The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading

(a) Framework data and line model. Adopt P1, P2, P4 and P5 (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, Autonomous dynamics postulate P4: self-adjoint generators and unitary evolution, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators), the state and observable readings cited there, and the declared effective one-dimensional line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions with a mass parameter $m>0$, spin label $s=0$, action constant $\mathrm{\hslash }>0$ and units from Dimensions, SI unit conventions and quantity types of the framework (The massive spin-s particle model of the framework: state space, operators, units and frame readings). Assume the Axiom of Countable Choice and the Axiom of Choice as the choice hypotheses of the cited mathematical suppliers (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Fix a real potential $V\in C_c^5(\mathbb{R};\mathbb{R})$ (five times continuously differentiable with compact support); it is real, bounded and measurable, so the line model's assigned energy operator $H_V$ of its clause (c) is defined with domain $D(H_V)=W^{2,2}(\mathbb{R};\mathbb{C})$, acts by $H_{V}f=-\frac{{\mathrm{\hslash }}^2}{2m}{D}^{2}f+Vf$, and its stationary equation at energy $E$ reads $-\frac{{\mathrm{\hslash }}^2}{2m}{\psi }^{\mathrm{\prime }\mathrm{\prime }}+V\psi =E\psi$ for classically $C^2$ solutions.

(b) Reference scales and nondimensionalisation. As an additional model assumption of the pair fix a reference length $\mathrm{\ell }>0$ (metres) and a reference energy $E_{\star }>0$ (joules), and put $W(\xi ):=\frac{V(\mathrm{\ell }\xi )}{E_{\star }},\lambda :=\frac{E}{E_{\star }},h:=\frac{\mathrm{\hslash }}{\mathrm{\ell }\sqrt{2m{E}_{\star }}}\mathrm{.}$ Then $W\in C_c^5(\mathbb{R};\mathbb{R})$ is dimensionless with support scaled by $\mathrm{\ell }$, and $h>0$ is dimensionless. By Nondimensionalisation of the one-dimensional stationary equation and of the WKB exponent the physical stationary equation of clause (a) at energy $E$ is equivalent, under $x=\mathrm{\ell }\xi$, $\psi (\mathrm{\ell }\xi )=u(\xi )$, to the dimensionless WKB equation $u^{\mathrm{\prime }\mathrm{\prime }}(\xi )=\frac{W(\xi )-\lambda }{h^2}u(\xi ),\xi \in I,$ on every interval $I$. The exact transfer of the square-root action integrals over a subinterval $[x_a,x_b]=\mathrm{\ell }[a,b]$ is, when the indicated strict sign holds throughout $[a,b]$, $\frac{1}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(V(x)-E)}dx=\frac{1}{h}{\int }_a^b\sqrt{W(\xi )-\lambda }d\xi \text{if }W-\lambda >0,$ $\frac{1}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(E-V(x))}dx=\frac{1}{h}{\int }_a^b\sqrt{\lambda -W(\xi )}d\xi \text{if }\lambda -W>0;$ these are respectively clauses (b)(i) and (b)(ii) of Nondimensionalisation of the one-dimensional stationary equation and of the WKB exponent. The pair calls $h$ the model's semiclassical parameter and calls $h\ll 1$ the semiclassical regime; the physical content of the regime is $\mathrm{\hslash }\ll \mathrm{\ell }\sqrt{2m{E}_{\star }}$, and every statement of the page's theorems is an exact statement about the family indexed by $h$, with the error bounds stated at each $h$. For the mathematical small-$h$ limit, hold the dimensionless profile $W$ and energy $\lambda$ fixed and vary the parameter in $u^{\mathrm{\prime }\mathrm{\prime }}=(W-\lambda )u\mathrm{/}{h}^2$. A corresponding physical family with fixed $m,E_{\star },\mathrm{\hslash }$ is obtained by taking ${\mathrm{\ell }}_h=\mathrm{\hslash }\mathrm{/}(h\sqrt{2m{E}_{\star }})$ and $V_h(x)=E_{\star }W(x\mathrm{/}{\mathrm{\ell }}_h)$, with $E=E_{\star }\lambda$; the physical width then changes, while the natural action constant does not. A single originally chosen $V,\mathrm{\ell },E_{\star }$ supplies one member of this family.

