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The energy observable of the oscillator model: spectrum, eigenbasis and outcome law

Physical scope

The energy and excitation-number observables of the quadratic-potential oscillator model: for the one-dimensional realization with parameters m>0, omega>0 and hbar>0 the exact energy eigenvalue list hbar omega (n+1/2) under the convention that subtracts the potential-energy value V(0), with quantum ground energy hbar omega/2, one-dimensional eigenspaces spanned by the Hermite eigenbasis, completeness of that basis, the energy gap hbar omega, the assigned dimensionless excitation-number observable in unit 1 and the atomic outcome laws; and for the isotropic three-dimensional realization of the spinless massive particle the energy spectrum hbar omega (M+3/2) with multiplicity (M+1)(M+2)/2. The statement is a conditional consequence of the postulates P1, P2, P4 and P5, of the adopted oscillator model and of the cited mathematics; it asserts no measured spectral line, no value of m, omega or hbar and no empirical result.

Statement

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 $m > 0$ , $ω > 0$ and action constant $ℏ > 0$ , and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Let $ℓ = \sqrt{ℏ / (m ω)}$ , let $(g_{n})$ , $D$ , $A$ , $(ψ_{n})$ 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 $ψ_{n} = D g_{n}$ and $(H_{1}, J)$ is the model's one-dimensional assigned energy observable of the model definition clause (c). Put $N_{ψ} : = D N D^{- 1}, a_{ψ} : = D a D^{- 1}, a_{ψ}^{†} : = D a^{†} D^{- 1},$ where $N$ is the number operator of Closed ladder domains and $N_{ψ}$ 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 $D (N_{ψ}) = {D x : x \in D (N)}$ , $D (a_{ψ}) = {D x : x \in D (a)}$ and $D (a_{ψ}^{†}) = {D x : x \in D (a^{†})}$ . Write $E_{n} : = ℏ ω (n + \frac{1}{2})$ and, for the three-dimensional realization of the model definition clause (d), let $(H_{3}, J)$ and $(ψ_{N})_{N \in N^{3}}$ be its assigned energy observable and its product eigenbasis. Then:

(a) Spectrum, eigenbasis and simplicity. $D (H_{1}) = {f \in H_{1} : \sum_{n} n^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty}$ , the family $(ψ_{n})$ is a complete orthonormal family of $H_{1}$ and its algebraic span is a core for $H_{1}$ , $H_{1} ψ_{n} = E_{n} ψ_{n}$ , and the spectrum and eigenspaces are $σ (H_{1}) = {E_{n} : n \in N}, \ker (H_{1} - E_{n} I) = C ψ_{n} (n \in N);$ the ground energy is $E_{0} = ℏ ω / 2$ and the gap between consecutive levels is $E_{n + 1} - E_{n} = ℏ ω$ ;

(b) Excitation reading. $H_{1} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ with common domain $D (H_{1}) = D (N_{ψ})$ ; $N_{ψ}$ is self-adjoint with spectrum $N$ and eigenbasis $(ψ_{n})$ ; $a_{ψ}$ and $a_{ψ}^{†}$ are closed with $a_{ψ}^{*} = a_{ψ}^{†}$ , and on the eigenbasis $a_{ψ} ψ_{n} = \sqrt{n} ψ_{n - 1} (n \geq 1), a_{ψ} ψ_{0} = 0, a_{ψ}^{†} ψ_{n} = \sqrt{n + 1} ψ_{n + 1}, N_{ψ} ψ_{n} = n ψ_{n};$ the eigenspace of $N_{ψ}$ at each $n \in N$ is exactly $C ψ_{n}$ , so its nonzero eigenvectors are precisely the nonzero scalar multiples of the $ψ_{n}$ ; the outcome law of $N_{ψ}$ in the pure preparation of $ψ_{n}$ is the point mass $δ_{n}$ ;

(c) Energy outcome law. For every unit $f \in H_{1}$ the outcome law of the assigned energy observable $(H_{1}, J)$ in the pure preparation of $f$ is purely atomic with $μ_{f}^{H_{1}} ({E_{n}}) = ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} (n \in N), μ_{f}^{H_{1}} (B) = \sum_{n : E_{n} \in B} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2}$ for every Borel set $B \subseteq R$ , and $μ_{f}^{H_{1}} (σ (H_{1})) = 1$ ; in particular the law has no mass outside ${E_{n} : n \in N}$ , and $E_{n}$ is an atom of the law of $f$ exactly when $⟨ f, ψ_{n} ⟩ \neq 0$ ;

(d) Three-dimensional isotropic realization. The energy observable $(H_{3}, J)$ of the model definition clause (d) has spectrum $σ (H_{3}) = {ℏ ω (M + \frac{3}{2}) : M \in N}$ ; for every $M \in N$ the eigenspace $\ker (H_{3} - ℏ ω (M + \frac{3}{2}) I)$ has the orthonormal basis ${ψ_{N} : N \in N^{3}, n_{1} + n_{2} + n_{3} = M}$ and dimension $(M + 1) (M + 2) / 2$ ; the ground energy is $\frac{3}{2} ℏ ω$ ;

(e) Status. Clauses (a)-(d) are conditional consequences of P1, P2, P4, P5, the adopted oscillator model and the cited mathematics. They concern the eigenvalue list, the eigenbasis, the excitation operators and the atomic outcome law of the model's assigned energy observable; they assert no measured spectral line, no numerical value of $m$ , $ω$ or $ℏ$ , no apparatus realization, no selection rule and no transition rate. The identification of a laboratory resonance or spectral line with an eigenvalue difference of this model is a tested identification under the preparation and apparatus postulate P7 and is not a consequence of this theorem.

Facts & Assumptions

Given: the adopted postulates P1, P2, P4, P5; the Axiom of Countable Choice and the Axiom of Choice; the oscillator model of The quadratic-potential oscillator model of the framework; the Hermite data, the unitary $D$ , the diagonal operator $A$ , the eigenbasis $(ψ_{n})$ and the energy operator $H_{1}$ of Oscillator scaling and exact realization; the closed ladder and number operators of Closed ladder domains and their transports; a unit vector $f \in H_{1}$ ; a Borel set $B \subseteq R$ ; a nonnegative integer $n$ and a nonnegative integer $M$ .

[F1]

The model's assigned observables. $(H_{1}, J)$ is the model's assigned energy observable, $(Q, m)$ and $(P, k g m s^{- 1})$ are its assigned position and momentum observables, and $(N_{ψ}, 1)$ is its assigned dimensionless excitation-number observable; the flow of the isolated model is $U (t) = \exp (- i t H_{1} / ℏ)$ , so $H_{1}$ is a self-adjoint operator with a spectral projection-valued measure $E_{H_{1}}$ and outcome law $μ_{f}^{A} (B) = ∥ E_{A} (B) f ∥^{2}$ for the assigned observable $(A, u_{A})$ in the pure preparation of a unit vector $f$ (The quadratic-potential oscillator model of the framework clauses (b), (c); Assigned sharp observables and the outcome law of the framework clause (b); Assigned energy observable, unitary flow and the two pictures of autonomous evolution clauses (a), (b)).

[F2]

The scaling theorem. $D : L^{2} (R, d y) \to H_{1}$ is a bijective linear isometry with $D g_{n} = ψ_{n}$ ; $(ψ_{n})$ is a complete orthonormal family of $H_{1}$ ; $H_{1} = \frac{ℏ ω}{2} D A D^{- 1}$ is self-adjoint with $D (H_{1}) = {f : \sum_{n} ℏ^{2} ω^{2} (n + \frac{1}{2})^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty}, H_{1} f = \sum_{n} ℏ ω (n + \frac{1}{2}) ⟨ f, ψ_{n} ⟩ ψ_{n},$ spectrum $σ (H_{1}) = {ℏ ω (n + \frac{1}{2}) : n \in N}$ , ground energy $ℏ ω / 2$ , core the span of $(ψ_{n})$ and pointwise differential identity $H_{1} F = - \frac{ℏ^{2}}{2 m} F^{''} + \frac{m ω^{2}}{2} x^{2} F$ for $F$ in that span (Oscillator scaling and exact realization clauses (a), (b), (c)).

[F3]

The closed ladder operators. For the complete orthonormal family $(g_{n})$ with coefficients $c_{n} = ⟨ x, g_{n} ⟩$ one has $D (a) = {x : \sum_{n} n ∣ c_{n} ∣^{2} < \infty}$ , $D (a^{†}) = {x : \sum_{n} (n + 1) ∣ c_{n} ∣^{2} < \infty}$ , $D (N) = {x : \sum_{n} n^{2} ∣ c_{n} ∣^{2} < \infty}$ , $a$ and $a^{†}$ are closed with $a^{*} = a^{†}$ and $(a^{†})^{*} = a$ , $D (a a^{†}) = D (a^{†} a) = D (N)$ , $a^{†} a = N$ and $a a^{†} = N + I$ on $D (N)$ , $N$ is self-adjoint with $N g_{n} = n g_{n}$ and spectrum $N$ , and $a g_{n} = \sqrt{n} g_{n - 1}$ for $n \geq 1$ , $a g_{0} = 0$ , $a^{†} g_{n} = \sqrt{n + 1} g_{n + 1}$ , $a^{†} a g_{n} = n g_{n}$ (Closed ladder domains clauses (a), (b), (d), (e)).

