Constants of Motion of the Harmonic Oscillator
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Constants of Motion of the Harmonic Oscillator ´ Belmonte1 · Sebastian ´ Cuellar ´ 1 Fabian Received: 23 June 2020 / Accepted: 11 September 2020 / © Springer Nature B.V. 2020
Abstract We prove that Weyl quantization preserves constant of motion of the Harmonic Oscillator. We also prove that if f is a classical constant of motion and Op(f ) is the corresponding operator, then Op(f ) maps the Schwartz class into itself and it defines an essentially self-adjoint operator on L2 (Rn ). As a consequence, we obtain detailed spectral information of Op(f ). A complete characterization of the classical constants of motion of the Harmonic Oscillator is given and we also show that they form an algebra with the integral Moyal star product. We give some interesting examples of constants of motion and we analyze Weinstein-Guillemin average method within our framework. Keywords Constant of motion · Harmonic oscillator · Weyl calculus · Moyal product Mathematics Subject Classification (2010) 81S08
1 Introduction Let Σ be a symplectic manifold and h0 ∈ C ∞ (Σ) be a classical Hamiltonian. A classical constant of motion for h0 is a classical observable f ∈ C ∞ (Σ) such that {h0 , f } = 0, where {·, ·} is the Poisson bracket on C ∞ (Σ). Similarly, if H0 is a quantum Hamiltonian, a quantum constant of motion for H0 is a quantum observable F such that [H0 , F ] = 0, which means that H0 and F are strongly commuting self-adjoint operators on certain Hilbert space H. Equivalently, a constant of motion
Sebasti´an Cu´ellar
[email protected] Fabi´an Belmonte [email protected] 1
Universidad Cat´olica del Norte, Angamos 0610, Antofagasta, Chile
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Math Phys Anal Geom
(2020) 23:35
is an observable invariant by the evolution determined by the Hamiltonian. Moreover, classical and quantum constants of motion admit a decomposition through the reduction and diagonalization processes respectively (see Section 2.1). Heuristically, a canonical quantization is a map Op that sends suitable classical observables into quantum observables in a physically meaningful manner. The analogy between classical and quantum constant of motion might suggest that a canonical quantization Op should map classical constants of motion of h0 into quantum constants of motion of H0 = Op(h0 ). In such case we say that Op preserves constants of motion of h0 . However, at least when Op is the canonical Weyl quantization, since Op does not interchange the Poisson bracket of classical observables with the commutator of the corresponding operators (Groenewold-Van Hove’s no go Theorem [12]), we should expect that preservation of constants of motion occurs only for suitable Hamiltonians. We prove that Weyl quantization preserves constants of motion of the Harmonic Oscillator (Theorem 1). We combine the latter result with the N−representation Theorem to obtain important properties about the classical constants of motion of the Harmonic Oscillator and the corresponding operators, which we shall briefly summarize in the following paragraphs. In Section 2,
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