(c) Turning points, regions and phase. For the data $(W,\lambda )$ put $\tau :=\{\xi \in \mathbb{R}:W(\xi )=\lambda \}$ for the set of turning points. A point ${\xi }_0\in \tau$ is simple when $W^{\mathrm{\prime }}({\xi }_0)\mathrm{\ne }0$. An open interval $J$ is an allowed interval when $W(\xi )-\lambda <0$ on $J$ (the dimensionless kinetic energy $-(W-\lambda )$ is positive there), a forbidden interval when $W(\xi )-\lambda >0$ on $J$. On an allowed interval the model uses the positive scaled momentum $p(\xi ):=\sqrt{\lambda -W(\xi )}>0$ with local wavenumber $k_h(\xi ):=p(\xi )\mathrm{/}h$ in the coordinate $\xi$ and WKB phase ${\theta }_h(\xi ):=h^{-1}{\int }_{{\xi }_{\mathrm{r}\mathrm{e}\mathrm{f}}}^{\xi }p(s)ds$ with a declared base point; on a forbidden interval the model uses $\kappa (\xi ):=\sqrt{W(\xi )-\lambda }>0$. The dimensionless forbidden action of a forbidden interval $[{\xi }_a,{\xi }_b]$ is $S:={\int }_{{\xi }_a}^{{\xi }_b}\sqrt{W(\xi )-\lambda }d\xi =\frac{h}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(V(x)-E)}dx,$ The WKB exponent is $S\mathrm{/}h$, the physical action in units of $\mathrm{\hslash }$. For a barrier whose endpoints are simple turning points, the strict-sign hypothesis of clause (b)(i) fails at those endpoints; apply that clause on $[{\xi }_a+\delta ,{\xi }_b-\delta ]$ and let $\delta \downarrow 0$. Both integrands extend continuously with value zero at the endpoints, so the omitted endpoint integrals tend to zero. The three declared regime hypotheses used by the page are:

(W1) Allowed-interval regularity: $W\in C^2$ on a neighbourhood of $[a,b]$ and $\lambda -W\ge \delta >0$ on $[a,b]$ for some $\delta >0$ (no turning point in the interval; the interval is allowed);

(W2) Separated simple turning points: $W-\lambda$ has exactly two zeros ${\xi }_a<{\xi }_b$ in $\mathbb{R}$, both simple, is positive on $({\xi }_a,{\xi }_b)$ and negative on the complement of $[{\xi }_a,{\xi }_b]$ (equivalently the physical barrier $V-E$ has exactly two simple zeros surrounding a forbidden region and is negative outside);

(W3) Single simple turning point: on an interval $[{\xi }_0-r,{\xi }_0+s]$ the equation $W-\lambda$ has the single zero ${\xi }_0$, it is simple, and $W-\lambda <0$ on one side and $>0$ on the other.

(d) Declared barrier transmission reading. Under (W2), $\lambda >0$ because $W=0$ outside its compact support. Choose $R>0$ with $\mathrm{supp}W\subset (-R,R)$ and put $p_{\mathrm{\infty }}:=\sqrt{\lambda }$. The declared scattering solution is the exact $C^2$ solution of the dimensionless equation with boundary conditions $u_h(\xi )=p_{\mathrm{\infty }}^{-1\mathrm{/}2}(e^{i{p}_{\mathrm{\infty }}\xi \mathrm{/}h}+r(h)e^{-i{p}_{\mathrm{\infty }}\xi \mathrm{/}h})(\xi \le -R),u_h(\xi )=p_{\mathrm{\infty }}^{-1\mathrm{/}2}t(h)e^{i{p}_{\mathrm{\infty }}\xi \mathrm{/}h}(\xi \ge R)\mathrm{.}$ Thus there is one incoming wave from the left and none from the right. These data determine unique amplitudes by Two-turn barrier tunnelling error, instantiated with its mass parameter $1\mathrm{/}2$, potential $W$ and energy $\lambda$. Define the reference current $J_h(u):=h\mathrm{Im}(\bar{u}{u}^{\mathrm{\prime }})$: the normalized exterior waves have currents $\pm 1$. The transmission coefficient is $T(h):=\mathrm{\mid }t(h){\mathrm{\mid }}^2$, the ratio of transmitted to incoming reference current. The physical line-model current is a positive common multiple of $J_h$ under clause (b), so its ratio is also $T(h)$. Airy/WKB matching computes this exact boundary-defined quantity; it does not define an additional approximate scattering solution. No detector, wave packet, scattering experiment or measured count is asserted, and this clause does not by itself assert an asymptotic formula for $T(h)$.

(e) Scope. The model concerns a declared spinless massive particle in the declared one-dimensional line model with a compactly supported $C^5$ potential, at a fixed energy, in the nondimensionalised family of clause (b). It asserts nothing about two- or three-dimensional scattering, about singular or non-smooth potentials, about wave packets, detectors or measurements, and about any laboratory realization of the line model, the potential $V$ or the energy $E$.