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Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value

Physical scope

A conditional consequence of P1, P2, P5 and P6 for the massive spin-s model and its orbital generators: for a smooth compactly supported radial profile times a spherical harmonic, the model vector is an eigenvector of the third orbital generator with eigenvalue hbar m, the Casimir quadratic form takes the value hbar^2 l(l+1), and the orbital quantum numbers are those of the spherical harmonic family. The statement concerns the spatial factor only, uses no dynamics and no measurement model, and asserts no spectral decomposition of the full spatial Hilbert space, no hydrogen spectrum and no empirical result.

Statement

Adopt the state and composition postulate P1, the observable and probability postulate P2, the kinematics and massive-particle model postulate P5 and the internal-factor assignment of the spin postulate P6, read for the massive spin- $s$ model of The massive spin-s particle model of the framework: state space, operators, units and frame readings and the angular momentum readings of Orbital, spin and total angular momentum of the massive spin-s model: assigned sharp observables, units and 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, Kinematics and massive-particle model postulate P5: Newtonian frames and one-particle operators, Identical-particle and spin assignments P6: permutation sectors and finite-spin representations), and assume the Axiom of Choice and the Axiom of Countable Choice (The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ )). Fix $l \geq 0$ , $m \in {- l, \dots, l}$ , a unit vector $w \in C^{r}$ and a complex-valued $R \in C_{c}^{\infty} ((0, \infty))$ , and let $h_{l m}$ be the harmonic homogeneous polynomial of degree $l$ with $h_{l m} ∣_{S^{2}} = Y_{l m}$ of Harmonic polynomial dimension clause (c) (Associated Legendre function normalization). Define the model vector $ψ (x) : = R (∣ x ∣) h_{l m} (x) w (x \in R^{3}, x \neq 0), ψ (0) : = 0.$ Then:

(a) $ψ$ is a well-defined class in the model's state space $H_{s}$ and in fact $ψ \in C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ ; it lies in $D (L_{i})$ for each $i$ , and for $x \neq 0$ $(L_{i} ψ) (x) = R (∣ x ∣) (- i ℏ D_{i} h_{l m}) (x) w, D_{i} h_{l m} = (x \times \nabla h_{l m})_{i},$ so that $- i ℏ D_{i} h_{l m}$ is the polynomial representative of the sphere function $L_{i}^{s p h} Y_{l m}$ of Orbital self-adjoint generators; in particular $L_{3} ψ = ℏ m ψ$ ;

(b) with $∥ R ∥_{l}^{2} : = \int_{0}^{\infty} ∣ R (r) ∣^{2} r^{2 l + 2} d r$ one has $∥ ψ ∥^{2} = ∥ R ∥_{l}^{2}$ and $∥ L_{i} ψ ∥^{2} = ∥ R ∥_{l}^{2} ∥ L_{i}^{s p h} Y_{l m} ∥^{2}, \sum_{i = 1}^{3} ∥ L_{i} ψ ∥^{2} = ℏ^{2} l (l + 1) ∥ ψ ∥^{2},$ so the Casimir quadratic form of the model's orbital generators on $ψ$ takes the value $ℏ^{2} l (l + 1)$ ;

(c) consequently, for the pure preparation of the unit vector $ψ / ∥ ψ ∥$ (for $R \equiv̸ 0$ ), the mean and the second moment of the outcome law of $(L_{3}, J s)$ are finite and equal $ℏ m$ and $ℏ^{2} m^{2}$ , so the variance vanishes; for each $i$ the mean and second moment of the law of $L_{i}$ are finite and the second moment equals $∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ ; the model reads $(l, m)$ as the orbital angular momentum quantum numbers of the preparation, in the sense of the eigenvalue of the angular momentum-squared form and the eigenvalue of $L_{3}$ ;

(d) Scope. Clauses (a)-(c) are conditional consequences of P1, P2, P5, P6 and the cited mathematics, and concern the spatial factor of the massive model. The item asserts no dynamics, no measurement or apparatus model, no joint sharp values of two different components $L_{i}, L_{j}$ , no empirical value and no spectral resolution of $L^{2} (R^{3})$ : the identity $\sum_{i} L_{i}^{2} = ℏ^{2} l (l + 1)$ is a quadratic-form statement on the displayed smooth vectors and is not read as a statement about a self-adjoint operator $\sum_{i} L_{i}^{2}$ on the full spatial Hilbert space.

Facts & Assumptions

Given: The postulates P1, P2, P5 and P6 in the stated readings; the Axiom of Choice and the Axiom of Countable Choice; the model with parameters $m > 0$ , $s$ , $r = 2 s + 1$ ; the harmonic polynomial $h_{l m}$ and the sphere functions $Y_{l m}$ ; the radial function $R$ ; the unit vector $w$ ; the vector $ψ$ ; indices $i, k$ .