[F4]

Eigenspaces of a diagonal operator. For a complete orthonormal family $(e_{n})$ and pairwise distinct real eigenvalues $λ_{n}$ , the diagonal operator with this data and domain ${x : \sum_{n} λ_{n}^{2} ∣ ⟨ x, e_{n} ⟩ ∣^{2} < \infty}$ has $\ker (A - λ_{n} I) = C e_{n}$ for every $n$ (Eigenspaces of the diagonal operator clauses (a), (b)).

[F5]

Atoms and support of a spectral law. For a self-adjoint operator $A$ with spectral projection-valued measure $E_{A}$ : $E_{A} ({λ})$ is the orthogonal projection onto $\ker (A - λ I)$ , so for a unit vector $f$ the law satisfies $μ_{f}^{A} ({λ}) = ∥ E_{A} ({λ}) f ∥^{2}$ ; and $μ_{f}^{A} (B) = μ_{f}^{A} (B \cap σ (A))$ with $μ_{f}^{A} (σ (A)) = 1$ for every Borel $B$ , while the scalar law of a pure preparation is a probability measure and is countably additive (Atoms of the spectral measure and eigenvalues of a self-adjoint operator clauses (a), (b); Support of the spectral measure, the spectrum and the values of the scalar law clauses (a), (b); The Born outcome law: probability measure, mixtures, atoms and spectral support clause (a)).

[F6]

The multidimensional theorem. For the three-dimensional isotropic realization the energy operator $H_{3}$ of the model definition clause (d) is self-adjoint with the domain and core displayed in Multidimensional oscillator completeness clause (c), and its spectrum is ${ℏ ω (M + \frac{3}{2}) : M \in N}$ with the eigenvalue $ℏ ω (M + \frac{3}{2})$ having multiplicity $(\binom{M + 2}{2}) = (M + 1) (M + 2) / 2$ and eigenspace spanned by the orthonormal family ${ψ_{N} : \sum_{j} n_{j} = M}$ (Multidimensional oscillator completeness clauses (c), (d)).

[F7]

Transport along isometric bijections. For complex Hilbert spaces $H, K$ , a bijective linear isometry $U : H \to K$ and a self-adjoint $T$ on $H$ , the operator $U T U^{- 1}$ with domain $U D (T)$ is self-adjoint, has the same spectrum as $T$ , and satisfies $μ_{U x}^{U T U^{- 1}} (B) = μ_{x}^{T} (B)$ for every unit $x \in H$ and Borel $B$ (Transport of spectral projection-valued measures along isometric bijections clauses (a), (c), (d)).

[F8]

Adjoint-domain test. For a densely defined operator $S$ , $y \in D (S^{*})$ exactly when $x \mapsto ⟨ S x, y ⟩$ is bounded on $D (S)$ in the ambient Hilbert norm; then $⟨ S x, y ⟩ = ⟨ x, S^{*} y ⟩$ for all $x \in D (S)$ (Adjoint of a densely defined operator).

[F9]

Status of laboratory identifications. P7 requires declared preparation, isolation, detection and sampling models for comparisons with recorded data; identifications of apparatus with the model are tested, uncertain physical assumptions, and agreement is evidence rather than proof (Preparation and apparatus postulate P7: declared preparations, isolation and sampling assumptions clauses (i), (iii), (v)). This supplies only the status qualification in clause (e), not a premise for clauses (a)-(d).

Proof

Given: the data of the Statement together with [F1]-[F9].

1.1givenF2F4algebra

[The diagonal form of $H_{1}$ ] By [F2], $(ψ_{n})$ is complete orthonormal, $H_{1}$ has domain ${f : \sum_{n} E_{n}^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty}$ and acts by $H_{1} f = \sum_{n} E_{n} ⟨ f, ψ_{n} ⟩ ψ_{n}$ ; since $E_{n} = ℏ ω (n + \frac{1}{2})$ satisfies $n^{2} \leq 4 (n + \frac{1}{2})^{2}$ and $(n + \frac{1}{2})^{2} \leq 2 n^{2} + \frac{1}{2}$ for every $n \in N$ , the domain condition is equivalent to $\sum_{n} n^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty$ , and the real numbers $E_{n}$ are pairwise distinct positive numbers with $E_{0} = ℏ ω / 2$ and $E_{n + 1} - E_{n} = ℏ ω$ . Thus $H_{1}$ is the self-adjoint diagonal operator of [F4] for the family $(ψ_{n})$ and the eigenvalues $E_{n}$ , and the span of $(ψ_{n})$ is a core.

