Qm Controlled Approximations — Examples

Adopt P1, P2 and P4 for a declared two-level system (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) with state space $\mathfrak{H}={\mathbb{C}}^2$ in the standard orthonormal basis $(e_1,e_2)$ and with the observable reading of Assigned sharp observables and the outcome law of the framework; read all energies in joules as in Dimensions, SI unit conventions and quantity types of the framework. Fix real gap and coupling parameters $g>0$ and $c>0$, and declare the model's energy observable family by the perturbed-family model of The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster with $H_0:=\left(\begin{array}{cc}0& 0\\ 0& g\end{array}\right),V:=\left(\begin{array}{cc}0& c\\ c& 0\end{array}\right),H(\lambda )=H_0+\lambda V(\lambda \in \mathbb{R})\mathrm{.}$ Then $H_0$ has the isolated simple eigenvalue $E=0$ with gap $g$, $v=\mathrm{\parallel }V\mathrm{\parallel }=c$, ${\psi }_0=e_1$, and the reduced inverse of the model clause (c)(iii) is $S=\mathrm{diag}(0,1\mathrm{/}g)$ on $Q_0\mathfrak{H}=\mathbb{C}{e}_2$. Then:

(a) for every real $\lambda$ the matrix $H(\lambda )$ is Hermitian with exactly two eigenvalues $E_{\pm }(\lambda )=\frac{g\pm \sqrt{g^2+4{\lambda }^2{c}^2}}{2},$ both simple for $\lambda \mathrm{\ne }0$ and spanning the spectrum for every $\lambda$; in the model's valid contour range $2\mathrm{\mid }\lambda \mathrm{\mid }c\mathrm{/}g<1$, its distinguished label is $E_{\lambda }=E_{-}(\lambda )$. The formula $E_{-}$ remains an exact continuous lower spectral branch for all real couplings, even beyond that fixed-contour range;

(b) the exact perturbation coefficients are $E^{\mathrm{\prime }}(0)=0,E^{\mathrm{\prime }\mathrm{\prime }}(0)=-\frac{2c^2}{g},{\psi }^{\mathrm{\prime }}(0)=-SV{\psi }_0=-\frac{c}{g}{e}_2,{\psi }_0=e_1,$ and the exact eigenvalue has the convergent expansion $E_{-}(\lambda )=-\frac{{\lambda }^2{c}^2}{g}+\frac{{\lambda }^4{c}^4}{g^3}-\frac{2{\lambda }^6{c}^6}{g^5}+O({\lambda }^8);$

(c) the cubic remainder bound of Nondegenerate stationary perturbation of a simple gapped label: first- and second-order coefficients and an explicit cubic remainder clause (e) holds in this model for every $\mathrm{\mid }\lambda \mathrm{\mid }<R=g\mathrm{/}(8c)$, and at the test point $\lambda =R\mathrm{/}2=g\mathrm{/}(16c)$ the exact remainder and its bound are $E_{-}\text{\hspace{0.17em}⁣}\left(\frac{g}{16c}\right)+\frac{g}{256}=\frac{g}{2}(1-\frac{\sqrt{65}}{8})+\frac{g}{256}\approx 1.51\cdot {10}^{-5}g\le \frac{g}{32},$ so the finite check is satisfied with a wide margin;

(d) the exact spectrum computed in (a) is a spectral statement about the declared two-level family; it identifies no physical system, asserts no measured splitting, and asserts nothing about which of the two spectral subspaces a preparation occupies (that is fixed by the preparation and the evolution of P4, not by the perturbative expansion).

• Why this is a "finite-gap" check. The complementary subspace is one-dimensional with distance $g$, so the reduced inverse is the explicit number $1\mathrm{/}g$ and every coefficient of the page's theorem can be compared against a closed-form eigenvalue. The second-order coefficient is strictly negative for the followed lower level, the signature of a level repulsion from the complementary state.
• The cubic bound is not sharp. The exact difference from the second-order truncation is $O({\lambda }^4)$, and the page's bound at the test point is more than three orders of magnitude larger; the check verifies the inequality, not optimality.

Adopt P1, P2 and P4 for the declared two-body Coulomb model of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (d), in the relative-coordinate factor $L^2({\mathbb{R}}^3;\mathbb{C})$ with the dipole operator $D_j=-q{Q}_j$ and the compressed finite-level coupling of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field. Use the Coulomb mass parameter $m=\mu :=m_1{m}_2\mathrm{/}(m_1+m_2)$ and $a_{\mu }:={\mathrm{\hslash }}^2\mathrm{/}(\mu \kappa )>0$. Choose as the finite-level subspace the declared two-dimensional subspace $W:=\mathrm{span}\{{\psi }_1,{\psi }_2\}\subseteq L^2({\mathbb{R}}^3;\mathbb{C}),$ where ${\psi }_1$ and ${\psi }_2$ are normalized bound states of the model with angular labels $(n,l,m)=(1,0,0)$ and $(2,1,0)$ supplied by The one-centre Coulomb model: bound energies, n^2-degeneracy, completeness of the bound states in the negative spectral subspace and the continuum, so that $H_{\kappa }{\psi }_1=E_1{\psi }_1$, $H_{\kappa }{\psi }_2=E_2{\psi }_2$ with $E_1=-\frac{m{\kappa }^2}{2{\mathrm{\hslash }}^2},E_2=-\frac{m{\kappa }^2}{8{\mathrm{\hslash }}^2},\mathrm{\Delta }:=E_2-E_1=\frac{3m{\kappa }^2}{8{\mathrm{\hslash }}^2}>0.$ Choose the external field along the third axis of the declared frame, ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}=(0,0,\mathcal{E})$ with $\mathcal{E}>0$ in $\mathrm{V}{\mathrm{m}}^{-1}$, and use the dimensionless coupling $\lambda :=\frac{\mathcal{E}}{1\mathrm{V}{\mathrm{m}}^{-1}},V:=-(1\mathrm{V}{\mathrm{m}}^{-1})D_3^W,$ so that the Stark model of the postulate reads $H_S(\mathcal{E})=H_0+\lambda V,H_0:={\mathrm{\Pi }}_W{H}_{\kappa }{\mid }_W=\left(\begin{array}{cc}{E}_1& 0\\ 0& E_2\end{array}\right)$ in the declared orthonormal basis. Then:

(a) the compression of the dipole matrix has the form $D_1^W=D_2^W=0,D_3^W=D_{\mathrm{r}\mathrm{e}\mathrm{l}}\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),D_{\mathrm{r}\mathrm{e}\mathrm{l}}:=\mathrm{\mid }⟨{\psi }_2,D_3{\psi }_1⟩\mathrm{\mid }>0,$ with unit $\mathrm{C}\mathrm{m}$, where the phase of ${\psi }_2$ is chosen to make the off-diagonal entry positive. The local computation below gives $D_{\mathrm{r}\mathrm{e}\mathrm{l}}=128\sqrt{2}q{a}_{\mu }\mathrm{/}243>0$ from the declared model parameters; this is an exact model identity, with no measured value of any parameter. The diagonal entries vanish by angular degree selection, and only $Q_0$ can contribute between the chosen $m=0$ states; nonvanishing is proved by explicit radial and angular integration;

(b) $v=\mathrm{\parallel }V\mathrm{\parallel }=D_{\mathrm{r}\mathrm{e}\mathrm{l}}(1\mathrm{V}{\mathrm{m}}^{-1})$ in joules and the family $H_S$ has the exact eigenvalues $F_{\pm }(\mathcal{E})=\frac{E_1+E_2}{2}\pm \sqrt{(\frac{\mathrm{\Delta }}{2}{)}^2+(D_{\mathrm{r}\mathrm{e}\mathrm{l}}\mathcal{E}{)}^2},$ the lower branch $F_{-}$ being the model's followed label for dimensionless couplings $0\le \lambda <R:=\mathrm{\Delta }\mathrm{/}(8v)$, equivalently $0\le \mathcal{E}<\mathrm{\Delta }\mathrm{/}(8D_{\mathrm{r}\mathrm{e}\mathrm{l}})$, with $\mathrm{\Delta }$ playing the role of the gap $g$;

(c) the page's perturbation theorem applies with nondegenerate label $E_1$ (gap $\mathrm{\Delta }$), and gives the vanishing first-order shift $E^{\mathrm{\prime }}(0)=0$ and the exact second derivative at zero $E^{\mathrm{\prime }\mathrm{\prime }}(0)=-\frac{2D_{\mathrm{r}\mathrm{e}\mathrm{l}}^2(1\mathrm{V}{\mathrm{m}}^{-1}{)}^2}{\mathrm{\Delta }},$ equivalently the quadratic Stark shift $F_{-}(\mathcal{E})=E_1-\frac{D_{\mathrm{r}\mathrm{e}\mathrm{l}}^2{\mathcal{E}}^2}{\mathrm{\Delta }}+O({\mathcal{E}}^4);$

(d) the finite remainder check at half the perturbation radius $\lambda =R\mathrm{/}2$, equivalently $\mathcal{E}=\mathrm{\Delta }\mathrm{/}(16D_{\mathrm{r}\mathrm{e}\mathrm{l}})$, gives the exact remainder $\mid F_{-}+\frac{\mathrm{\Delta }}{256}-E_1\mid =\mathrm{\Delta }\frac{129-16\sqrt{65}}{256}=\frac{\mathrm{\Delta }}{256(129+16\sqrt{65})}<\frac{\mathrm{\Delta }}{32768}<\frac{\mathrm{\Delta }}{32},$ so the page's cubic bound is satisfied at this coupling with a wide margin;

(e) scope: the computation is conditional on the declared truncated two-state model and the adopted coupling; it identifies no laboratory atom, reports no measured shift and no polarizability value, asserts no transition rate, and does not treat the uncompressed linear potential $-{\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}\cdot \mathbf{D}$. The postulate excludes that model and proves its common-test-domain expectations unbounded below for every nonzero uniform field; no full energy-form domain is used here.

• Why the first-order shift vanishes. The two declared states have angular degrees $0$ and $1$; the coordinate multiplier changes the degree by exactly one, so the diagonal matrix elements of $D_j$ vanish and the leading shift is quadratic, exactly as in the textbook quadratic Stark effect. The nonzero dipole modulus is computed locally as $D_{\mathrm{r}\mathrm{e}\mathrm{l}}=128\sqrt{2}q{a}_{\mu }\mathrm{/}243$ from the model's parameters; no measured charge, length or dipole value is supplied.
• Shared two-by-two identity. The eigenvalue formula for $\left(\begin{array}{cc}a& w\\ w& b\end{array}\right)$ used in step 1.2 was verified directly on this page in A two-level finite-gap model: exact eigenvalues, the second-order correction and a finite check of the cubic remainder bound; the present example re-derives it by the same substitution, so no dependency on the sibling example is claimed.
• Relation to the linear Stark effect. A degenerate truncation (for instance the four states with $n=2$) would instead be governed by Degenerate stationary perturbation: the perturbed eigenvalues stay within 4 lambda^2 v^2/g of the first-order shifts, and no branch relaxation is asserted and could produce a first-order shift; the present two-state nondegenerate truncation produces the quadratic shift, and no statement about a laboratory line is made in either case.

Adopt P1, P2, P4 and P6 for the massive spin-$s$ particle model of The massive spin-s particle model of the framework: state space, operators, units and frame readings (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, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), together with the effective magnetic-moment coupling Effective magnetic-moment coupling of the finite-spin model: gyromagnetic parameter, magnetic moment and Zeeman energy operator with real gyromagnetic parameter $\gamma :={\gamma }_1$ and the spin-one internal factor ${\mathbb{C}}^3$ with spin operators $S_i=\mathrm{\hslash }{J}_i$ of Finite spin matrices at $s=1$; read energies in joules as in Dimensions, SI unit conventions and quantity types of the framework. Declare the finite-level Zeeman model by the perturbed-family model of The declared perturbed-family model: bounded self-adjoint perturbation, isolated gapped label and followed spectral cluster with $\mathfrak{H}:={\mathbb{C}}^3,H_0:=\epsilon I_3,V:=-\gamma B_z{S}_3,$ where $\epsilon \in \mathbb{R}$ is a declared multiplet energy, $B_z>0$ is the field magnitude in tesla of the uniform field $\mathbf{B}=(0,0,B_z)$, so that the family is $H(\lambda )=H_0+\lambda V$ with real coupling $\lambda$ (dimensionless, equal to $1$ for the physical field) and $v=\mathrm{\parallel }V\mathrm{\parallel }=\mathrm{\mid }\gamma \mathrm{\mid }{B}_z\mathrm{\hslash }$. For the physical value $\lambda =1$ the family is the model's Zeeman operator on the internal multiplet. Then:

(a) the perturbation preserves the multiplet: $[H_0,V]=0$, the first-order operator is $V_E=V$, and its eigenvalues counted with multiplicity are $-\mathrm{\hslash }\gamma B_z,0,+\mathrm{\hslash }\gamma B_z$ in joules. If $\gamma \mathrm{\ne }0$, these are three distinct simple eigenvalues; if $\gamma =0$, the only eigenvalue is $0$ with multiplicity three. Thus the first-order operator describes a triply degenerate level when the adopted magnetic coupling vanishes;

(b) the exact energy eigenvalues of the family, counted with multiplicity, are $\epsilon -\lambda \mathrm{\hslash }\gamma B_z,\epsilon ,\epsilon +\lambda \mathrm{\hslash }\gamma B_z\mathrm{.}$ They are three distinct simple levels exactly when $\lambda \gamma \mathrm{\ne }0$; when $\lambda =0$ or $\gamma =0$, the only energy is $\epsilon$ with multiplicity three. Each eigenvalue is affine in $\lambda$, so its second-order and higher Taylor coefficients vanish identically;

(c) when $v>0$, the page's degenerate theorem applies for every $g>0$ (the gap condition is vacuous because the spectrum of $H_0$ is the single value $\epsilon$) and every real $\lambda$ with $\mathrm{\mid }\lambda \mathrm{\mid }v<g\mathrm{/}4$; each eigenvalue $z$ then satisfies $\mathrm{dist}(z,\epsilon +\lambda \mathrm{spec}(V_E))\le 4{\lambda }^2{v}^2\mathrm{/}g$, and the exact spectrum in (b) shows that this distance is $0$. If $v=0$, equivalently $\gamma =0$, then $H(\lambda )=\epsilon I_3$ for every real $\lambda$, so the same distance is $0$ directly, without invoking the theorem's positive-norm hypothesis. In both cases the first-order eigenvalue set is exact, with multiplicity retained as in (a)-(b);

(d) for a unit preparation vector $u\in {\mathbb{C}}^3$, put $u(t)=e^{-itH(\lambda )\mathrm{/}\mathrm{\hslash }}u$ and $s_i(t)=⟨S_{i}u(t),u(t)⟩$. With $\omega =\lambda \gamma B_z$ (in ${\mathrm{s}}^{-1}$), $s_3(t)=s_3(0),s_1(t)=\cos (\omega t)s_1(0)+\sin (\omega t)s_2(0),s_2(t)=\cos (\omega t)s_2(0)-\sin (\omega t)s_1(0)\mathrm{.}$ Thus the transverse means rotate by $-\omega t$, with the signed Larmor parameter $\omega$ equal to $\gamma B_z$ at $\lambda =1$;

(e) no branch selection: the energy eigenspaces of (b) are spectral data of the family; a preparation in any eigenstate is stationary, and a superposition evolves with the relative phases of (b). The example asserts no instantaneous transition of any preparation to the lowest sublevel, no relaxation and no measurement; which sublevel a preparation occupies is fixed by the preparation and by P4.

• Degenerate perturbation theory is exact here. The perturbation commutes with $H_0$ and acts inside the multiplet, so the first-order operator is the whole perturbation and there is no second-order shift; when $\lambda \gamma \mathrm{\ne }0$ it gives three distinct levels, while zero coupling leaves one triply degenerate level. The example exhibits the degenerate theorem's first-order content on a clean exactly solvable family. This is the Zeeman side of the page's examples; the Stark side uses a nondegenerate truncation with a genuine second-order shift.
• No relaxation. When distinct levels occur, they do not force a preparation to move to the lowest one; the affine spectral shifts, with degeneracy retained at zero coupling, are statements about the spectrum, in agreement with the corrected reading of the textbook degenerate-perturbation conjecture recorded in What the controlled-approximations page adopts, proves and does not claim: gaps, branch populations, turning points and the static-Stark limitation.

Adopt the declared one-dimensional line model The declared one-dimensional line model: state space, Hamiltonian, bound states and generalized stationary solutions and the WKB regime model The declared WKB regime model: nondimensionalised line equation, turning points, phase integrals and the declared barrier transmission reading with their framework postulates (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), with units as in Dimensions, SI unit conventions and quantity types of the framework. Then:

(a) Smooth-barrier semiclassical limit. Let $V,m,\mathrm{\hslash },\mathrm{\ell },E_{\star },E,W,\lambda ,h$ be data of the WKB regime model satisfying its hypothesis (W2), with exponent $S={\int }_{{\xi }_a}^{{\xi }_b}\sqrt{W-\lambda }d\xi$ and declared transmission coefficient $T(h)$. At fixed dimensionless barrier data $(W,\lambda )$, there are $h_0>0$, $0\le \epsilon (h)\le C_{1}h$ for $0<h<h_0$, and $0\le C_0<\mathrm{\infty }$, depending only on the fixed barrier data and matching sections, such that $\mid T(h)e^{2S\mathrm{/}h}-1\mid \le \frac{C_0[\epsilon (h)+e^{-2S\mathrm{/}h}]}{1-C_0[\epsilon (h)+e^{-2S\mathrm{/}h}]}$ for $0<h<h_0$ whenever the denominator is positive; in particular ${\lim }_{h\downarrow 0}T(h)e^{2S\mathrm{/}h}=1,$ and the physical reading of the limit is the exponential law $T\approx \exp (-\frac{2}{\mathrm{\hslash }}{\int }_{x_a}^{x_b}\sqrt{2m(V-E)}dx)$ with prefactor tending to one;

(b) Rectangular thick-barrier limit. Fix $m,\mathrm{\hslash },E,U$ with $U>E>0$ and vary the wall width $a\to \mathrm{\infty }$ (equivalently, $\kappa a\to \mathrm{\infty }$). For the rectangular wall model excluded by the smooth theorem, take the exact scattering solution of the line model with $V(x)=U$ for $0<x<a$ and $V=0$ otherwise, $U>E>0$, $k=\sqrt{2mE}\mathrm{/}\mathrm{\hslash }$, $\kappa =\sqrt{2m(U-E)}\mathrm{/}\mathrm{\hslash }$ and exact transmission $T_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}=[1+\frac{U^2{\sinh }^2(\kappa a)}{4E(U-E)}{]}^{-1};$ then, in this fixed-parameter limit, $\frac{T_{\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{t}}}{e^{-2\kappa a}}⟶\frac{16E(U-E)}{U^2}=\frac{16k^2{\kappa }^2}{(k^2+{\kappa }^2{)}^2},$ so in the rectangular model the exponential law holds with the profile-dependent prefactor $16E(U-E)\mathrm{/}{U}^2$; the exponent $2\kappa a$ equals the classical WKB exponent $2B\mathrm{/}\mathrm{\hslash }$ of the rectangular barrier, and the limit shows that the exponential alone does not determine the prefactor;

(c) Reading of the comparison. Statement (a) is a theorem about smooth compactly supported barriers with separated simple turning points, and its prefactor-one conclusion uses the Airy matching of that class; statement (b) is an exact model computation for a piecewise constant wall, whose sharp discontinuities are precisely why it is excluded from (a). Neither statement asserts a measured transmission, a detector response, a wave packet or an apparatus; both are conditional statements about the declared one-dimensional models.