[F1]

Harmonic polynomial and spherical harmonic. For $l \geq 0$ and $- l \leq m \leq l$ the polynomial $h_{l m}$ is homogeneous of degree $l$ , harmonic, nonzero, with $h_{l m} ∣_{S^{2}} = Y_{l m}$ , and $D_{3} h_{l m} = i m h_{l m}$ where $D_{3} = x_{1} \partial_{2} - x_{2} \partial_{1}$ ; moreover $L_{3}^{s p h} Y_{l m} = ℏ m Y_{l m}$ and $L_{i}^{s p h} Y_{l m} = {r e s}_{l} (- i ℏ D_{i} h_{l m})$ for $D_{i} p = (x \times \nabla p)_{i}$ (Harmonic polynomial dimension clause (c), Orbital self-adjoint generators clauses (a),(c), Associated Legendre function normalization clause (c)).

[F2]

Core action of the orbital generators. For every componentwise Schwartz class $φ \in S^{(r)}$ the orbital generators satisfy $L_{i} φ \in H_{s}$ , $S^{(r)} \subseteq D (L_{i})$ , and $L_{i} φ = - i ℏ \sum_{j} (e_{i} \times x)_{j} D_{j} φ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} φ,$ with $D_{j}$ the componentwise weak derivative (Coordinate-rotation generators on the Schwartz classes and their commutators clauses (a),(b), Componentwise position and momentum operators on the component-valued wave-function space clauses (a),(c)).

[F3]

Orbital Casimir identity. For $F \in V_{l}$ one has $\sum_{i} L_{i}^{2} F = ℏ^{2} l (l + 1) F$ on the harmonic span, and the on-shell values are exactly the squared norms $\sum_{i} ∥ L_{i}^{s p h} Y_{l m} ∥^{2} = ℏ^{2} l (l + 1)$ (The orbital Casimir identity: the restrictions of the orbital generators to the degree-l blocks are spin-l models clauses (b),(c), Orbital self-adjoint generators).

[F4]

Polar integration and the sphere measure. For a Borel $f \geq 0$ on $R^{3}$ one has $\int_{R^{3}} f d x = \int_{0}^{\infty} \int_{S^{2}} f (r Ω) r^{2} d σ (Ω) d r$ , and $\int_{S^{2}} F d σ = \int_{(0, π) \times (0, 2 π)} F \circ Φ \sin θ d (θ, φ)$ with $σ (S^{2}) = 4 π$ (Polar coordinates decompose Lebesgue measure into r^{n-1} dr d sigma, Sphere integrals in angle coordinates).

[F5]

Schwartz calculus and the class reading. $C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ ; a classical partial derivative of a $C^{1}$ function that is locally integrable is its weak derivative; the model space consists of almost-everywhere classes (Schwartz space and its seminorms, Componentwise position and momentum operators on the component-valued wave-function space, The space $L^{p} (μ)$ as the quotient by null functions, Complex Lp classes and Euclidean test-function conventions, Hilbert space, Real and complex inner product spaces, with the inner product linear in the first argument).

Proof

Proof technique: direct; the vector is a compactly supported smooth class with a homogeneous polynomial angular factor, so the core identity of the generators applies; the radial and angular integrals separate by polar integration and the Casimir value is transported from the sphere.

Given: The data of the statement and [F1]-[F5].

1.1F1F5algebra

[Smoothness and the polynomial identity] Since $supp R \subseteq [a, b]$ with $0 < a \leq b < \infty$ and $h_{l m}$ is a polynomial, the function $ψ = R (∣ x ∣) h_{l m} (x) w$ vanishes identically near $0$ and is smooth on $R^{3}$ with compact support, so $ψ \in C_{c}^{\infty} (R^{3}; C^{r}) \subseteq S^{(r)}$ by [F5]; for $x \neq 0$ and each $i$ , the product rule gives $(x \times \nabla)_{i} ψ = R (∣ x ∣) (D_{i} h_{l m}) (x) w$ because $(x \times \nabla)_{i} R (∣ x ∣) = R^{'} (∣ x ∣) (x \times x)_{i} / ∣ x ∣ = 0$ , and by [F1] the function $- i ℏ D_{i} h_{l m}$ is the polynomial representative of $L_{i}^{s p h} Y_{l m}$ ; moreover [F1] gives $D_{3} h_{l m} = i m h_{l m}$ .

