Qm Observables And Born Probabilities — Examples

Let $H={\mathbb{C}}^2$ carry the standard inner product $⟨z,w⟩=z_1\overline{w_1}+z_2\overline{w_2}$ (Real and complex inner product spaces, with the inner product linear in the first argument, Hilbert space, The induced length is a norm, Conjugation is an involutive real-field automorphism, $z\bar{z}=\mathrm{\mid }z{\mathrm{\mid }}^2$, and modulus is definite, multiplicative, and subadditive), with orthonormal basis $e_1=(1,0)$, $e_2=(0,1)$. Fix two distinct energies $E_1\mathrm{\ne }{E}_2$ in joules (model parameters, not measured values), and consider the assigned sharp observable $A:=E_1{T}_{e_1,e_1}+E_2{T}_{e_2,e_2},u_A=\mathrm{J},$ with $T_{u,v}x=⟨x,v⟩u$ the rank-one calculus of Rank-one operator calculus (Assigned sharp observables and the outcome law of the framework). Then:

(i) $A$ is bounded and self-adjoint with $D(A)=H$; its spectral projection-valued measure is $E_A(B)={\mathbf{1}}_{E_1\in B}{T}_{e_1,e_1}+{\mathbf{1}}_{E_2\in B}{T}_{e_2,e_2}(B\subseteq \mathbb{R}\text{ Borel}),$ so the spectrum is $\sigma (A)=\{E_1,E_2\}$ and the outcome lies in it almost surely: ${\mu }_{\rho }^A(\{E_1,E_2\})=1$ for every preparation $\rho$. This probability-one statement does not logically exclude realizations in a null event.

(ii) For the pure preparation of the unit vector $\psi =c_1{e}_1+c_2{e}_2$, $\mathrm{\mid }{c}_1{\mathrm{\mid }}^2+\mathrm{\mid }{c}_2{\mathrm{\mid }}^2=1$, the outcome law is ${\mu }_{\psi }^A(\{E_1\})=\mathrm{\mid }{c}_1{\mathrm{\mid }}^2,{\mu }_{\psi }^A(\{E_2\})=\mathrm{\mid }{c}_2{\mathrm{\mid }}^2,{\mu }_{\psi }^A(B)={\sum }_{i:E_i\in B}\mathrm{\mid }{c}_i{\mathrm{\mid }}^2,$ and for the statistical mixture $\rho =w{T}_{e_1,e_1}+(1-w)T_{e_2,e_2}$ with $0\le w\le 1$ the law is ${\mu }_{\rho }^A(\{E_1\})=w$, ${\mu }_{\rho }^A(\{E_2\})=1-w$. This preparation is mixed exactly when $0<w<1$; at $w=0$ it is the pure state $T_{e_2,e_2}$, and at $w=1$ it is the pure state $T_{e_1,e_1}$.

(iii) With $p:=\mathrm{\mid }{c}_1{\mathrm{\mid }}^2$ and $q:=\mathrm{\mid }{c}_2{\mathrm{\mid }}^2=1-p$, the mean, second moment and variance of the pure preparation are $⟨E{⟩}_{\psi }=E_{1}p+E_{2}q,⟨E^2{⟩}_{\psi }=E_1^{2}p+E_2^{2}q,{\mathrm{Var}}_{\psi }=pq(E_1-E_2{)}^2,$ and for the mixture replace $p$ by $w$ and $q$ by $1-w$; all three numbers are finite and exact because $A$ is bounded.

The computation is a model-internal readout: the $E_i$ are parameters of the assigned observable, the probabilities are those of the framework's outcome law, and no laboratory measurement, apparatus or empirical value is asserted.

• The two eigenvalues $E_1\mathrm{\ne }{E}_2$ are distinct, so the eigenspaces are one-dimensional and the spectral projections are exactly the rank-one projections $T_{e_i,e_i}$; a degenerate energy would replace them by the projection onto the eigenspace and the exact-value probabilities by its trace weight, in agreement with Atoms of the spectral measure and eigenvalues of a self-adjoint operator.
• The example is a finite model computation, not a reported experiment: nothing here identifies $E_1,E_2$ with the levels of a specific physical system, and no counts, uncertainty or apparatus behaviour is asserted.

