Rotational Heisenberg Inequalities

Since their discovery in 1927, the Heisenberg Inequalities have become an icon of quantum mechanics [1]. Often inappropriately referred to as the Uncertainty Principle, these inequalities relating the standard deviations of the position and momentum observables to Planck’s constant are one of the cornerstones of the quantum formalism even if the physical interpretation of quantum mechanics remains still open to controversy nowadays [2]. The Heisenberg Inequalities governing translational motion are well understood. However, the corresponding inequalities pertaining to rotational motion have not been established so far. To fill this gap, we present here the Rotational Heisenberg Inequalities relating the standard deviations of the orientation axis and orbital angular momentum observables of an isolated molecule. The reason for choosing this system is that a molecule separated from its environment corresponds to a bound system preserving the orbital angular momentum. Relative and rest observables. – The quantum dynamics of a molecular system consisting of N nuclei and n electrons is obtained from the classical dynamics by applying the “correspondence principle”. The position and momentum observables of the nucleus μ are characterised respectively by the self-adjoint operators Rμ ⊗ 1e and P μ ⊗ 1e, where μ = 1, .., N , acting trivially on the Hilbert subspace He associated to the electrons. Similarly, the position and momentum observables of the electron ν are respectively characterised respectively by the self-adjoint operators 1N ⊗ rν and 1N ⊗ pν where ν = 1, .., n, acting trivially on the Hilbert subspace HN associated to the nuclei. These operators satisfy the canonical commutation relations, i.e. [ ej · P μ, e ·Rμ ] = − i~ ( ej · e ) 1N [ ej · pν , e · rν ] = − i~ ( ej · e ) 1e , (1) where where ej are the units vectors of an orthonormal basis and e are the units vectors of the dual orthonormal basis. In order to treat molecular rotations as genuine quantum degrees of freedom, we introduce explicitly the rotation group and the associated Lie algebra [3]. In such a treatment, we introduce a molecular orientation operator ω that is fully determined by the position operators Rμ and rν , which ensures that it is a self-adjoint operator. Since the position operators commute, the components of the operator ω satisfy the trivial commutation relations, [ e · ω, e · ω ] = 0 . (2) The orientation operator ω should not be confused with the angle velocity operator or the phase operator. The orientation operator ω is related to the molecular rotation operator R (ω) that belongs to the rotation group by exponentiation, i.e. R (ω) = exp (ω · G) , (3) taking into account the commutation relation (2). The components of the vector G are rank-2 tensors and generators of the rotation group. The action of the rotation group is locally defined as, (ej · G) x = ej × x . (4) The action of the rotation operator on a vectorial observable A ∈ L (H) is expressed in terms of the unitary representation of the rotation group U (ω) acting on the Hilbert space H by the well-known relation [4], U (ω)−1 ( e ·A ) U (ω) = ( e · R (ω) · ej ) ( e ·A ) . (5) p-1 ar X iv :1 50 3. 03 05 2v 1 [ qu an tph ] 1 0 M ar 2 01 5 S. D. Brechet et al. We also introduce the operators n(j) (ω) that are Killing vectors [5] of the Lie algebra of the rotation group, which are related to the rotation operator R (ω) and the rotation generators G as, R (ω)−1 · (ej · ∂ω) R (ω) = n(j) (ω) · G . (6) The dual operator m (ω) satisfies the duality condition, n(j) (ω) ·m (ω) = ej · e . (7) The operators n(j) (ω) and m (k) (ω) determine the structure of the Lie algebra of the rotation group. The description of molecular dynamics in a classical framework would be much simpler than in a quantum framework since in the former a rest frame could be attached easily to the physical system. In quantum physics, the approach is slightly different because observables are described mathematically by operators, which implies that there exists no rest frame and no centre of mass frame associated to the molecular system. However, even in the absence of a centre of mass frame, the position and momentum observables of the centre of mass can be expressed mathematically as self-adjoint operators. This enables us to define other position and momentum observables with respect to the centre of mass. We shall refer to them as “relative” position and momentum observables because they are the quantum equivalent of the classical relative position and momentum variables defined with respect to the center of mass frame. Then, using the rotation operator, we define the “rest” position and momentum observables, which are the quantum equivalent of the classical position and momentum variables defined in the molecular rest frame. The description of molecular dynamics is illustrated in Fig 1.