfest identities and the conditional timescale bound is a conditional mathematical statement about a curve $\psi :I\to H$ that is $C^1$ into the graph norm of a self-adjoint $A$, stays in $D(AK)$ (and in $D(KA)$ where stated), has unit norm and satisfies ${\psi }^{\mathrm{\prime }}=-\frac{i}{\mathrm{\hslash }}K\psi$. Under those hypotheses, with $f(t):=⟨A\psi (t),\psi (t)⟩$ and $f^{\mathrm{\prime }}(t_0)\mathrm{\ne }0$, the timescale ${\tau }_A(t_0):=\mathrm{\Delta }A(t_0)\mathrm{/}\mathrm{\mid }{f}^{\mathrm{\prime }}(t_0)\mathrm{\mid }$ satisfies $\mathrm{\Delta }K(t_0){\tau }_A(t_0)\ge \frac{\mathrm{\hslash }}{2}\mathrm{.}$ With $K=H$ the energy, the Heisenberg-picture theorem identifies $f^{\mathrm{\prime }}(t_0)$ with the derivative of the mean of the representative observable when $A$ is bounded and preserves the energy domain, $AD(H_{\mathrm{o}\mathrm{p}})\subseteq D(H_{\mathrm{o}\mathrm{p}})$ (The Heisenberg picture: equivalent predictions and the domain-qualified equation of motion clause (c)). Boundedness alone does not imply this domain-invariance condition. For a domain-qualified, possibly unbounded $A$, the graph-domain theorem still gives the Schroedinger-picture derivative stated above, but no Heisenberg-picture identification is asserted here. The spreads $\mathrm{\Delta }A(t_0)$ and $\mathrm{\Delta }K(t_0)$ are preparation spreads in the sense of Preparation variance and spread of an assigned sharp observable, and the theorem's clause (d) records that when $f^{\mathrm{\prime }}(t_0)=0$ the quotient supplies no timescale at all.

What is not claimed. The timescale ${\tau }_A$ depends on the chosen observable $A$ and on the state: different choices give different numbers, and no universal "lifetime" or "duration" of the system is defined. No time operator, no energy-time Robertson inequality, no claim that an energy uncertainty is a linewidth and no relation such as ${\gamma }_E\approx \mathrm{\hslash }\mathrm{\Gamma }$ is asserted by P1-P6; such statements require additional model assumptions (an apparatus or clock model, a decay model or an analytic continuation) that are outside this page. The heuristic accounts of measurement disturbance in the literature are not identities of the framework and are not used as premises here.

• Relation to the preparation-spread results. The timescale bound is a Robertson-type estimate applied to the pair (energy, chosen observable) along a curve; it does not weaken or strengthen the preparation-spread relation of The Robertson relation for preparation spreads of assigned sharp observables, which concerns a single preparation and two observables.
• No empirical content. No measured lifetime, linewidth or energy uncertainty is recorded here, and no comparison with data is attempted.

Preparation spread. For an assigned sharp observable $(A,u_A)$ and a preparation $\rho$, the spread ${\mathrm{\Delta }}_{\rho }A$ of Preparation variance and spread of an assigned sharp observable is the standard deviation of the outcome law that the observable and probability postulate assigns to the pair $(\rho ,A)$: it is a property of the preparation and the assigned observable, computed from the Born law alone, and it exists only under the stated moment hypotheses. The inequalities of The Robertson relation for preparation spreads of assigned sharp observables concern these spreads and nothing else.

Apparatus resolution or error. A finite-resolution apparatus is modeled in this category by a declared instrument with effects and an update rule (Declared ideal measurements and instruments of the framework; clause (iv) of P2 and the ideal conditional measurement postulate P3 in the readings of Ideal conditional measurement postulate P3: the projective Lueders update and discarded outcomes). Whether a declared effect family approximates a target observable, and by how much, is a property of the chosen instrument: the postulates do not attach a universal error parameter to a sharp observable, and the effects alone do not determine the update (Outcome effects do not determine the conditional update, and general instruments need not be repeatable).

Disturbance. The state change produced by a measurement is likewise part of the declared instrument: for the framework's declared ideal sharp measurement the conditional and discarded-outcome updates are specified in Consequences of a declared ideal sharp measurement: repeatability and discarded ensembles, and they are idealizations, not claims about every apparatus. The framework's postulates assign no universal disturbance to "a measurement of position" as such.

