Qm Harmonic Oscillator

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H$ be a complex Hilbert space (Hilbert space), let $(e_n{)}_{n\in \mathbb{N}}$ be a complete orthonormal family of $H$ (Orthonormal families, complete orthonormal systems and Hilbert bases), let $({\lambda }_n{)}_{n\in \mathbb{N}}\subseteq \mathbb{R}$, and let $A$ be the self-adjoint diagonal operator of Diagonal adjoint and graph-core lemma: $D(A)=\{x\in H:{\sum }_n{\lambda }_n^2\mathrm{\mid }⟨x,e_n⟩{\mathrm{\mid }}^2<\mathrm{\infty }\},Ax={\sum }_n{\lambda }_n⟨x,e_n⟩e_n\mathrm{.}$ Then:

(a) for every $\lambda \in \mathbb{R}$ the eigenspace of $A$ at $\lambda$ is the closed linear span of the basis vectors carrying the eigenvalue $\lambda$: $\ker (A-\lambda I)=\stackrel{}{\mathrm{span}}\{e_n:{\lambda }_n=\lambda \};$ in particular $\lambda$ is an eigenvalue of $A$ exactly when $\lambda ={\lambda }_n$ for some $n$;

(b) if ${\lambda }_m\mathrm{\ne }{\lambda }_n$ whenever $m\mathrm{\ne }n$, then $\ker (A-{\lambda }_{n}I)=\mathbb{C}{e}_n$ for every $n$: every eigenspace is one-dimensional with eigenvector $e_n$, the eigenspaces are pairwise orthogonal, and every nonzero eigenvector of $A$ is a nonzero multiple of some $e_n$;

(c) in the situation of (b) the linear span of all eigenspaces of $A$ is $\mathrm{span}\{e_n:n\in \mathbb{N}\}$, which is dense in $H$.

Assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $H,K$ be complex Hilbert spaces (Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument) and let $U:H\to K$ be a bijective linear map with $⟨Ux,Uy{⟩}_K=⟨x,y{⟩}_H\text{for all }x,y\in H\mathrm{.}$ Let $T$ be a self-adjoint operator on $H$ with spectral projection-valued measure $E_T$ on the Borel $\sigma$-algebra of $\mathbb{R}$ (Spectral theorem for unbounded self-adjoint operators (PVM form), Projection valued measure, Symmetric, self-adjoint and essentially self-adjoint operators), and define $D(T^{\mathrm{\prime }}):=UD(T),T^{\mathrm{\prime }}(Ux):=U(Tx)(x\in D(T))\mathrm{.}$ Then:

(a) $T^{\mathrm{\prime }}$ is well defined, linear, densely defined and self-adjoint on $K$;

(b) the assignment $E(B):=U{E}_T(B)U^{-1}$ on Borel sets $B\subseteq \mathbb{R}$ is a regular projection-valued measure on the Borel $\sigma$-algebra of $\mathbb{R}$, and it is the spectral projection-valued measure of $T^{\mathrm{\prime }}$;

(c) for every unit vector $x\in H$ the scalar laws of the two operators agree, ${\mu }_{Ux}^{T^{\mathrm{\prime }}}(B):=\mathrm{\parallel }E(B)Ux{\mathrm{\parallel }}_K^2=\mathrm{\parallel }{E}_T(B)x{\mathrm{\parallel }}_H^2=:{\mu }_x^T(B)(B\subseteq \mathbb{R}\text{ Borel});$

(d) $\sigma (T^{\mathrm{\prime }})=\sigma (T)$.

(a) Adopted framework assumptions. Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5, in the readings of The state space of the framework: preparations, pure rays and density operators, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and 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, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators). 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). Fix real model parameters $m>0$ (mass parameter, unit $\mathrm{k}\mathrm{g}$), $\omega >0$ (angular frequency, unit ${\mathrm{s}}^{-1}$) and the action constant $\mathrm{\hslash }>0$ (unit $\mathrm{J}\mathrm{s}$) of Dimensions, SI unit conventions and quantity types of the framework clause (d)4, and put $\mathrm{\ell }:=\sqrt{\frac{\mathrm{\hslash }}{m\omega }}>0\text{(unit }\mathrm{m}\text{),}$ the model's length scale.

(b) One-dimensional realization. Write ${\mathcal{H}}_1:=L^2(\mathbb{R},dx;\mathbb{C})$ for the model's one-dimensional state space, the complex Hilbert space of almost-everywhere classes with inner product linear in the first argument, taken in the scalar reading $r=1$ of the component-valued space of Componentwise position and momentum operators on the component-valued wave-function space for $d=1$. The model's position and momentum are the operators of that lemma at $d=1$, $r=1$, namely $Q:=M_x,D(Q)=\{f\in {\mathcal{H}}_1:xf\in L^2\},(Qf)(x)=xf(x),$ $P:=-i\mathrm{\hslash }{D}_x,D(P)=\{f\in {\mathcal{H}}_1:\text{the weak derivative }{D}_{x}f\text{ lies in }{L}^2\},Pf=-i\mathrm{\hslash }{D}_{x}f,$ which are densely defined and self-adjoint with exactly these domains; the model assigns to $(Q,\mathrm{m})$ and to $(P,\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1})$ the status of assigned sharp observables in the sense of Assigned sharp observables and the outcome law of the framework clause (a), with outcome laws read from P2.

(c) Energy and excitation-number observables. Let $g_0,(g_n{)}_{n\in \mathbb{N}}$ and the unitary $D:L^2(\mathbb{R},dy)\to L^2(\mathbb{R},dx)$ be the Hermite data and scaling map of Oscillator scaling and exact realization clauses (a), (b) for the parameters $m,\omega ,\mathrm{\hslash }$. Put ${\mathcal{P}}_y:=\{p{g}_0:p\in \mathbb{C}[y]\}\subseteq L^2(\mathbb{R},dy)$; the Hermite recurrence Hermite recurrence identifies this polynomial-Gaussian space with $\mathrm{span}\{g_n:n\in \mathbb{N}\}$, and its scaled image ${\mathcal{P}}_x:=D({\mathcal{P}}_y)=\mathrm{span}\{{\psi }_n:n\in \mathbb{N}\}$ is the core supplied by the scaling theorem. Let $A$ be the diagonal operator of Diagonal adjoint and graph-core lemma for the family $(g_n)$ and eigenvalues $2n+1$. The model's energy operator is $H_1:=\frac{\mathrm{\hslash }\omega }{2}DA{D}^{-1},D(H_1):=\{f\in {\mathcal{H}}_1:D^{-1}f\in D(A)\},$ the operator of Oscillator scaling and exact realization clause (b); its exact domain, action, spectrum and eigenbasis are those established there. The model assigns to $(H_1,\mathrm{J})$ the status of assigned energy observable of the isolated system in the sense of Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (a) and (b): the model's flow is $U(t)=\exp (-it{H}_1\mathrm{/}\mathrm{\hslash })$ and the model's Schroedinger dynamics is the one of that definition. Let $N$ be the self-adjoint number operator on $L^2(\mathbb{R},dy)$ of Closed ladder domains for the Hermite family $(g_n)$, and set $N_{\psi }:=DN{D}^{-1}$. Its transported domain and action are $D(N_{\psi })=\{f\in {\mathcal{H}}_1:{\sum }_n{n}^2\mathrm{\mid }⟨f,{\psi }_n⟩{\mathrm{\mid }}^2<\mathrm{\infty }\},N_{\psi }f={\sum }_{n}n⟨f,{\psi }_n⟩{\psi }_n,$ with the series converging in ${\mathcal{H}}_1$; it is self-adjoint by Transport of spectral projection-valued measures along isometric bijections clauses (a), (d). The model assigns to $(N_{\psi },1)$ the status of its dimensionless excitation-number observable; here $1$ is the dimensionless unit, and the spectral coordinate $n$ records the number of excitations, and its outcome laws are read by P2. On the scaled polynomial-Gaussian core ${\mathcal{P}}_x$ the operator $H_1$ is the self-adjoint realization of the quadratic expression: for every $f\in {\mathcal{P}}_x$, $H_{1}f=-\frac{{\mathrm{\hslash }}^2}{2m}{f}^{\mathrm{\prime }\mathrm{\prime }}+\frac{m{\omega }^2}{2}{x}^{2}f$ pointwise, and for every $f\in D(H_1)$ the weak identity $\int (H_{1}f)\phi dx=\int f(-\frac{{\mathrm{\hslash }}^2}{2m}{\phi }^{\mathrm{\prime }\mathrm{\prime }}+\frac{m{\omega }^2}{2}{x}^2\phi )dx$ holds for every $\phi \in C_c^{\mathrm{\infty }}(\mathbb{R})$ (Oscillator scaling and exact realization clauses (c), (d)). The formal differential expression $-\frac{{\mathrm{\hslash }}^2}{2m}\frac{d^2}{d{x}^2}+\frac{m{\omega }^2}{2}{x}^2$ is the model's quadratic potential energy plus kinetic energy reading; no identification of $D(H_1)$ with any independently defined maximal differential-operator domain, and no separate quadratic-form-domain characterization, is asserted.

(d) Three-dimensional isotropic realization. For the spinless reading $s=0$ of the massive-particle model of The massive spin-s particle model of the framework: state space, operators, units and frame readings, with state space ${\mathcal{H}}_3:=L^2({\mathbb{R}}^3,dx;\mathbb{C})$, position operators $Q_j$ and momentum operators $P_k$ ($j,k\in \{1,2,3\}$) with the domains and units of that definition, the model with common frequency $\omega$ has energy operator $H_3$ equal to the diagonal operator $H$ of Multidimensional oscillator completeness clause (c) for $d=3$, $m_j=m$, ${\omega }_j=\omega$, a self-adjoint operator on ${\mathcal{H}}_3$ with the domain, spectrum and core displayed there. The model assigns to $(H_3,\mathrm{J})$ the status of assigned energy observable in the reading of clause (c); its eigenbasis and the identity $H_3={\sum }_{j=1}^3(-\frac{{\mathrm{\hslash }}^2}{2m}{\mathrm{\partial }}_{x_j}^2+\frac{m{\omega }^2}{2}{x}_j^2)$ on the core are those of that theorem, and the one-dimensional realization of clauses (b) and (c) is the coordinate specialization of this one.

(e) Physical reading and scope. Clause (c) defines the model quantities that the page calls the energy and excitation number; it is a model assumption that the energy of the declared isolated particle is the assigned observable $(H_1,\mathrm{J})$ and that $(N_{\psi },1)$ is its excitation-number observable. The additive constant of the potential energy is removed at the minimum: for a one-dimensional potential-energy function $V:\mathbb{R}\to \mathbb{R}$ of class $C^2$, with values in joules for position $x$ in metres, suppose $V$ has a non-degenerate local minimum at $0$, meaning $V^{\mathrm{\prime }}(0)=0$ and $V^{\mathrm{\prime }\mathrm{\prime }}(0)>0$. Taylor's theorem gives $V(x)-V(0)=\frac{1}{2}{V}^{\mathrm{\prime }\mathrm{\prime }}(0)x^2+o(x^2)$ as $x\to 0$; the oscillator model approximates this shifted potential-energy function by its quadratic term and sets $m{\omega }^2=V^{\mathrm{\prime }\mathrm{\prime }}(0)$. The oscillator's quantum ground energy in this convention is $\mathrm{\hslash }\omega \mathrm{/}2$. With only the stated $C^2$ regularity, the remainder is $o(x^2)$; no quantitative finite-amplitude error bound is asserted on this page (Binney §3.1, cited in the references). No measured value, no laboratory identification and no derivation of $m$ or $\omega$ from any microscopic or environmental theory is asserted. The model contains no damping, no external driving, no time-dependent parameters, no coupling to electromagnetic fields or radiation, no spin-dependent force, no anharmonic term, no relativistic correction and no boundary or many-body effect; those are separate model assumptions belonging to their own items. The parameters $m$ and $\omega$ are model parameters of dimensions $\mu$ and ${\tau }^{-1}$ in the conventions of Dimensions, SI unit conventions and quantity types of the framework, and all displayed quantities are read in the units fixed there.

(f) Status. Clauses (a)-(e) fix the objects used by the theorems of this page and record the readings of P1, P2, P4 and P5 in the model's terminology. They assert no preparation or apparatus model, no detection model, no joint outcome law of position and momentum and no empirical result.

• Why the model is fixed through the scaling theorem. The operator $H_1$ is not introduced as "the differential expression with some self-adjoint extension": it is the unique self-adjoint extension of the symmetric operator on the polynomial-Gaussian core that is fixed outright by Oscillator scaling and exact realization. The weak identity of clause (c) then records that this operator is the realization of the quadratic expression in the sense of distributions, without claiming that its domain coincides with any maximally defined differential domain.
• The two realizations are separate adoptions. Clause (b) declares a particle confined to a line; P5's spacetime arena is $\mathbb{R}\times {\mathbb{R}}^3$, so the one-dimensional model is an additional idealization, not a restriction of P5. Clause (d) is the isotropic three-dimensional model of P5 at $s=0$. Later items of this page state their hypotheses for one or the other realization explicitly; results about the one-dimensional realization are not transferred to the three-dimensional one or conversely, except where the multidimensional theorem supplies the identification.
• Units. The quantities $\mathrm{\ell }$, $Q$, $P$, $H_1$, $H_3$, $m$, $\omega$ and $\mathrm{\hslash }$ are read as dimensioned model quantities: $[\mathrm{\ell }]=\mathrm{\ell }$, $[Q]=\mathrm{\ell }$, $[P]=\mu \mathrm{\ell }{\tau }^{-1}$, $[H]=\mu {\mathrm{\ell }}^2{\tau }^{-2}$, $[m]=\mu$, $[\omega ]={\tau }^{-1}$ and $[\mathrm{\hslash }]=\mu {\mathrm{\ell }}^2{\tau }^{-1}$. The excitation number $N_{\psi }$ is dimensionless and is read in the declared unit $1$; $V(0)$ is subtracted only as a potential-energy offset, and the quantum energy eigenvalues $\mathrm{\hslash }\omega (n+\frac{1}{2})$ are read in joules.

Adopt the state and composition postulate P1, the observable and probability postulate P2, the autonomous-dynamics postulate P4 and the kinematics and massive-particle model postulate P5, read through the oscillator model of The quadratic-potential oscillator model of the framework with parameters $m>0$, $\omega >0$ and action constant $\mathrm{\hslash }>0$, and 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). Let $\mathrm{\ell }=\sqrt{\mathrm{\hslash }\mathrm{/}(m\omega )}$, let $(g_n)$, $D$, $A$, $({\psi }_n)$ and $H_1$ be the Hermite family, the unitary, the dimensionless diagonal operator, the scaled eigenbasis and the energy operator of Oscillator scaling and exact realization clauses (a) and (b), so that ${\psi }_n=D{g}_n$ and $(H_1,\mathrm{J})$ is the model's one-dimensional assigned energy observable of the model definition clause (c). Put $N_{\psi }:=DN{D}^{-1},a_{\psi }:=Da{D}^{-1},a_{\psi }^{†}:=D{a}^{†}{D}^{-1},$ where $N$ is the number operator of Closed ladder domains and $N_{\psi }$ is the assigned excitation-number observable with unit $1$ of the model definition clause (c); these are the number and ladder operators transported from that lemma, with domains $D(N_{\psi })=\{Dx:x\in D(N)\}$, $D(a_{\psi })=\{Dx:x\in D(a)\}$ and $D(a_{\psi }^{†})=\{Dx:x\in D(a^{†})\}$. Write $E_n:=\mathrm{\hslash }\omega (n+\frac{1}{2})$ and, for the three-dimensional realization of the model definition clause (d), let $(H_3,\mathrm{J})$ and $({\psi }_N{)}_{N\in {\mathbb{N}}^3}$ be its assigned energy observable and its product eigenbasis. Then:

(a) Spectrum, eigenbasis and simplicity. $D(H_1)=\{f\in {\mathcal{H}}_1:{\sum }_n{n}^2\mathrm{\mid }⟨f,{\psi }_n⟩{\mathrm{\mid }}^2<\mathrm{\infty }\}$, the family $({\psi }_n)$ is a complete orthonormal family of ${\mathcal{H}}_1$ and its algebraic span is a core for $H_1$, $H_1{\psi }_n=E_n{\psi }_n$, and the spectrum and eigenspaces are $\sigma (H_1)=\{E_n:n\in \mathbb{N}\},\ker (H_1-E_{n}I)=\mathbb{C}{\psi }_n(n\in \mathbb{N});$ the ground energy is $E_0=\mathrm{\hslash }\omega \mathrm{/}2$ and the gap between consecutive levels is $E_{n+1}-E_n=\mathrm{\hslash }\omega$;

(b) Excitation reading. $H_1=\mathrm{\hslash }\omega (N_{\psi }+\frac{1}{2}I)$ with common domain $D(H_1)=D(N_{\psi })$; $N_{\psi }$ is self-adjoint with spectrum $\mathbb{N}$ and eigenbasis $({\psi }_n)$; $a_{\psi }$ and $a_{\psi }^{†}$ are closed with $a_{\psi }^{\ast }=a_{\psi }^{†}$, and on the eigenbasis $a_{\psi }{\psi }_n=\sqrt{n}{\psi }_{n-1}(n\ge 1),a_{\psi }{\psi }_0=0,a_{\psi }^{†}{\psi }_n=\sqrt{n+1}{\psi }_{n+1},N_{\psi }{\psi }_n=n{\psi }_n;$ the eigenspace of $N_{\psi }$ at each $n\in \mathbb{N}$ is exactly $\mathbb{C}{\psi }_n$, so its nonzero eigenvectors are precisely the nonzero scalar multiples of the ${\psi }_n$; the outcome law of $N_{\psi }$ in the pure preparation of ${\psi }_n$ is the point mass ${\delta }_n$;

(c) Energy outcome law. For every unit $f\in {\mathcal{H}}_1$ the outcome law of the assigned energy observable $(H_1,\mathrm{J})$ in the pure preparation of $f$ is purely atomic with ${\mu }_f^{H_1}(\{E_n\})=\mathrm{\mid }⟨f,{\psi }_n⟩{\mathrm{\mid }}^2(n\in \mathbb{N}),{\mu }_f^{H_1}(B)={\sum }_{n:E_n\in B}\mathrm{\mid }⟨f,{\psi }_n⟩{\mathrm{\mid }}^2$ for every Borel set $B\subseteq \mathbb{R}$, and ${\mu }_f^{H_1}(\sigma (H_1))=1$; in particular the law has no mass outside $\{E_n:n\in \mathbb{N}\}$, and $E_n$ is an atom of the law of $f$ exactly when $⟨f,{\psi }_n⟩\mathrm{\ne }0$;

(d) Three-dimensional isotropic realization. The energy observable $(H_3,\mathrm{J})$ of the model definition clause (d) has spectrum $\sigma (H_3)=\{\mathrm{\hslash }\omega (M+\frac{3}{2}):M\in \mathbb{N}\}$; for every $M\in \mathbb{N}$ the eigenspace $\ker (H_3-\mathrm{\hslash }\omega (M+\frac{3}{2})I)$ has the orthonormal basis $\{{\psi }_N:N\in {\mathbb{N}}^3,n_1+n_2+n_3=M\}$ and dimension $(M+1)(M+2)\mathrm{/}2$; the ground energy is $\frac{3}{2}\mathrm{\hslash }\omega$;