Constants of Motion of the Harmonic Oscillator

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 $\mathfrak {Op}(f)$ is the corresponding operator, then $\mathfrak {Op}(f)$ maps the Schwartz class into itself and it defines an essentially self-adjoint operator on $L^{2}(\mathbb {R}^{n})$. As a consequence, we obtain detailed spectral information of $\mathfrak {Op}(f)$. 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.

Appendix A: Weyl Quantization

In this appendix we recall the definition of Weyl quantization and some of its main features.

Weyl quantization [12, 29] (or Weyl calculus) is a map meant to transform functions in the canonical phase space ℝ^{sp 2n} (classical observables) into operators on L^{sp 2}(ℝ^{sp n}) (quantum observables) in a physically meaningful manner, that is, taking into account the analogies between the descriptions of classical and quantum mechanics. There are several approaches to introduce Weyl quantization, for instance [25, 27, 30], but for the purposes of this articles we will mainly follow [11]. We also recommend [2] for a quite complete review of quantization theory.

Formally, for certain function f on phase space, we define the operator 𝔒𝔭_ℏ(f) acting on L^{sp 2}(ℝ^{sp n}) given by

$$ \mathfrak{O}\mathfrak{p}_{\hbar}(f)u(x)={\int}_{\mathbb{R}^{n}}{\int}_{\mathbb{R}^{n}}f\left( \frac{x+y}{2},\xi\right)e^{\frac{i}{\hbar}(x-y)\cdot \xi}u(y)\mathrm{d}\xi\mathrm{d} y, $$

(19)

where ℏ is a positive parameter interpreted as Planck’s constant. In this article, we will not need to consider the roll played by ℏ, so we will take ℏ = 1 and 𝔒𝔭 := 𝔒𝔭₁.

The meaning of the expression (19) changes depending on the class of functions we are considering. More precisely, if $f\in S^{\prime }(\mathbb{R} ^{2n})$, then $\mathfrak{O}\mathfrak{p} (f):S(\mathbb{R}^{n})\to S^{\prime }(\mathbb{R}^{n})$, so (19) defines a tempered distribution (in particular, the evaluation on x is not well-defined). The standard way to rigorously define 𝔒𝔭(f) is the following: Notice that, the kernel of 𝔒𝔭(f) should be

$$ \begin{array}{@{}rcl@{}} K_{f}(x,y)&=&(I\otimes \mathcal F) f\left( \frac{x+y}{2},y-x\right) \\ &=&{\int}_{\mathbb{R}^{n}}f\left( \frac{x+y}{2},\xi\right)e^{i(x-y)\cdot \xi}\mathrm{d}\xi, \end{array} $$

(20)

where $(I\otimes \mathcal F)$ is the Fourier transform in the momentum variable. The change of variables $(x,y)\to \left (\frac {x+y}{2},y-x\right)$ is linear, injective and preserves measure. Thus composing with the latter change of variable maps S(ℝ^{sp 2n}) into itself continuously, and it can be extended to $S^{\prime }(\mathbb{R} ^{2n})$. Since the $(I\otimes \mathcal F)$ is well-defined on $S^{\prime }(\mathbb{R} ^{2n})$, we can define K_f for $f\in S^{\prime }(\mathbb{R} ^{2n})$ using the middle expression in (20). In fact, the map f → K_f is an automorphism of the locally convex space $S^{\prime }(\mathbb{R} ^{2n})$. Finally, for u,v ∈ S(ℝ^{sp n}), we define

$$ [\mathfrak{O}\mathfrak{p}(f)u](v)= K_{f}(v\otimes u) $$

The Wigner transform is a fundamental object in the description of Weyl quantization. It is defined by the map W : S(ℝ^{sp n}) × S(ℝ^{sp n}) → S(ℝ^{sp 2n}) given by

$$ W(u,v)={\int}_{\mathbb{R}^{n}}e^{-i\xi\cdot p}u\left( x+\frac{p}{2}\right)\overline{v\left( x-\frac{p}{2}\right)}\mathrm{d} p. $$

The Wigner transform can be regarded as the restriction to functions of two variables of the form $g(x,y)=u(x)\overline {v(y)}$ of the map $\tilde W$ defined by

$$ \tilde W g(x,\xi)={\int}_{\mathbb{R}^{n}}e^{-i\xi \cdot p}g\left( x+\frac{p}{2}, x-\frac{p}{2}\right). $$

(21)

Using the arguments in the definition of the kernel K_f, we can define $\tilde W$ on $S^{\prime }(\mathbb{R} ^{2n})$. In fact, $\tilde W$ is the inverse of the map f → K_f, and it is unitary on L^{sp 2}(ℝ^{sp 2n}). In particular, we have that

$$ \langle\mathfrak{O}\mathfrak{p}(f)u,v\rangle=\langle f,W(v,u)\rangle. $$

(22)

For instance, if f is polynomially bounded, then

$$ \langle Op(f)u,v\rangle ={\int}_{\mathbb{R}^{2n}}f W(u,v). $$

One of the main properties of Weyl quantization is its relation to the so called metaplectic representation. Let Sp(2n) be the real symplectic group, i.e. the group formed by all the linear and symplectic maps S : ℝ^{sp 2n} → ℝ^{sp 2n}. There is a map $\mu : Sp(2n)\to \mathcal U(L^{2}(\mathbb{R} ^{n}))$, characterized up to a ± 1 factor by the identity:

$$ \mathbb{R}ho(S(q,p))=\mu(S)\mathbb{R}ho(q,p)\mu(S)^{-1}, $$

where $(q,p)\in \mathbb {H}_{n}$ is an element of Heisenberg group and ρ is the Schrödinger representation. The map μ is called the metaplectic representation (see [11] for details). As consequences, we have the very important (and beautiful) identities

$$ W(\mu(S)\phi,\mu(S)\psi)=W(\phi,\psi)\circ S^{*} $$

(23)

$$ \mathfrak{O}\mathfrak{p}(f\circ S^{*})=\mu(S)\mathfrak{O}\mathfrak{p}(f)\mu(S)^{-1} $$

(24)

The metaplectic representation is not a true representation of Sp(2n), because in general we only have that μ(ST) = ±μ(S)μ(T), so it defines a representation of the double covering group Sp₂(n) of Sp(2n). However, restricted to the complex unitary group U(n), μ does define a representation (notice that the Hamiltonian flow of the Harmonic Oscillator belongs to U(n)).