The mean position of a normalized state need not exist

Given: 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 one-dimensional instance of P1 with $H=L^2(\mathbb{R};\mathbb{C})$ and a normalized vector $\psi$; the P2 position observable $Q$ as multiplication by $x$, with the spectral PVM and law of [L6]; and the quantity conventions of Dimensions, SI unit conventions and quantity types of the framework.

Let $L_0>0$ be a length and define ${\psi }_{L_0}(x)=\frac{1}{2\sqrt{L_0}}$ for $\mathrm{\mid }x\mathrm{\mid }\le L_0$ and ${\psi }_{L_0}(x)=\frac{\sqrt{L_0}}{2\mathrm{\mid }x\mathrm{\mid }}$ for $\mathrm{\mid }x\mathrm{\mid }\ge L_0$. The formulas agree at $\mathrm{\mid }x\mathrm{\mid }=L_0$, and $[{\psi }_{L_0}]={\mathrm{\ell }}^{-1\mathrm{/}2}$. It is measurable and represents an almost-everywhere class; its $L^2$ membership is established in step 1.1 using [L5] and monotone convergence. Its squared tail is $L_0\mathrm{/}(4x^2)$, whose improper Riemann integral converges by the rational $p$-test (Improper integrals over unbounded intervals, The improper $p$-test for rational exponents, The space $L^p(\mu )$ as the quotient by null functions).

For $N>L_0$ the fundamental theorem and power rule give ${\int }_{L_0}^N{x}^{-2}dx=L_0^{-1}-N^{-1}$, ${\int }_{L_0}^N{x}^{-1}dx=\ln (N\mathrm{/}{L}_0)$, and ${\int }_{L_0}^{N}1dx=N-L_0$; the improper integrals are the limits as $N\to \mathrm{\infty }$ (Every continuous function on an interval has a primitive; two primitives differ by a constant; and ${\int }_a^{b}f=G(b)-G(a)$ for any primitive $G$, For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, Improper integrals over unbounded intervals).

Monotone convergence for the truncated absolute moments: ${\int }_{-N}^N\mathrm{\mid }\psi {\mathrm{\mid }}^2\to {\int }_{\mathbb{R}}\mathrm{\mid }\psi {\mathrm{\mid }}^2$ and ${\int }_{-N}^N\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }\psi {\mathrm{\mid }}^2\to {\int }_{\mathbb{R}}\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }\psi {\mathrm{\mid }}^2\in [0,+\mathrm{\infty }]$ along the increasing sequence of sets $[-N,N]$, and similarly for the second moments (Monotone convergence for the integral).

Second moments and Cauchy-Schwarz: $\psi \in D(Q)$ means ${\int }_{\mathbb{R}}{x}^2\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx<\mathrm{\infty }$, and for a probability density with finite second moment the first absolute moment is finite by the Cauchy-Schwarz inequality applied to the probability measure $\mathrm{\mid }\psi (x){\mathrm{\mid }}^{2}dx$ (Position and momentum operators on Euclidean L2, Cauchy–Schwarz: $\mathrm{\mid }⟨u,v⟩\mathrm{\mid }\le \parallel u\parallel \parallel v\parallel$, with equality exactly for linearly dependent vectors).

For each $N>L_0$, the three global integrands $\mathrm{\mid }{\psi }_{L_0}{\mathrm{\mid }}^2$, $\mathrm{\mid }x\mathrm{\mid }\mathrm{\mid }{\psi }_{L_0}{\mathrm{\mid }}^2$ and $x^2\mathrm{\mid }{\psi }_{L_0}{\mathrm{\mid }}^2$ are real and continuous on $[-N,N]$, including at $x=\pm L_0$, because their defining formulas agree there. The tail functions $x^{-2}$, $x^{-1}$ and $1$ of [L2] are real and continuous on $[L_0,N]$, which excludes zero. Each of these restrictions is therefore bounded and Riemann integrable by A continuous function on $[a,b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion. Under the stated Countable Choice hypothesis, A bounded Riemann integrable function on a closed bounded interval is Lebesgue measurable and has the same integral identifies its finite-interval Riemann and Lebesgue integrals. Thus [L2] evaluates the truncated Lebesgue integrals used by [L3]; the tail notation ${\int }_{L_0}^{\mathrm{\infty }}$ below denotes the monotone limit of these nonnegative Lebesgue integrals, including $+\mathrm{\infty }$.

Choice is carried from P2 and is an explicit hypothesis of Spectral measure of a real L2 multiplication operator clauses (a)-(c). On the positive, $\sigma$-finite Lebesgue measure space $\mathbb{R}$, the measurable finite multiplier $m(x)=x$ satisfies that lemma's hypotheses. Under the stated Choice assumption (The Axiom of Choice), it identifies the self-adjoint position operator $Q$ on $D(Q)=\{f:xf\in L^2\}$ and its spectral PVM $E_Q(B)=M_{{\mathbf{1}}_B}$, with pure-state law ${\mu }_{\psi }^Q(B)=\mathrm{\parallel }{E}_Q(B)\psi {\mathrm{\parallel }}^2={\int }_B\mathrm{\mid }\psi {\mathrm{\mid }}^{2}dx$ for every unit $\psi$. The explicit moment calculations require no additional arbitrary choice.