2.1step 1.1F1F2F5algebra

[Core action and $L_{3}$ eigenvalue] By step 1.1 and [F2], $ψ \in D (L_{i})$ and $L_{i} ψ = \sum_{j, k} ε_{i j k} Q_{j} P_{k} ψ = - i ℏ (x \times \nabla)_{i} ψ = R (∣ x ∣) (- i ℏ D_{i} h_{l m}) (x) w$ for almost every $x$ , where the weak derivatives agree with the classical ones by [F5]; here $P_{k} = - i ℏ \partial_{k}$ on the Schwartz class, so the factor $- i ℏ$ occurs once. For $i = 3$ , [F1] gives $L_{3} ψ = - i ℏ R D_{3} h_{l m} w = ℏ m R h_{l m} w = ℏ m ψ$ , proving (a).

3.1step 2.1F1F3F4givenalgebra

[Norms and the Casimir form] Put $q_{i} : = D_{i} h_{l m}$ . Each term $x_{j} \partial_{k} h_{l m}$ is homogeneous of degree $l$ or zero, so $q_{i} (r Ω) = r^{l} q_{i} (Ω)$ for $r > 0$ ; for $l = 0$ , $q_{i} = 0$ . By step 2.1, $∥ w ∥ = 1$ and polar integration [F4], $∥ L_{i} ψ ∥^{2} = ℏ^{2} \int_{R^{3}} ∣ R (∣ x ∣) ∣^{2} ∣ q_{i} (x) ∣^{2} d x = ℏ^{2} (\int_{0}^{\infty} ∣ R (r) ∣^{2} r^{2 l + 2} d r) \int_{S^{2}} ∣ q_{i} (Ω) ∣^{2} d σ (Ω)$ . By [F1], $L_{i}^{s p h} Y_{l m} = - i ℏ q_{i} ∣_{S^{2}}$ , hence $∥ L_{i} ψ ∥^{2} = ∥ R ∥_{l}^{2} ∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ . Likewise homogeneity of $h_{l m}$ and [F4] give $∥ ψ ∥^{2} = ∥ R ∥_{l}^{2} \int_{S^{2}} ∣ Y_{l m} ∣^{2} d σ = ∥ R ∥_{l}^{2}$ , using the normalization in Associated Legendre function normalization clause (d). Finally, on the finite invariant block $V_{l}$ , self-adjointness and [F3] give $\sum_{i} ∥ L_{i}^{s p h} Y_{l m} ∥^{2} = ⟨ \sum_{i} (L_{i}^{s p h})^{2} Y_{l m}, Y_{l m} ⟩ = ℏ^{2} l (l + 1)$ , so $\sum_{i} ∥ L_{i} ψ ∥^{2} = ℏ^{2} l (l + 1) ∥ ψ ∥^{2}$ , proving (b), including $l = 0$ and $R = 0$ .

4.1step 2.1step 3.1F3algebra

[Readings] Let $R \equiv̸ 0$ and put $φ : = ψ / ∥ ψ ∥$ ; by step 2.1, $L_{3} φ = ℏ m φ$ with $φ \in D (L_{3})$ , so the mean and the second moment of the law of $L_{3}$ in the pure preparation $[φ]$ are $⟨ L_{3} φ, φ ⟩ = ℏ m$ and $∥ L_{3} φ ∥^{2} = ℏ^{2} m^{2}$ , with variance $0$ ; for each $i$ , step 2.1 gives $φ \in D (L_{i})$ and step 3.1 gives $∥ L_{i} φ ∥^{2} = ∥ L_{i}^{s p h} Y_{l m} ∥^{2}$ , so the mean and second moment of the law of $L_{i}$ are finite with that second moment, and the quadratic form is $\sum_{i} ∥ L_{i} φ ∥^{2} = ℏ^{2} l (l + 1)$ , which is (c).

5.1step 1.1step 2.1step 3.1step 4.1F2∎

[Assembly and scope] Steps 1.1 and 2.1 prove (a), step 3.1 proves (b) and step 4.1 proves (c); only the spatial core identities, the sphere data and polar integration were used, no dynamics, apparatus or empirical input entered, and no operator identity for $\sum_{i} L_{i}^{2}$ on the full space was asserted, which is the scope statement (d).

Remarks

The density of the family. The vectors $R (∣ x ∣) h_{l m} (x) w$ with $R \in C_{c}^{\infty} ((0, \infty))$ span a dense subspace of the angular sector $L^{2} ((0, \infty), r^{2} d r) \otimes C Y_{l m} \otimes C w$ in the polar decomposition; this theorem uses only their eigenproperty and makes no completeness claim.
No hidden operator $(L^{2})$ . The Casimir value is stated as a quadratic-form identity. Defining a self-adjoint operator $\sum_{i} L_{i}^{2}$ on $L^{2} (R^{3})$ would require a separate closed-form and domain argument, which is not part of this page.

Orbital angular momentum eigenstates of the massive model: spherical harmonics, the l,m quantum numbers and the Casimir value

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources