Qm Symmetries And Conservation — Examples

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m>0$ and spin label $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$ and the parity action $\mathrm{\Pi }$ (The spatial parity action on the component-valued wave-function space); assume the Axiom of Countable Choice, the Axiom of Choice and the Axiom of Dependent Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The Axiom of Choice, The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $V:{\mathbb{R}}^3\to \mathbb{R}$ be real, bounded and measurable with $V(-x)=V(x)\text{for almost every }x\in {\mathbb{R}}^3,$ let $H_V=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2+M_V,D(H_V)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ and put $V_P(x):=V(-x)$. Then:

(a) if $\psi \in D(H_V)$ is an eigenvector of $H_V$ with eigenvalue $E\in \mathbb{R}$, $H_V\psi =E\psi$ and $\psi \mathrm{\ne }0$, then $\mathrm{\Pi }\psi$ is again an eigenvector of $H_V$ with the same eigenvalue $E$; consequently every eigenspace of $H_V$ is invariant under $\mathrm{\Pi }$;

(b) if the eigenspace of $E$ is one-dimensional, then $\mathrm{\Pi }\psi =\lambda \psi$ with $\lambda \in \{-1,1\}$: the state has definite parity;

(c) if $\psi ,{\psi }^{\mathrm{\prime }}\in H_s$ are unit vectors with $\mathrm{\Pi }\psi =\sigma \psi$ and $\mathrm{\Pi }{\psi }^{\mathrm{\prime }}=\tau {\psi }^{\mathrm{\prime }}$ for signs $\sigma ,\tau \in \{-1,1\}$ and $\psi \in D(Q_j)$, then $\sigma =\tau ⟹⟨Q_j\psi ,{\psi }^{\mathrm{\prime }}⟩=0;$ the position matrix element of two states of equal parity vanishes, while no nonvanishing statement is made for opposite parities;

(d) Scope. The example is conditional on the postulates and the cited mathematics; the matrix element of (c) is an identity about the model's assigned position observables. No transition rate, no spectral intensity, no time-dependent coupling, no measurement model and no empirical consequence is computed; the hypothesis that the relevant eigenvalue is nondegenerate is essential for (b).

• Why the same-parity case vanishes. The parity action conjugates the position operator to its negative, so a matrix element between two states of equal parity equals its own negative; the argument uses the involution structure and self-adjointness of $\mathrm{\Pi }$, not the details of the potential beyond its evenness.
• What a selection rule is not. The vanishing statement is an identity for the model's assigned position observables in the declared even-potential model. Turning it into a statement about observed transition intensities requires a time-dependent coupling model, the corresponding transition-rate analysis, a detection model and a sampling model, none of which is supplied here.

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m>0$ and spin label $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$ and translations $T_a$ (Componentwise position and momentum operators on the component-valued wave-function space clause (c)); assume the Axiom of Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Consider the free model, the bounded-potential model with $V\equiv 0$: $H_0:=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2,D(H_0)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ with flow $U(t)=\exp (-it{H}_0\mathrm{/}\mathrm{\hslash })$, momentum unitary $U^{(r)}$ (The momentum representation of the componentwise kinematics) and $f_t(p):=e^{-it\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m\mathrm{\hslash })}$. Then:

(a) for every $a\in {\mathbb{R}}^3$ the translation $T_a$ is a symmetry of $H_0$ in the sense of Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables clause (b), and $T_a{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}{T}_a^{-1}=e^{-it{H}_0\mathrm{/}\mathrm{\hslash }}(a\in {\mathbb{R}}^3,t\in \mathbb{R});$

(b) in the momentum representation the two actions are the multipliers $U^{(r)}{T}_a{U}^{(r)-1}=M_{e^{-ia\cdot p\mathrm{/}\mathrm{\hslash }}},U^{(r)}{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}{U}^{(r)-1}=M_{f_t},$ which commute; consequently $T_{a}U(t)=U(t)T_a$ for all $a,t$;

(c) Momentum conservation. For every $k$ the momentum component $P_k$ is a conserved observable of the free model: for every preparation $\rho$, Borel $B\subseteq \mathbb{R}$ and $t$, ${\mu }_{\rho (t)}^{P_k}(B)={\mu }_{\rho (0)}^{P_k}(B),$ and for a unit vector $\psi (0)\in D(P_k)$ the mean and second moment of the law of $P_k$ are finite and independent of $t$, with time-zero values $⟨P_k\psi ,\psi ⟩$ and $\mathrm{\parallel }{P}_k\psi {\mathrm{\parallel }}^2$; the coordinate translations satisfy $T_{t{e}_k}=e^{-it{P}_k\mathrm{/}\mathrm{\hslash }}$, so the momentum components are the generators of the translational symmetry groups;

(d) the modulus of the momentum representative is invariant in time: $\mid \left(U^{(r)}{e}^{-it{H}_0\mathrm{/}\mathrm{\hslash }}\psi \right)(p)\mid =\mid \left(U^{(r)}\psi \right)(p)\mid \text{for almost every }p\text{ and every }\psi \in H_s;$ consequently the squared modulus of the momentum representative is the same almost-everywhere function of $p$ at every time;

(e) Scope. The statement concerns the free model $V\equiv 0$ only. For a nonconstant bounded potential the translations need not be symmetries and momentum need not be conserved; that failure is exhibited on this page. No trajectory in space, no classical limit, no measurement model and no empirical result is asserted.