1.2givenF2F3F7F8

[Transported ladder and number operators] The map $D$ is a bijective linear isometry by [F2], and the operators $N_{ψ} = D N D^{- 1}$ , $a_{ψ} = D a D^{- 1}$ , $a_{ψ}^{†} = D a^{†} D^{- 1}$ have the transported domains $D (N_{ψ}) = D D (N)$ , $D (a_{ψ}) = D D (a)$ , $D (a_{ψ}^{†}) = D D (a^{†})$ ; because $ψ_{n} = D g_{n}$ and $D$ preserves the inner product, the coefficients satisfy $⟨ D x, ψ_{n} ⟩ = ⟨ x, g_{n} ⟩$ , so by [F3] $D (N_{ψ}) = {f : \sum_{n} n^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty}$ and the identities of [F3] transport: $D (a_{ψ} a_{ψ}^{†}) = D (a_{ψ}^{†} a_{ψ}) = D (N_{ψ})$ , on which $a_{ψ}^{†} a_{ψ} = N_{ψ}$ and $a_{ψ} a_{ψ}^{†} = N_{ψ} + I$ , and $N_{ψ} ψ_{n} = n ψ_{n}$ , $a_{ψ} ψ_{n} = \sqrt{n} ψ_{n - 1}$ ( $n \geq 1$ ) with $a_{ψ} ψ_{0} = 0$ , $a_{ψ}^{†} ψ_{n} = \sqrt{n + 1} ψ_{n + 1}$ . For the non-self-adjoint pair, closedness and the adjoint relations transport by the direct pairing computation: for $x \in D (a_{ψ})$ and $y \in D (a_{ψ}^{†})$ , $⟨ a_{ψ} x, y ⟩ = ⟨ a D^{- 1} x, D^{- 1} y ⟩ = ⟨ D^{- 1} x, a^{†} D^{- 1} y ⟩ = ⟨ x, a_{ψ}^{†} y ⟩$ , so $a_{ψ} \subseteq (a_{ψ}^{†})^{*}$ and $a_{ψ}^{†} \subseteq a_{ψ}^{*}$ ; and if $x_{k} \to x$ with $a_{ψ} x_{k} \to y$ , then $D^{- 1} x_{k} \to D^{- 1} x$ and $a D^{- 1} x_{k} = D^{- 1} a_{ψ} x_{k} \to D^{- 1} y$ , so closedness of $a$ gives $D^{- 1} x \in D (a)$ and $a D^{- 1} x = D^{- 1} y$ , that is $x \in D (a_{ψ})$ and $a_{ψ} x = y$ : $a_{ψ}$ is closed, and the same argument with $a^{†}$ makes $a_{ψ}^{†}$ closed. By [F8], for $y \in H_{1}$ , the functional $x \mapsto ⟨ a_{ψ} x, y ⟩ = ⟨ a D^{- 1} x, D^{- 1} y ⟩$ on $D (a_{ψ})$ is bounded in the ambient norm exactly when $D^{- 1} y \in D (a^{*}) = D (a^{†})$ , because $D$ is a surjective isometry between the domains. Thus $D (a_{ψ}^{*}) = D D (a^{†})$ and $a_{ψ}^{*} y = D a^{*} D^{- 1} y = a_{ψ}^{†} y$ ; interchanging $a$ and $a^{†}$ proves the other adjoint equality. By [F7] applied with $U = D$ and $T = N$ , the operator $N_{ψ}$ is self-adjoint with $σ (N_{ψ}) = σ (N) = N$ and $μ_{D x}^{N_{ψ}} (B) = μ_{x}^{N} (B)$ for every unit $x$ and Borel $B$ .

1.3givenF6

[The three-dimensional realization] By [F6] the spectrum of $H_{3}$ is ${ℏ ω (M + \frac{3}{2})} = {E_{M}^{(3)} : M \in N}$ , the eigenspace at $E_{M}^{(3)}$ has the orthonormal basis ${ψ_{N} : n_{1} + n_{2} + n_{3} = M}$ and dimension $(M + 1) (M + 2) / 2$ , and the least eigenvalue is $\frac{3}{2} ℏ ω$ ; this is clause (d), whose self-adjointness, domain and core assertions are those of [F6].