No conflation. The Robertson relation bounds the product of two preparation spreads; it does not bound an apparatus error, it does not bound a disturbance, and it is not a statement that measuring one quantity "must" perturb another by a fixed amount. The heuristic accounts of measurement intrusion in the literature are accordingly used on this page at most as motivation: the exact statements are the preparation-spread results, and the identification of a concrete apparatus with a declared instrument is a tested identification with uncertainty under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions).

No universal error-disturbance relation is recorded. This page and the measurement page contain no theorem of the form "error times disturbance is at least $\mathrm{\hslash }\mathrm{/}2$": such a statement would require declared error and disturbance quantifiers for an instrument and its own proof. What is recorded here is only the separation of the three objects and the conditional results actually proved on the two pages.

• Why the distinction is load-bearing. An experiment that reports a finite resolution, a detection efficiency or a background model is reporting apparatus properties; a preparation spread is a model quantity of the state and the assigned observable. Agreement of a measured histogram with a predicted distribution tests the combined preparation, apparatus and observable identification, not the inequality alone.
• Disturbance is model-dependent. Two declared instruments with the same outcome effects can have different updates and hence different disturbances, as Outcome effects do not determine the conditional update, and general instruments need not be repeatable records; a disturbance claim must therefore name its instrument.

One exact model computation. The Gaussian family attains the position-momentum bound in the massive model computes the position and momentum densities and spreads of the Gaussian family in the spinless massive model and verifies that the product equals $\mathrm{\hslash }\mathrm{/}2$ with the centered vectors linearly dependent and their pairing purely imaginary, realizing the equality case of The position-momentum preparation relation of the massive spin-s model. It is a statement about one family of preparations of the model; it contains no dynamics, no detection model and no measured number, and it does not characterize all equality states. The spectral outcome-law and moment steps use the Axiom of Choice, as recorded in the example's assumptions.

One finite internal-factor deduction. Spin one half: spread identity, incompatible component outcomes and saturated Robertson bounds derives the spread identity ${\sum }_i(\mathrm{\Delta }{S}_i{)}^2=3{\mathrm{\hslash }}^2\mathrm{/}4-\mathrm{\mid }⟨\mathbf{S}⟩{\mathrm{\mid }}^2$ for the spin-$\frac{1}{2}$ internal factor, the lower bound ${\mathrm{\hslash }}^2\mathrm{/}2$, the failure of spectral compatibility for every pair of components, and the saturation of the Robertson bound in the $S_z$ eigenstates. It is a conditional deduction inside the model's internal factor; it asserts no Stern-Gerlach apparatus, no dynamics and no measurement.

One purely mathematical counterexample. A commutator vanishing on a dense invariant domain does not imply strong spectral commutation is a statement of pure mathematics about two self-adjoint extensions of the interval momentum operator: it refutes the implication from a commutator identity on a dense invariant domain to strong spectral commutation. It carries no physical premise and makes no physical claim.

Three remark items. Time is not an observable of the framework: the conditional timescale bound records that time is a parameter and not an observable, so no energy-time Robertson pair exists here and only the conditional timescale bound is available; Preparation spread, apparatus resolution and disturbance are different objects separates preparation spread, apparatus resolution and disturbance. Both are readings of other items, not new theorems. This item is the third remark item and audits the companion's inventory and evidence scope.

No experiment and no data. The companion records no preparation procedure, apparatus, calibration, trial count, count rate, histogram, uncertainty budget or statistical analysis, and it uses no empirical premise. Every quantity computed here is a model quantity. Any comparison with data would test the combined model and apparatus assumptions through the identification of a preparation and an apparatus with the model's objects, a tested identification with uncertainty under the preparation and apparatus postulate P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions), and would not turn the conditional results into unconditional statements about nature.

No universal claims. The Gaussian family is not all preparations; the spin deduction concerns the internal factor of the model; the counterexample concerns two specific operators. In particular the companion does not claim that every state attaining $\mathrm{\hslash }\mathrm{/}2$ is Gaussian, that every pair of incompatible components excludes all forms of joint measurement, or that the timescale bound is a universal energy-time relation.

• Reading order. The mathematical items precede the model computations on the main page; the companion consumes the main page's items and adds no new supplier to them.
• Status. As everywhere in this category, agreement of the conditional predictions with a laboratory result would be evidence for the combined model and apparatus assumptions, not a proof of the framework; the companion reports no such result.