• Momentum conservation is translation invariance. The conservation of $P_k$ in (c) is the special case $V\equiv 0$ of the translation-invariance theorem; the point of the example is the explicit multiplier form: both the translations and the free flow act in the momentum representation by unimodular multipliers, and multipliers of the same variable commute.
• No trajectory. The invariance of $\mathrm{\mid }{U}^{(r)}\psi \mathrm{\mid }$ in the momentum variable is a statement about the momentum representative's class, not a statement that the particle moves along a classical straight line; no position-space trajectory is asserted.

Let $V:{\mathbb{R}}^3\to \mathbb{R}$ be any real, bounded, measurable, nonconstant potential energy, expressed in joules. Then for every nonzero $a\in {\mathbb{R}}^3\setminus \{0\}$ the translation $T_a$ is a symmetry of the declared bounded-potential dynamics with energy $H_V$.

• Where the failure comes from. The criterion [F1] is an almost-everywhere identity of the shifted potential classes, and the example realizes the failure on a set of positive measure. Boundedness and measurability suffice for the model to be well posed; they do not produce any translation symmetry.
• What is not claimed. The counterexample refutes one universal statement about bounded nonconstant potentials; it does not claim that every bounded nonconstant potential has no nontrivial translational symmetry (potentials constant along a direction do have the corresponding translations as symmetries), and it makes no statement about any laboratory potential or measurement.

Adopt the postulates P1, P2, P4, P5 and P6 in the readings of The massive spin-s particle model of the framework: state space, operators, units and frame readings, Assigned sharp observables and the outcome law of the framework, Assigned energy observable, unitary flow and the two pictures of autonomous evolution and Symmetries of the declared dynamics, unitary symmetry groups and conserved sharp observables for a declared particle of mass parameter $m>0$ and spin label $s$, with state space $H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^r)$, $r=2s+1$, componentwise operators $Q_j,P_k$, rotation action $U(R)$ and coordinate-rotation generators $L_i$ (The spatial rotation action on the component-valued wave-function space, Coordinate-rotation generators on the Schwartz classes and their commutators); assume the Axiom of Choice, the Axiom of Countable Choice and the Axiom of Dependent Choice (The Axiom of Choice, The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain). Let $v:[0,\mathrm{\infty })\to \mathbb{R}$ be bounded and measurable, and let $V:{\mathbb{R}}^3\to \mathbb{R}$ be real, bounded and measurable with $V(x)=v(\mathrm{\mid }x\mathrm{\mid })\text{for almost every }x\in {\mathbb{R}}^3,$ a radial potential almost everywhere (if $V$ is the composition $x\mapsto v(\mathrm{\mid }x\mathrm{\mid })$ everywhere, this holds with empty exceptional set). Let $H_V=\frac{1}{2m}{\sum }_{k=1}^3{P}_k^2+M_V,D(H_V)=W^{2,2}({\mathbb{R}}^3;{\mathbb{C}}^r),$ be the bounded-potential energy operator. Then:

(a) for every $R\in SO(3)$ one has $V(R^{-1}x)=V(x)$ for almost every $x$, and $V(-x)=V(x)$ for almost every $x$;

(b) every $U(R)$ is a symmetry of $H_V$, the map $R\mapsto U(R)$ is a strongly continuous unitary symmetry group of the declared dynamics, and each generator $L_i$ is a conserved observable: for every preparation $\rho$, Borel $B\subseteq \mathbb{R}$ and $t$, ${\mu }_{\rho (t)}^{L_i}(B)={\mu }_{\rho (0)}^{L_i}(B),$ with conserved mean and second moment for unit vectors in $D(L_i)$ and time-zero values $⟨L_i\psi ,\psi ⟩$, $\mathrm{\parallel }{L}_i\psi {\mathrm{\parallel }}^2$, in the unit $\mathrm{J}\mathrm{s}$ of the model's action (Dimensions, SI unit conventions and quantity types of the framework);

(c) the parity action $\mathrm{\Pi }$ is a symmetry of $H_V$, and its outcome law in every preparation is time-independent;

(d) Scope. The example is conditional on the postulates and the bounded radial class; it computes no eigenvalue, no bound state, no degeneracy and no radial solution, and the boundedness of $v$ excludes the Coulomb potential and other unbounded radial models, which require their own self-adjointness and spectral analysis. No claim is made that any particular physical potential is radial, and no empirical result is recorded.

• What makes the example work. Only two ingredients are used: the radial identity $V(x)=v(\mathrm{\mid }x\mathrm{\mid })$ almost everywhere and the orthogonality of rotations through $\mathrm{\mid }{R}^{-1}x\mathrm{\mid }=\mathrm{\mid }x\mathrm{\mid }$; the boundedness of $v$ keeps the model inside the bounded-potential class of the page, whose self-adjointness and conservation results are already established.
• What an angular-momentum conservation law is here. The conserved object is the outcome law of the assigned generator observable $L_i$ (with its moments where they exist); no classical angular-momentum vector, no definite value and no measured spectral line is asserted.

The free model: translation symmetries and conserved momentum in the momentum representation is the exact free-model computation: with $V\equiv 0$ every translation is a symmetry, the free flow in the momentum representation is the unimodular multiplier $M_{f_t}$, $f_t(p)=e^{-it\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}(2m\mathrm{\hslash })}$, the translations are the multipliers $M_{e^{-ia\cdot p\mathrm{/}\mathrm{\hslash }}}$, and the momentum components are conserved sharp observables with the law, moment and generator statements of Translation invariance of bounded potentials and conservation of momentum; the modulus of the momentum representative is invariant in time almost everywhere. A bounded nonconstant potential with no nontrivial translational symmetry shows that translations need not remain symmetries for nonconstant potentials: for the bounded measurable potential energy $E_{\ast }{1}_{B(0,R)}$, with $E_{\ast }>0$ in J and $R=1\mathrm{m}$, no nonzero translation is a symmetry, because the shifted potential differs from the original on a set of positive measure, so the symmetry criterion of the translation theorem is not met. For a specified translation $a$, the criterion is exactly $V(\cdot -a)=V$ almost everywhere; nonconstancy in a direction alone does not exclude discrete translations along that direction. Definite parity of nondegenerate eigenstates and the position-matrix-element selection rule derives, under its stated model and choice hypotheses for a real bounded measurable potential even almost everywhere, that every energy eigenspace is invariant under the parity action, that a nondegenerate eigenstate has definite parity, and that the position matrix element $⟨Q_j\psi ,{\psi }^{\mathrm{\prime }}⟩$ vanishes for unit vectors $\psi ,{\psi }^{\mathrm{\prime }}\in H_s=L^2({\mathbb{R}}^3;{\mathbb{C}}^{2s+1})$ of equal definite parity with $\psi \in D(Q_j)$, as in clause (c). Here $j\in \{1,2,3\}$, $Q_j\psi (x)=x_j\psi (x)$ on the maximal domain $D(Q_j)=\{f\in H_s:x_{j}f\in H_s\}$, and equal definite parity means $\mathrm{\Pi }\psi =\sigma \psi$ and $\mathrm{\Pi }{\psi }^{\mathrm{\prime }}=\sigma {\psi }^{\mathrm{\prime }}$ for a common $\sigma \in \{-1,1\}$, with $\mathrm{\Pi }f(x)=f(-x)$. No position-domain condition is required on ${\psi }^{\mathrm{\prime }}$; evenness of the potential supplies none on $\psi$. A radial bounded potential: rotational and parity symmetry and conservation of the rotation generators verifies the almost-everywhere rotational and parity invariance of a bounded radial potential and instantiates Rotational and parity invariance of bounded-potential models: symmetries and conservation of angular momentum and parity laws: all rotations are symmetries, the rotation action is a strongly continuous unitary symmetry group, and the coordinate-rotation generators $L_i$ are conserved sharp observables, with the underlying conservation mechanism of Conservation of a symmetry generator: commuting flows, energy compatibility and conserved outcome laws.

None of the four items is an experiment: no apparatus, preparation map, calibration, detection model, efficiency, resolution, background model, sampling model, uncertainty budget, count or dataset appears, and no measured value of any model parameter is asserted. The objects $m$, $q$ when present, $s$, the potentials and the conserved laws are model data and model statements in the units of Dimensions, SI unit conventions and quantity types of the framework; deciding that a laboratory system realizes the model, that its potential is approximately radial or even, or that a transition amplitude is the matrix element computed here, is a tested identification with uncertainty under the preparation and apparatus assumptions, and it is external to this page. In particular the parity selection rule is an identity about the model's assigned position observables: turning it into a statement about observed intensities requires a time-dependent coupling model, the corresponding transition-rate analysis, a detection model and a sampling model, none of which is supplied here.

The free computation is conditional on $V\equiv 0$; the counterexample concerns one potential inside the bounded measurable class and refutes only the universal claim that every bounded nonconstant potential has all nonzero translations as symmetries (potentials constant along a direction keep the corresponding translations). The central-potential example assumes the radial identity almost everywhere and a bounded radial function; it asserts no eigenvalue, bound state, degeneracy, radial solution or spectral line, and its boundedness hypothesis excludes the Coulomb and other unbounded radial models. The parity example requires the evenness hypothesis and, for definite parity, the nondegeneracy of the eigenvalue; at a degenerate eigenvalue only the eigenspace invariance is asserted for arbitrary eigenvectors. The equal-parity matrix-element identity separately requires the displayed $D(Q_j)$ condition and directly assumed parity signs; it requires neither an eigenvector hypothesis nor nondegeneracy. No item of this page makes a claim about antiunitary or time-reversal symmetries, about superselection, about spinorial rotations or the internal factor, or about relativistic and field-theoretic structures.