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A confined model with Dirichlet walls: the naive interval momentum is not a sharp observable

Physical scope

A conditional warning about a hypothetical confined-model extension of the massive-particle kinematics: if a model confines a spinless particle to an interval and declares the formal momentum -i hbar d/dx on the Dirichlet domain, then that operator is symmetric but not self-adjoint and hence is not an assigned sharp observable of the observable and probability postulate; no self-adjoint extension keeps the Dirichlet domain, and the Dirichlet eigenfunctions are not momentum eigenfunctions of any such extension. The item asserts no experiment, no apparatus realization and no claim that confinement models are physically realized; it excludes any assertion about which self-adjoint realization, if any, describes a laboratory device.

Statement

Adopt the observable and probability postulate P2 in the reading of Assigned sharp observables and the outcome law of the framework and the kinematics and massive-particle model postulate P5 (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), and assume the Axiom of Countable Choice and the Axiom of Choice (The Axiom of Countable Choice ( ${A C}_{ω}$ ), The Axiom of Choice). Consider the following hypothetical confined-model extension of the kinematics: a spinless particle of mass $m > 0$ is restricted to an interval $(0, L)$ of length $0 < L < \infty$ in one spatial direction, with Hilbert space $L^{2} (0, L; C)$ , and the model declares its momentum to be the formal differential expression $T : = - i ℏ D, D (T) : = {u \in H^{1} (0, L; C) : u^{*} (0) = u^{*} (L) = 0},$ with $H^{1}$ and the continuous trace $u^{*}$ as in One-dimensional Sobolev spaces on intervals. Then, in that hypothetical model:

(a) $T$ is densely defined and symmetric, but not self-adjoint; its adjoint is $T^{*} = - i ℏ D, D (T^{*}) = H^{1} (0, L; C),$ without boundary condition;

(b) no self-adjoint extension $S \supseteq T$ has domain contained in the Dirichlet class $D (T)$ : there is no self-adjoint realization of the formal momentum on $(0, L)$ that keeps the Dirichlet condition on its domain;

(c) for every self-adjoint extension $S \supseteq T$ and every $n \geq 1$ , the normalized Dirichlet function $u_{n} (x) = \sqrt{2 / L} \sin (n π x / L)$ lies in $D (S)$ but is not an eigenvector of $S$ ; consequently a momentum quantization read off from the Dirichlet eigenfunctions is not a statement about any self-adjoint realization of the interval momentum;

(d) therefore $T$ is not an assigned sharp observable of the model in the sense of P2(i), and does not supply the sharp spectral outcome law of P2(ii); assigning such a momentum law requires the model to declare a self-adjoint realization. This obstruction does not exclude the unsharp finite outcome probabilities of P2(iv): a separately declared apparatus model may assign its momentum channels bounded effects $F_{a}$ on $L^{2} (0, L; C)$ , with $0 \leq F_{a} \leq I$ and $\sum_{a} F_{a} = I$ , and use $p_{a} = tr (ρ F_{a})$ for a preparation $ρ$ as in P2(iv), without declaring a self-adjoint momentum realization. Here an effect is a bounded self-adjoint operator with $0 \leq ⟨ F_{a} v, v ⟩ \leq ∥ v ∥^{2}$ for every vector $v$ , as defined in Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes. An explicit family of self-adjoint extensions of $T$ is $ℏ P_{θ}$ , $θ \in [0, 2 π)$ , where $P_{θ} = - i D$ is the self-adjoint operator of The twisted-periodic momentum operator on an interval; the scaled operator $ℏ P_{θ}$ acts as $- i ℏ D$ on the domain ${u \in H^{1} (0, L) : u^{*} (L) = e^{i θ} u^{*} (0)}$ and has units of momentum. These domains impose a twisted-periodic condition rather than Dirichlet conditions; nothing here claims that any member of that family, or any other realization, describes a laboratory apparatus.

Facts & Assumptions

Given: The postulates P2 and P5 read as above; the Axiom of Countable Choice and the Axiom of Choice; $0 < L < \infty$ ; the operator $T$ of the statement; classes $u, v \in H^{1} (0, L; C)$ ; a self-adjoint extension $S$ of $T$ ; an integer $n \geq 1$ ; the Dirichlet function $u_{n}$ .

[F1]

Interval Sobolev space and integration by parts. $H^{1} (0, L)$ is the space of $L^{2}$ classes with an $L^{2}$ weak derivative; each class has a continuous representative, its trace $u^{*}$ is defined at the endpoints, and $- \int_{0}^{L} u \overline{v^{'}} = u^{*} (0) \overline{v^{*} (0)} - u^{*} (L) \overline{v^{*} (L)} + \int_{0}^{L} u^{'} \overline{v}$ for all $u, v \in H^{1} (0, L)$ ; the test classes $C_{c}^{\infty} (0, L)$ have zero endpoint traces and lie in $D (T)$ (One-dimensional Sobolev spaces on intervals, Integration by parts and the endpoint-vanishing space on intervals). Under the assumed Countable Choice, these test classes are dense in $L^{2} (0, L; C)$ by Smooth compactly supported functions of an open set are dense in $L^{2}$ applied to the open set $Ω = (0, L) \subset R$ . In particular $D (T) \supseteq C_{c}^{\infty} (0, L)$ is dense in $L^{2} (0, L; C)$ and $D (T)$ is a linear subspace of $H^{1}$ .

[F2]

Adjoints and self-adjointness. A densely defined operator $B$ is symmetric when $⟨ B u, v ⟩ = ⟨ u, B v ⟩$ for all $u, v \in D (B)$ ; its adjoint $B^{*}$ has domain consisting of the $v$ for which $u \mapsto ⟨ B u, v ⟩$ is bounded on $D (B)$ , and $B$ is self-adjoint when $B^{*} = B$ including domains; a self-adjoint operator has no proper symmetric extension (Adjoint of a densely defined operator, Symmetric, self-adjoint and essentially self-adjoint operators, Weak derivative of a locally integrable function).

[F3]

Sharp laws and unsharp outcomes. An assigned sharp observable of the framework is a pair of a densely defined self-adjoint operator and a declared unit; a symmetric formal differential expression on a smaller domain does not by itself supply an assigned sharp observable or its spectral outcome law (Assigned sharp observables and the outcome law of the framework clauses (a)-(b)). Separately, P2(iv) permits a declared finite family of bounded effects $F_{a}$ with $0 \leq F_{a} \leq I$ and $\sum_{a} F_{a} = I$ , with probabilities $p_{a} = tr (ρ F_{a})$ for a preparation $ρ$ ; these effects and their apparatus interpretation need not arise from a declared self-adjoint momentum operator (Observable and probability postulate P2: self-adjoint observables, spectral probability and finite unsharp outcomes clause (iv), Assigned sharp observables and the outcome law of the framework clause (f)).

[F4]

Twisted-periodic realizations. For each $θ \in [0, 2 π)$ the operator $P_{θ} = - i D$ with domain $D (P_{θ}) = {u \in H^{1} (0, L) : u^{*} (L) = e^{i θ} u^{*} (0)}$ is densely defined, symmetric and self-adjoint (The twisted-periodic momentum operator on an interval clause (a)). Because $ℏ > 0$ is real and nonzero, the adjoint of $ℏ P_{θ}$ has the same domain as $P_{θ}$ and equals $ℏ P_{θ}^{*} = ℏ P_{θ}$ ; thus $ℏ P_{θ}$ is self-adjoint and acts as $- i ℏ D$ . Moreover $D (T) \subseteq D (P_{θ})$ , so $ℏ P_{θ}$ extends $T$ .

Proof

technique · direct; compute the adjoint of the Dirichlet momentum by integration by parts, show that no self-adjoint extension can keep the Dirichlet domain, test the Dirichlet functions against an arbitrary self-adjoint extension, and read the observability conclusion off the self-adjointness requirement of the observable postulate

Given: The data of the statement and [F1]-[F4].

1.1F1algebra

[Density and symmetry] $D (T)$ contains $C_{c}^{\infty} (0, L)$ , which is dense in $L^{2}$ [F1]; hence $T$ is densely defined, and it is linear on the linear subspace $D (T)$ [F1]. For $u, v \in D (T)$ , the integration-by-parts identity [F1] gives $⟨ T u, v ⟩ - ⟨ u, T v ⟩ = - i ℏ (u^{*} (L) \overline{v^{*} (L)} - u^{*} (0) \overline{v^{*} (0)}) = 0,$ because both traces of both functions vanish. Hence $T$ is symmetric.

2.1step 1.1F1F2algebra

[The adjoint] Let $v \in H^{1} (0, L)$ . For every $u \in D (T)$ , [F1] gives $⟨ T u, v ⟩ = - i ℏ \int_{0}^{L} D u \overline{v} = - i ℏ [u^{*} \overline{v^{*}}]_{0}^{L} + \int_{0}^{L} u \overline{- i ℏ D v} = ⟨ u, - i ℏ D v ⟩$ , since $u^{*}$ vanishes at both endpoints; so $u \mapsto ⟨ T u, v ⟩$ is the restriction of a bounded functional and $v \in D (T^{*})$ with $T^{*} v = - i ℏ D v$ [F2]. Conversely, if $v \in D (T^{*})$ , testing the identity $⟨ T u, v ⟩ = ⟨ u, T^{*} v ⟩$ against $u \in C_{c}^{\infty} (0, L) \subseteq D (T)$ shows that the distribution $- i ℏ D v$ equals the $L^{2}$ class $T^{*} v$ ; hence $v$ has an $L^{2}$ weak derivative, that is $v \in H^{1} (0, L)$ , and $T^{*} v = - i ℏ D v$ . Thus $D (T^{*}) = H^{1} (0, L)$ . The constant class $1$ lies in $H^{1} (0, L)$ with $1^{*} (0) = 1 \neq 0$ , so $D (T^{*}) \neq D (T)$ and $T$ is not self-adjoint. This proves (a).

2.2step 1.1F1algebra

[Dirichlet functions are not momentum eigenvectors] Fix $n \geq 1$ and put $u_{n} (x) : = \sqrt{2 / L} \sin (n π x / L)$ , a unit class in $H^{1} (0, L)$ with $u_{n}^{*} (0) = u_{n}^{*} (L) = 0$ , so $u_{n} \in D (T)$ and its weak derivative is the classical derivative $u_{n}^{'} (x) = \sqrt{2 / L} (n π / L) \cos (n π x / L)$ [F1]. Let $S \supseteq T$ be any self-adjoint extension. Then $u_{n} \in D (S)$ and $S u_{n} = T u_{n} = - i ℏ u_{n}^{'} = - i ℏ \sqrt{2 / L} \frac{n π}{L} \cos (n π x / L) .$ Suppose $S u_{n} = λ u_{n}$ for some $λ \in R$ ; then the continuous functions $\cos (n π x / L)$ and $\sin (n π x / L)$ would be proportional on $[0, L]$ . At $x = 0$ one has $\sin 0 = 0$ and $\cos 0 = 1$ , so proportionality would force the sine multiple to vanish identically, a contradiction. Hence $u_{n}$ is not an eigenvector of $S$ , for any self-adjoint extension $S$ . This proves (c).

3.1step 1.1step 2.1F2algebra

[No Dirichlet self-adjoint extension] Let $S \supseteq T$ be self-adjoint with $D (S) \subseteq D (T)$ . Since $T \subseteq S$ , one has $D (T) \subseteq D (S)$ ; with the assumed inclusion this gives $D (S) = D (T)$ , and since $S$ agrees with $T$ on $D (T)$ , the operators coincide: $S = T$ . But then $T$ would be self-adjoint, contradicting step 2.1. Hence no such $S$ exists, which is (b). Equivalently, any self-adjoint extension of $T$ must contain a vector of $H^{1} (0, L)$ whose trace at $0$ or $L$ is nonzero.

4.1step 1.1step 2.1step 3.1step 2.2F3F4∎

[Observability and scope] By [F3], assigned sharp observables of the model are densely defined self-adjoint operators; by steps 1.1 and 2.1 the Dirichlet momentum $T$ is densely defined and symmetric but not self-adjoint, so $T$ cannot supply an assigned sharp momentum observable or its P2(ii) spectral law. A sharp momentum law requires a separately declared self-adjoint realization. This does not exclude P2(iv)'s finite unsharp probabilities: by [F3] a separately declared family of effects and apparatus interpretation supplies them without a self-adjoint momentum assignment; $T$ alone supplies no such family or interpretation. [F4] gives the explicit family $ℏ P_{θ}$ of self-adjoint extensions of the same formal expression $- i ℏ D$ , distinct from the Dirichlet realization; the statement neither selects a member of that family nor asserts any laboratory realization, and it makes no claim about the framework's own momentum $P$ of P5, which is defined on the full line and is not obtained by restricting $T$ . This proves (d).

Remarks

The obstruction is the boundary form. Symmetry of $- i ℏ D$ on a domain imposes $u^{*} (0) \overline{v^{*} (0)} = u^{*} (L) \overline{v^{*} (L)}$ for all pairs in the domain; the Dirichlet domain satisfies this only in the degenerate way, and enlarging it to any self-adjoint domain necessarily relates the endpoint values, as the twisted-periodic family illustrates (The twisted-periodic momentum operator on an interval).
What is not claimed. The thought experiment does not assert that confined models are physically realized, does not compare models with data, and does not claim that box energy levels lack a self-adjoint energy realization; the Dirichlet Laplacian is a self-adjoint energy operator, and the warning concerns the momentum observable only.

A confined model with Dirichlet walls: the naive interval momentum is not a sharp observable

Statement

Facts & Assumptions

Proof

Remarks

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