2.1step 1.1F2F4

[Spectrum and eigenspaces of $H_{1}$ ] By step 1.1 the operator $H_{1}$ is the diagonal operator of [F4] with the pairwise distinct eigenvalues $E_{n}$ , and by [F2] its spectrum is ${E_{n} : n \in N}$ , a closed discrete set whose least element is $E_{0}$ and whose consecutive gaps are $ℏ ω$ ; [F4] gives $\ker (H_{1} - E_{n} I) = C ψ_{n}$ for every $n$ , and [F2] gives that $(ψ_{n})$ is a complete orthonormal family and its algebraic span is a core. This is clause (a).

2.2step 1.1step 1.2F4F5F7

[Number-operator identity] By steps 1.1 and 1.2, $D (H_{1}) = {f : \sum_{n} n^{2} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2} < \infty} = D (N_{ψ})$ and both operators act diagonally on the family, with $H_{1}$ having eigenvalues $E_{n} = ℏ ω (n + \frac{1}{2})$ and $N_{ψ}$ having eigenvalues $n$ ; hence $H_{1} = ℏ ω (N_{ψ} + \frac{1}{2} I)$ with common domain $D (H_{1}) = D (N_{ψ})$ . By step 1.2 and [F4] the eigenspaces of $N_{ψ}$ are $\ker (N_{ψ} - n I) = C ψ_{n}$ , so its eigenvectors are exactly the nonzero multiples of the $ψ_{n}$ , and its spectrum is $N$ by [F7]. For the preparation of the unit eigenvector $ψ_{n}$ , [F5] and the eigenspace identity give $μ_{ψ_{n}}^{N_{ψ}} ({n}) = ∥ E_{N_{ψ}} ({n}) ψ_{n} ∥^{2} = ∥ ψ_{n} ∥^{2} = 1$ ; since the scalar law of a pure preparation is a probability measure [F5], the law is the point mass $δ_{n}$ . Together with the ladder actions of step 1.2 this is clause (b).

3.1step 2.1F1F5

[Atoms of the energy law] By [F1], $(H_{1}, J)$ is an assigned sharp observable of the model whose outcome law in the pure preparation of a unit vector $f$ is $μ_{f}^{H_{1}} (B) = ∥ E_{H_{1}} (B) f ∥^{2}$ . For every $n$ , [F4] and step 2.1 give $\ker (H_{1} - E_{n} I) = C ψ_{n}$ , so the orthogonal projection $E_{H_{1}} ({E_{n}})$ is the projection onto $C ψ_{n}$ by the first clause of [F5]; for a unit $f$ this gives $μ_{f}^{H_{1}} ({E_{n}}) = ∥ E_{H_{1}} ({E_{n}}) f ∥^{2} = ∣ ⟨ f, ψ_{n} ⟩ ∣^{2}$ .

4.1step 2.1step 3.1F5

[The full law and its support] Let $B \subseteq R$ be Borel. By the support clause of [F5], $μ_{f}^{H_{1}} (B) = μ_{f}^{H_{1}} (B \cap σ (H_{1}))$ and $μ_{f}^{H_{1}} (σ (H_{1})) = 1$ ; by step 2.1, $σ (H_{1}) = {E_{n} : n \in N}$ , so $B \cap σ (H_{1})$ is the countable union of the disjoint singletons ${E_{n}}$ with $E_{n} \in B$ , and countable additivity of the probability measure $μ_{f}^{H_{1}}$ gives $μ_{f}^{H_{1}} (B) = \sum_{n : E_{n} \in B} μ_{f}^{H_{1}} ({E_{n}}) = \sum_{n : E_{n} \in B} ∣ ⟨ f, ψ_{n} ⟩ ∣^{2}$ by step 3.1. In particular the law has no mass on Borel sets disjoint from ${E_{n}}$ , and $E_{n}$ is an atom of the law of $f$ exactly when $⟨ f, ψ_{n} ⟩ \neq 0$ . This is clause (c).

5.1step 1.3step 2.1step 2.2step 4.1F9∎

[Status] Clauses (a), (b) and (c) were obtained from the postulates, the adopted model, the scaling theorem, the closed ladder operators and the general spectral law facts; clause (d) from the multidimensional theorem. No measured spectral line, no numerical value of $m$ , $ω$ or $ℏ$ , no apparatus model, no selection rule and no transition rate entered, and the identification of a laboratory resonance with an eigenvalue difference is a tested identification under P7 as specified in [F9], not a consequence of the argument. This is clause (e).

The energy observable of the oscillator model: spectrum, eigenbasis and outcome law

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources