Qm Light Matter And Spectral Transitions — Examples

Adopt the prescribed dipole-drive model The declared prescribed dipole-drive model and its long-wavelength regime with the following realized two-level matter data. For any declared ${\omega }_0>0$ (${\mathrm{s}}^{-1}$) and $d>0$ ($\mathrm{C}\mathrm{m}$), use the positive framework constant ${\epsilon }_0$ of The declared Coulomb and two-body hydrogen model: one-centre operator, relative problem, reduced mass and scope exclusions clause (d) and define $q:={\left(\frac{8\sqrt{2}{\pi }^{3\mathrm{/}2}{\epsilon }_0\mathrm{\hslash }{\omega }_{0}d}{3}\right)}^{1\mathrm{/}3},a:=\frac{\sqrt{2}d}{q},\mu :=\frac{9\mathrm{\hslash }}{10{\omega }_0{a}^2},\kappa :=\frac{q^2}{4\pi {\epsilon }_0}\mathrm{.}$ These have units $\mathrm{C}$, $\mathrm{m}$, $\mathrm{k}\mathrm{g}$ and $\mathrm{J}\mathrm{m}$, respectively. Declare the charged pair of Static finite-level electric-dipole coupling: compressed dipole of the declared two-body Coulomb model in a uniform field with $m_1=m_2=2\mu$, charges $-q,+q$ and relative-coordinate energy $H_{\kappa }=-({\mathrm{\hslash }}^2\mathrm{/}(2\mu )){\mathrm{\Delta }}_w-\kappa \mathrm{/}\mathrm{\mid }x\mathrm{\mid }$ on $H^2({\mathbb{R}}^3;\mathbb{C})$; ${\mathrm{\Delta }}_w$ is the sum of the three second weak coordinate derivatives. Its dipoles are $D_j=-q{Q}_j$, $Q_{j}f(x)=x_{j}f(x)$ on $D(Q_j)=\{f\in L^2:x_{j}f\in L^2\}$. Here $L^2=L^2({\mathbb{R}}^3;\mathbb{C})$ consists of Lebesgue square-integrable classes, and $H^2$ consists of those classes whose weak coordinate derivatives through order two also belong to $L^2$: $D^{\alpha }f$ is a weak derivative when $\int f{\mathrm{\partial }}^{\alpha }\phi =(-1{)}^{\mathrm{\mid }\alpha \mathrm{\mid }}\int D^{\alpha }f\phi$ for every smooth compactly supported test function $\phi :{\mathbb{R}}^3\to \mathbb{C}$. Here $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)\in {\mathbb{N}}_0^3$, $\mathrm{\mid }\alpha \mathrm{\mid }={\alpha }_1+{\alpha }_2+{\alpha }_3$ and ${\mathrm{\partial }}^{\alpha }$ denotes that sequence of classical coordinate derivatives. In this relative-coordinate space put $g_a(x):=(\pi a^2{)}^{-3\mathrm{/}4}{e}^{-\mathrm{\mid }x{\mathrm{\mid }}^2\mathrm{/}(2a^2)},w_{\downarrow }:=g_a,w_{\uparrow }:=-\frac{\sqrt{2}{x}_1}{a}{g}_a,W:=\mathrm{span}\{w_{\uparrow },w_{\downarrow }\}\mathrm{.}$ The verification below proves that this is an orthonormal pair in $D(H_{\kappa })\cap {\bigcap }_{j=1}^{3}D(Q_j)$ and that the actual compressions $H_a={\mathrm{\Pi }}_W{H}_{\kappa }{\mathrm{\mid }}_W$, $D_j^W={\mathrm{\Pi }}_W{D}_j{\mathrm{\mid }}_W$ have the following matrices. Identify $(w_{\uparrow },w_{\downarrow })$ with the coordinate basis $(e_{\uparrow },e_{\downarrow })$ of ${\mathbb{C}}^2$, using the Pauli matrices $({\sigma }_x,{\sigma }_y,{\sigma }_z)$ of Finite spin matrices clause (e): $H_a:=\frac{\mathrm{\hslash }{\omega }_0}{2}{\sigma }_z,D_1^W:=d{\sigma }_x,D_2^W:=D_3^W:=0,$ where ${\omega }_0,d$ are the arbitrary positive parameters fixed above. The two vectors are eigenvectors of the compression, not asserted to be Coulomb eigenfunctions; invariance of $W$ under the uncompressed operators is unnecessary. Declare the field ${\mathbf{E}}_{\mathrm{e}\mathrm{x}\mathrm{t}}(r):=E_0\cos (\omega r)e_1,r\in I=[0,\tau ],$ with $E_0>0$ in $\mathrm{V}{\mathrm{m}}^{-1}$ and $\omega >0$ in ${\mathrm{s}}^{-1}$, and $\tau >0$ in seconds; declare $L>0$ in metres, $x_0=0\in {\mathbb{R}}^3$, $k=0\in {\mathbb{R}}^3$ in inverse metres, and a dimensionless phase tolerance ${\delta }_{\mathrm{p}\mathrm{h}}=1\mathrm{/}10$. Use the spatially uniform template ${\mathbf{E}}^{\mathrm{s}\mathrm{p}}(x,r)=E_0\cos (\omega r)e_1$ of the dipole-drive definition (d), so $\mathrm{\mid }k\mathrm{\mid }L=0\le {\delta }_{\mathrm{p}\mathrm{h}}$ for every $L$. These are declared prescribed-field data, with no Maxwell dispersion relation; the temporal frequency $\omega >0$ does not determine $k$ in this model. Let $v_{\uparrow }:=e_{\uparrow }$, $v_{\downarrow }:=e_{\downarrow }$, so that $E_{\uparrow }=\mathrm{\hslash }{\omega }_0\mathrm{/}2$, $E_{\downarrow }=-\mathrm{\hslash }{\omega }_0\mathrm{/}2$ and $\mathrm{\Delta }=E_{\uparrow }-E_{\downarrow }=\mathrm{\hslash }{\omega }_0$. Write $\delta :={\omega }_0-\omega$ for the detuning frequency, distinct from the energy $\mathrm{\Delta }$. Then:

(a) (identification) $H_{\mathrm{d}}(r)=\frac{\mathrm{\hslash }}{2}({\omega }_0{\sigma }_z+{\mathrm{\Omega }}_1\cos (\omega r){\sigma }_x)$ with ${\mathrm{\Omega }}_1=-\frac{2d{E}_0}{\mathrm{\hslash }},$ so the model is exactly the declared linear two-level drive model of clause (f) of The declared driven model: prescribed time-dependent energy families, propagator evolution and the two-level drive models with splitting frequency ${\omega }_0$ and transverse coupling ${\mathrm{\Omega }}_1$;

(b) (resonance and first-order coefficient) at resonance $\omega ={\omega }_0$ the co-rotating first-order amplitude of Stimulated dipole-drive transitions of the finite-level model: first-order amplitudes, equal emission and absorption coefficients, and the exact two-level identification clause (b) for the transition $e_{\downarrow }\to e_{\uparrow }$ from time $0$ to time $\tau$ has modulus $\mathrm{\mid }{a}_{+}\mathrm{\mid }=\mid ⟨V_0{e}_{\downarrow },e_{\uparrow }⟩\mid \frac{\tau }{2\mathrm{\hslash }}=\frac{d{E}_0\tau }{2\mathrm{\hslash }},V_0=-d{E}_0{\sigma }_x,$ and the co-rotating component coefficient is ${\gamma }_{+}=\mathrm{\mid }{a}_{+}{\mathrm{\mid }}^2=(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }){)}^2$; the counter-rotating term is bounded by $\mathrm{\mid }⟨V_0{e}_{\downarrow },e_{\uparrow }⟩\mathrm{\mid }\mathrm{/}(\mathrm{\Delta }+\mathrm{\hslash }\omega )=d{E}_0\mathrm{/}(2\mathrm{\hslash }\omega )$ at resonance. The full first-order interaction-picture amplitude is $A_1=\frac{id{E}_0}{2\mathrm{\hslash }}[\tau +\frac{e^{2i\omega \tau }-1}{2i\omega }],{\gamma }_1:=\mathrm{\mid }{A}_1{\mathrm{\mid }}^2,$ including interference of both components; ${\gamma }_1$ is an unnormalized first-order approximation, not a normalized-state Born law;

(c) (comparison with the exact averaged Rabi probability) by The linearly driven two-level model: rotating-wave approximation with an explicit finite-time error contract and the averaged Rabi populations clause (c) the averaged down-population of the initial up-state at resonance is $p_{\downarrow }^{\mathrm{a}\mathrm{v}\mathrm{g}}(\tau )={\sin }^2(\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }\tau \mathrm{/}4)={\sin }^2(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }))$, whose small-$\tau$ expansion begins with $(d{E}_0\tau \mathrm{/}(2\mathrm{\hslash }){)}^2$; the co-rotating coefficient ${\gamma }_{+}$ of clause (b) is therefore the leading term of the exact averaged Rabi probability, and the rotating-wave error contract of the cited item bounds the difference between the exact linear-model population and this averaged probability by $2\epsilon (\tau )$ with $\epsilon (\tau )=\frac{\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{2\omega }[1+\tau (\frac{\mathrm{\mid }\delta \mathrm{\mid }}{2}+\frac{3\mathrm{\mid }{\mathrm{\Omega }}_1\mathrm{\mid }}{4})]$;

(d) (scope) the example is a conditional computation in the declared two-level drive model. The identification of a laboratory two-level atom, molecule or field with the declared data is a tested identification with uncertainty under P7; the model has no quantized field, no vacuum, no spontaneous emission, no damping, no measurement record and no empirical value, and the exact circular and linear families of the driven-dynamics page are models with their own scope.

• First order versus exact. The co-rotating component coefficient of clause (b) is the leading term of the exact averaged Rabi oscillation of clause (c) only at small $\tau$; the full linear-drive first-order coefficient is ${\gamma }_1=\mathrm{\mid }{a}_{+}+a_{-}{\mathrm{\mid }}^2$ and has four times that leading small-time coefficient, so no averaging of the full model is silently assumed; the exact linear model differs from the averaged model by the rotating-wave error, and the exact circular model of the driven-dynamics page is a different declared family, not an approximation proved here.
• No spontaneous emission. The example is stimulated: the field is prescribed and unaffected by the transition, and the model contains no vacuum. Spontaneous emission is treated only in the quantized cutoff dipole model of this pair, under the additional vacuum-preparation assumption of the postulate item.

Adopt the declared Coulomb pair and the model dipole components $D_q=-q_{\mathrm{e}\mathrm{l}}{Q}_q$ with $q_{\mathrm{e}\mathrm{l}}>0$ the charge magnitude with $Q_0=x_3$ and $Q_{\pm }=(x_1\pm i{x}_2)\mathrm{/}\sqrt{2}$ of Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence, and use the explicit spherical harmonics of Associated Legendre function normalization in the angle coordinates $\mathrm{\Phi }(\theta ,\phi )=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$ of Sphere integrals in angle coordinates, with $P_0=1$, $P_1(t)=t$ of Legendre Rodrigues equation: $Y_{00}=\frac{1}{\sqrt{4\pi }},Y_{10}=\sqrt{\frac{3}{4\pi }}\cos \theta ,Y_{1,\pm 1}=\mp \sqrt{\frac{3}{8\pi }}\sin \theta e^{\pm i\phi },$ so that $Q_0=\cos \theta$ and $Q_{\pm }=\sin \theta e^{\pm i\phi }\mathrm{/}\sqrt{2}$ on the chart. Then, with the inner product of $L^2(S^2,\sigma ;\mathbb{C})$ linear in the first argument:

(a) (allowed $\pi$ entry) $⟨Y_{10},Q_0{Y}_{00}⟩={\displaystyle \frac{1}{\sqrt{3}}}$ and $⟨Y_{00},Q_0{Y}_{10}⟩={\displaystyle \frac{1}{\sqrt{3}}}$;

(b) (allowed ${\sigma }^{\pm }$ entries) $⟨Y_{1,\pm 1},Q_{\pm }{Y}_{00}⟩=\mp {\displaystyle \frac{1}{\sqrt{3}}}$ and $⟨Y_{00},Q_{\pm }{Y}_{1,\mp 1}⟩=\pm {\displaystyle \frac{1}{\sqrt{3}}}$, all of modulus ${\displaystyle \frac{1}{\sqrt{3}}}$;

(c) (forbidden entries) $⟨Y_{00},Q_0{Y}_{00}⟩=0$, $⟨Y_{00},Q_{+}{Y}_{10}⟩=0$, $⟨Y_{00},Q_0{Y}_{11}⟩=0$, $⟨Y_{1,\pm 1},Q_{\mp }{Y}_{00}⟩=0$ and $⟨Y_{1,\pm 1},Q_{\pm }{Y}_{1,\pm 1}⟩=0$; each vanishing is the case $l^{\mathrm{\prime }}=l$ or $m^{\mathrm{\prime }}\mathrm{\ne }m+q$ of the selection rules of Electric-dipole selection rules of the declared hydrogenic finite-level model: necessary angular conditions and the parity (Laporte) consequence clause (b);

(d) (parity check) the nonzero entries of (a),(b) connect $l=0$ with $l^{\mathrm{\prime }}=1$, hence states of opposite parity, while the vanishing entries of (c) include the parity-preserving pair $l=l^{\mathrm{\prime }}=0$ and $l=l^{\mathrm{\prime }}=1$; this is the Laporte condition of clause (d) of the selection-rules theorem;

(e) (reading in the hydrogenic model) applied to the model's uncompressed bound-state matrix elements, the entries of (a),(b) give the angular factors of $⟨{\psi }_{n^{\mathrm{\prime }}1m^{\mathrm{\prime }}},D_q{\psi }_{n00}⟩=-q_{\mathrm{e}\mathrm{l}}\cdot (\text{radial factor})\cdot (\text{the computed number})$, and the entries of (c) show that the angular factor of $⟨{\psi }_{n^{\mathrm{\prime }}00},D_0{\psi }_{n00}⟩$ vanishes for every $n,n^{\mathrm{\prime }}$; the radial factors are not evaluated, so no transition probability or intensity is computed;

(f) (oscillator contrast) on the declared scaled oscillator family $({\psi }_n^{\mathrm{o}\mathrm{s}\mathrm{c}})$ of Atomic dipole matrix-element domain clause (d) with scale $\mathrm{\ell }>0$, the position matrix elements are $⟨{\psi }_1^{\mathrm{o}\mathrm{s}\mathrm{c}},X{\psi }_0^{\mathrm{o}\mathrm{s}\mathrm{c}}⟩=\mathrm{\ell }\mathrm{/}\sqrt{2}$ and $⟨{\psi }_2^{\mathrm{o}\mathrm{s}\mathrm{c}},X{\psi }_0^{\mathrm{o}\mathrm{s}\mathrm{c}}⟩=0$, so the oscillator's position selection rule is $\mathrm{\mid }\mathrm{\Delta }n\mathrm{\mid }=1$ rather than a rule on an angular quantum number;

(g) (scope) the example is a conditional computation of angular factors of declared model quantities. No rate, oscillator strength, line intensity, observed spectrum or laboratory identification is asserted, and no radial integral is evaluated.

• Angular nonvanishing is not transition occurrence. The 1s-to-2p angular factor $1\mathrm{/}\sqrt{3}$ is nonzero, but the matrix element also carries the radial factor, which this example does not compute; conversely the vanishing of the 1s-to-2s angular factor makes that electric-dipole matrix element vanish in the declared model, without asserting that no slower mechanism connects the levels.
• Two different selection rules. The hydrogenic rule constrains the angular quantum numbers; the oscillator rule constrains the excitation number. Both are properties of the respective declared model's matrix elements.

Adopt the declared finite-level cutoff dipole model The declared finite-level matter model and its cutoff dipole coupling to the transverse radiation sector and its spectral-transition reading Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection, the declared ideal energy-window detection model The declared ideal counting, energy-window and analyser detection models of the radiation sector clause (c) and the preparation and apparatus discipline P7 (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions). Assume the Axioms of Countable Choice and of Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice). Declare three models, all with the same matter data $H_a$, $D_j^{\mathrm{c}\mathrm{h}}$ and the same cutoff $\mathrm{\Lambda }$ and coupling ${\eta }_0$, and with a declared upward level pair $(a,b)$, $\mathrm{\Delta }\in (0,\mathrm{\Lambda })$:

1. model $M_1$: cutoff mode data whose single-vertex photon density is ${\nu }_b^{(1)}=J_1\lambda$ with a declared nonnegative Lebesgue-measurable integrable representative $J_1:\mathbb{R}\to [0,\mathrm{\infty })$ of its $L^1(\lambda )$ class, $J_1=0$ $\lambda$-almost everywhere off $(0,\mathrm{\Lambda }]$ and $J_1$ continuous at $\mathrm{\Delta }$ with $J_1(\mathrm{\Delta })>0$;
2. model $M_2$: same matter and cutoff data, single-vertex photon density ${\nu }_b^{(2)}=J_2\lambda$, with a declared representative $J_2:\mathbb{R}\to [0,\mathrm{\infty })$ having the same properties and $J_2(\mathrm{\Delta })=J_1(\mathrm{\Delta })$;
3. model $L$ (the declared ideal sharp-line model): a declared spectral model in which the energy of an emitted and detected quantum is the value $\mathrm{\Delta }$ with probability one, so that every energy window $B\subseteq \mathbb{R}$ has probability ${\mathbf{1}}_B(\mathrm{\Delta })$.

Assume in addition that $J_1$ and $J_2$ are not proportional: there is no $\gamma \ge 0$ with $J_1=\gamma J_2$ $\lambda$-almost everywhere. The single-vertex laws below are the formal normalized spectral laws of the cited definition for all real ${\eta }_0$. Their reading as normalized one-photon preparations uses ${\eta }_0\mathrm{\ne }0$; at zero coupling the physical single-vertex vector is zero and no such preparation is formed. No zero-coupling conditioning is asserted. Then:

(a) (equal rate coefficients) the model rate coefficients of clause (c) of The golden-rule limit of the single-vertex emission coefficient and its reading as a model rate coefficient coincide, ${\mathrm{\Gamma }}_1=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_1(\mathrm{\Delta })=\frac{2\pi {\eta }_0^2}{\mathrm{\hslash }}{J}_2(\mathrm{\Delta })={\mathrm{\Gamma }}_2;$ in particular a comparison of the leading rate coefficients alone does not discriminate $M_1$ from $M_2$;

(b) (different formal laws and conditional detected-energy reading) for every $t>0$ the normalised single-vertex photon-energy laws $P_t^{(1)}$ and $P_t^{(2)}$ of clause (e) of Spectral-transition readings of the coupled model: dipole coefficients, the single-vertex photon density and ideal energy-window detection differ: there is a Borel set $B\subseteq \mathbb{R}$ with $P_t^{(1)}(B)\mathrm{\ne }{P}_t^{(2)}(B)$; when ${\eta }_0\mathrm{\ne }0$, these are distinct predicted ideal energy-window probabilities in the declared normalized single-vertex one-photon preparations. This is a distinction between specified model laws, with no finite-sample discrimination guarantee; at ${\eta }_0=0$ only the formal laws are compared;

(c) (the sharp line is a limit, not a finite-time property) if $J_1>0$ $\lambda$-almost everywhere on a Borel set $B$ of positive Lebesgue measure with $\mathrm{\Delta }\mathrm{\notin }B$, then $P_t^{(1)}(B)>0$ for every $t>0$ whereas $L$ assigns probability $0$ to $B$; on the other hand $P_t^{(1)}(B)\to 0$ as $t\to \mathrm{\infty }$ for every Borel $B$ bounded away from $\mathrm{\Delta }$. Thus the declared sharp-line model is the idealisation that the single-vertex law approaches in the limit, not a property of the finite-time law;

(d) (what the comparison is) clauses (a)-(c) are statements about the three declared models. They construct no laboratory comparison and report no data: no laboratory apparatus or preparation protocol, calibration, detector response, efficiency, background, trial count, frequency or uncertainty budget is supplied, and none is invented. Comparing any of these predictions with reported data would test the combined model and apparatus assumptions under P7, would require a declared sampling and measurement model, and would carry finite-sample fluctuations and systematic uncertainties; it would not prove a model.