Work in the spinless massive-particle model of P5, that is $H=L^2({\mathbb{R}}^3;\mathbb{C})$ with position operators $Q_j$ (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Position and momentum operators on Euclidean L2), fix a width $\sigma >0$ in metres, and let ${\psi }_{\sigma }(t):=(2\pi {\sigma }^2{)}^{-1\mathrm{/}4}{e}^{-t^2\mathrm{/}(4{\sigma }^2)}(t\in \mathbb{R}),$ the Gaussian packet of width $\sigma$. Let $\phi \in L^2({\mathbb{R}}^2;\mathbb{C})$ satisfy ${\int }_{{\mathbb{R}}^2}\mathrm{\mid }\phi {\mathrm{\mid }}^2=1$ and put $\psi (x):={\psi }_{\sigma }(x_1)\phi (x_2,x_3)(x\in {\mathbb{R}}^3),$ so that $\psi \in H$ is a unit vector and the preparation $[\psi ]$ is a state of the model. Then:

(i) The probability that the first coordinate $Q_1$ lies in a Borel set $B\subseteq \mathbb{R}$ is ${\mu }_{\psi }^{Q_1}(B)={\int }_B\mathrm{\mid }{\psi }_{\sigma }(t){\mathrm{\mid }}^{2}dt=\frac{1}{\sqrt{2\pi {\sigma }^2}}{\int }_B{e}^{-t^2\mathrm{/}(2{\sigma }^2)}dt,$ independently of the transverse factor $\phi$; the law has density $\mathrm{\mid }{\psi }_{\sigma }{\mathrm{\mid }}^2$ and ${\mu }_{\psi }^{Q_1}(\mathbb{R})=1$.

(ii) For every $\lambda \in \mathbb{R}$ the probability of the exact value $\lambda$ is zero, ${\mu }_{\psi }^{Q_1}(\{\lambda \})=0$, while for every $a<b$ the bin $(a,b]$ has positive probability with the explicit bounds $\frac{b-a}{\sqrt{2\pi {\sigma }^2}}\exp (-\frac{\max \{\mathrm{\mid }a\mathrm{\mid },\mathrm{\mid }b\mathrm{\mid }{\}}^2}{2{\sigma }^2})\le {\mu }_{\psi }^{Q_1}((a,b])\le \frac{b-a}{\sqrt{2\pi {\sigma }^2}}\mathrm{.}$ In particular these bounds hold when $0\le a<b$. In the symmetric case $a=-\delta$, $b=\delta$ with $\delta >0$, the probability satisfies $\sqrt{\frac{2}{\pi e}}\le {\mu }_{\psi }^{Q_1}((-\sigma ,\sigma ])\le \sqrt{\frac{2}{\pi }},0<{\mu }_{\psi }^{Q_1}((-\delta ,\delta ])\le \frac{2\delta }{\sqrt{2\pi {\sigma }^2}}\mathrm{.}$ In particular a single recorded event localized in a bin of positive length is possible in the model, while an event at an exact coordinate value has probability zero.

(iii) For each $j\in \{1,2,3\}$ the same formulas hold for $Q_j$ in the unit product preparation ${\psi }^{(j)}(x):={\psi }_{\sigma }(x_j)\phi (x_{\perp })$, where $x_{\perp }$ lists the other two coordinates in increasing index order. For the original preparation $\psi ={\psi }^{(1)}$, the laws of $Q_2$ and $Q_3$ are the marginals of $\mathrm{\mid }\phi {\mathrm{\mid }}^2$ and need not be Gaussian. The width and bin endpoints are model parameters read in metres.

The computation is the framework's own outcome law applied to an explicit state; it records no measurement, no apparatus response and no empirical data.

• The transverse factor $\phi$ enters the computation only through its total mass $\int \mathrm{\mid }\phi {\mathrm{\mid }}^2=1$; the first-coordinate law of a product state is independent of the transverse profile, while a state that is not a product can correlate the coordinates and change the marginal, as the general density formula $f_j(t)={\int }_{{\mathbb{R}}^2}\mathrm{\mid }\psi (x_{\perp },t){\mathrm{\mid }}^{2}d{x}_{\perp }$ of Position and momentum outcome laws and the Born density of the massive particle model shows.
• The bounds are deliberately elementary: they neither sharpen the Gaussian integral nor claim an error analysis. They are finite checks of the model's law, not measured frequencies, and no apparatus resolution, binning procedure or count is asserted.

Adopt the state and composition postulate P1, the observable and probability postulate P2, and the massive-particle model postulate P5 of the framework (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), with Countable Choice and the Axiom of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice) for the cited spectral suppliers. Then: every bounded smooth function $\psi :{\mathbb{R}}^3\to \mathbb{C}$ that solves the free time-independent Schrödinger equation $-\frac{{\mathrm{\hslash }}^2}{2m}\mathrm{\Delta }\psi =E\psi$ pointwise determines a pure preparation of the model with position density $\mathrm{\mid }\psi {\mathrm{\mid }}^2$; in particular the plane wave ${\psi }_k(x)=e^{ik\cdot x}$ with $k\mathrm{\ne }0$ is a preparation with constant position density $\mathrm{\mid }{\psi }_k{\mathrm{\mid }}^2=1$, and the point-localized object ${\delta }^3(x-X)$ is the preparation of a particle exactly at $X$.

Point-localized symbol. Here ${\delta }^3(x-X)$ denotes the evaluation functional ${\delta }_X:C_c({\mathbb{R}}^3;\mathbb{C})\to \mathbb{C}$, ${\delta }_X(\eta ):=\eta (X)$, where $C_c$ is the space of continuous complex functions with compact support. An $L^2$ representation would be a class $f$ with $\int f(x)\eta (x)dx=\eta (X)$ for every such $\eta$; these integrals are defined by Cauchy–Schwarz because $\eta \in L^2$.

• The counterexample concerns the state assignment of this model, not the usefulness of plane waves or delta symbols as distributional tools: they are standard idealizations in formal manipulations, where any probability conclusion requires a separate mathematical statement about normalizable approximants. Nothing here asserts that such approximation schemes fail, only that the raw objects are not preparations and carry no Born law.
• Solving the free equation and being bounded do not by themselves imply membership in $H$; the plane wave above is one explicit bounded solution that is not square-integrable. Every proposed preparation must separately satisfy the model's state-space and normalization requirements.

Work in the spinless massive-particle model of P5 with $H=L^2({\mathbb{R}}^3;\mathbb{C})$ and position operator $Q_1$ (Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Position and momentum operators on Euclidean L2). Use the numerical SI spatial coordinates of Position and momentum outcome laws and the Born density of the massive particle model: $x_1,x_2,x_3,t$ are read in metres and the assigned observable is $(Q_1,\mathrm{m})$. Thus the cutoff $\mathrm{\mid }t\mathrm{\mid }\le 1$ below means a distance of at most one metre; powers of $t$ in the profile apply to its numerical value in that fixed unit. The unnormalized $g_0$ is dimensionless, $Z$ has length units, and $g$, $\phi$, $\psi$ carry units ${\mathrm{m}}^{-1\mathrm{/}2}$, ${\mathrm{m}}^{-1}$, ${\mathrm{m}}^{-3\mathrm{/}2}$ respectively, so the mean is read in metres and the second moment and variance in ${\mathrm{m}}^2$. Define the symmetric profile $g_0(t):=\{\begin{array}{ll}1,& \mathrm{\mid }t\mathrm{\mid }\le 1,\\ \mathrm{\mid }t{\mathrm{\mid }}^{-3\mathrm{/}2},& \mathrm{\mid }t\mathrm{\mid }>1,\end{array}Z:={\int }_{\mathbb{R}}\mathrm{\mid }{g}_0(t){\mathrm{\mid }}^{2}dt,g:=Z^{-1\mathrm{/}2}{g}_0,$ so that $g\in L^2(\mathbb{R};\mathbb{C})$ has $\mathrm{\parallel }g{\mathrm{\parallel }}_2=1$; fix a unit vector $\phi \in L^2({\mathbb{R}}^2;\mathbb{C})$, choose a finite Borel representative as in the cited outcome-law theorem, and put $\psi (x):=g(x_1)\phi (x_2,x_3)$. Then $\psi \in H$ is a unit vector, $[\psi ]$ is a preparation, and for its position law ${\mu }_{\psi }^{Q_1}$ with density $\mathrm{\mid }g{\mathrm{\mid }}^2$:

(i) The mean exists and equals $0$: ${\int }_{\mathbb{R}}\mathrm{\mid }t\mathrm{\mid }\mathrm{\mid }g(t){\mathrm{\mid }}^{2}dt<\mathrm{\infty },{\int }_{\mathbb{R}}t\mathrm{\mid }g(t){\mathrm{\mid }}^{2}dt=0,$ and $\psi \in D(\mathrm{\mid }{Q}_1{\mathrm{\mid }}^{1\mathrm{/}2})$ while $\psi \mathrm{\notin }D(Q_1)$.

(ii) The second moment diverges, so the variance of the position law does not exist: ${\int }_{\mathbb{R}}{t}^2\mathrm{\mid }g(t){\mathrm{\mid }}^{2}dt=+\mathrm{\infty },{\int }_{\mathbb{R}}(t-0{)}^{2}d{\mu }_{\psi }^{Q_1}(t)=+\mathrm{\infty }\mathrm{.}$

Thus a preparation of the model can have a finite mean position without a finite second moment: the moment identities of the framework are conditional on the domains, and a finite mean does not require membership in $D(Q_1)$.

The example is a model computation; it asserts no measured value and no apparatus behaviour.

• The profile is symmetric, so the finite mean is zero; a non-symmetric variant shifts the mean while keeping the second moment infinite, since the tail exponent $3$ is what decides finiteness: $\mathrm{\mid }g{\mathrm{\mid }}^2$ decays like $t^{-3}$, whose first moment $t\cdot t^{-3}=t^{-2}$ is integrable while the second is not.
• The example shows that the domain hypotheses of Mean, second moment and variance of an assigned observable in a preparation are not automatic: the state here is a perfectly legitimate preparation of the model, yet the variance of the position law is infinite and the statement "the spread $\mathrm{\Delta }{Q}_1$" has no finite value in this preparation.

Finite energy readout. An exact two-level energy readout and its moments computes, on the explicit two-dimensional state space, the spectral measure of the bounded observable $E_1{P}_1+E_2{P}_2$, the exact outcome probabilities of a pure and of a mixed preparation, and the mean, second moment and variance of the readout. The energies $E_1\mathrm{\ne }{E}_2$ are parameters of the assigned observable; the computation verifies the framework's law on finite data and asserts no apparatus realization, no measured level and no identification with a specific physical system.

Position bins. Position-bin probabilities of a Gaussian packet computes the first-coordinate law of a normalized Gaussian packet in the model's state space, its exact-value probability zero and explicit elementary bounds on the probability of a bin of positive length. The computation verifies the Born density reading of the model on a normalizable state; it reports no counts, no detector response and no resolution, and its bounds are mathematical estimates rather than uncertainty statements about an experiment.

Distribution caveats. The plane wave and the point-localized delta are not preparations of the model refutes the informal claim that a free plane wave or a point-localized delta symbol is a preparation of the model: both fail to be elements of the state space, and the model's exact-position statement is that the probability of an exact value is zero. The refutation concerns the state assignment; it does not deny the usefulness of these objects as distributional idealizations, and it makes no claim about approximation schemes in which they are formally employed.

Domains of the moment statements. A preparation with finite mean position and infinite variance exhibits a legitimate preparation whose position law has a finite mean and infinite second moment, so that the variance of the position law does not exist; the computation verifies that the domain hypotheses of the moment statements are genuine and not automatic. It asserts no measurement error, no disturbance and no apparatus behaviour.

Empirical neutrality. None of the four items is an experiment. Each is a finite or asymptotic computation inside the model's state space and outcome law, or a refutation of an informal claim about that model; none records data, calibration, sampling, uncertainty or a comparison with observation. Agreement or disagreement with data would test the combined model and apparatus assumptions and is not established or claimed here.