Abstract In the present paper we construct a properly defined quantum state ψ ( θ ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L = − i d / d θ in units of ℏ = 1 . The quantum uncertainties Δ θ and Δ L for the state are well-defined and are shown to give a lower value of the uncertainty product Δ θ Δ L in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨ L ⟩ of the state ψ ( θ ) is not required to be an integer. In the case of any half-integer mean value ⟨ L ⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δ θ = π 2 / 3 − 2 and Δ L = 1 / 2 .
Abstract In the present paper we construct a properly defined quantum state ψ ( θ ) expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L = − i d / d θ in units of ℏ = 1 . The quantum uncertainties Δ θ and Δ L for the state are well-defined and are shown to give a lower value of the uncertainty product Δ θ Δ L in contrast to the so called minimal uncertainty states as discussed in Franke-Arnold et al (2004 New J. Phys. 6 103-1-8). The mean value ⟨ L ⟩ of the state ψ ( θ ) is not required to be an integer. In the case of any half-integer mean value ⟨ L ⟩ the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δ θ = π 2 / 3 − 2 and Δ L = 1 / 2 .
In the present paper we construct a properly defined quantum state expressed in terms of elliptic Jacobi theta functions for the self-adjoint observables angular position θ and the corresponding angular momentum operator L=−id/dθ. The quantum uncertainties Δθ and ΔL for the state are well-defined and are, e.g., shown to give a lower value of the uncertainty product ΔθΔL than the minimal uncertainty states of Ref.\cite{Padgett_2004}. The mean value <L> of the state is not required to be an integer. In the case of any half-integer mean value <L> the state constructed exhibits a remarkable critical behavior with upper and lower bounds Δθ<π2/3−2 and ΔL>1/2.
The states of linear momentum that satisfy the equality in the Heisenberg uncertainty principle for position and momentum, that is the intelligent states, are also the states that minimize the uncertainty product for position and momentum. The corresponding uncertainty relation for angular momentum and angular position, however, is more complicated and the intelligent states need not be the constrained minimum uncertainty product states. In this paper, we investigate the differences between the intelligent and the constrained minimum uncertainty product states for the angular case by means of instructive approximations, a numerical iterative search and the exact solution. We find that these differences can be quite significant for particular values of angular position uncertainty and indeed may be amenable to experimental measurement with the present technology.
The Hamiltonian for the oscillator has earlier been written in the form where ν† and ν are raising and lowering operators for ν†ν, which has eigenvalues k (the ’’radial’’ quantum number), and λ† and λ are raising and lowering 3-vector operators for λ†⋅λ, which has eigenvalues l (the total angular momentum quantum number). A new set of coherent states for the oscillator is now defined by diagonalizing ν and λ. These states bear a similar relation to the commuting operators H, L2, and L3 (where L is the angular momentum of the system) as the usual coherent states do to the commuting number operators N1, N2, and N3. It is proposed to call them coherent angular momentum states. They are shown to be minimum-uncertainty states for the variables ν, ν† , λ, and λ†, and to provide a new quasiclassical description of the oscillator. This description coincides with that provided by the usual coherent states only in the special case that the corresponding classical motion is circular, rather than elliptical; and, in general, the uncertainty in the angular momentum of the system is smaller in the new description. The probabilities of obtaining particular values for k and l in one of the new states follow independent Poisson distributions. The new states are overcomplete, and lead to a new representation of the Hilbert space for the oscillator, in terms of analytic functions on C×K3, where K3 is the three-dimensional complex cone. This space is related to one introduced recently by Bargmann and Todorov, and carries a very simple realization of all the representations of the rotation group.
In the present study, we consider quasi-probability densities W[θ, p] and W1/2[θ, p] for pure states depending on two classical parameters θ and p, where −π ≤ θ ≤ π and p can take any real number. For integer values of p, the corresponding marginal distributions W[p] and W1/2[p] are positive and in accordance with Born’s rule in quantum mechanics for an angular momentum observable L. For the state ψ(θ) in Skagerstam and Rekdal [J. Opt. 26, 095201 (2024); arXiv:2312.16535v1 [quant-ph]], which can have half-integer angular momentum expectation values, negative values of W[θ, p] and W1/2[θ, p] can reveal non-classical features of ψ(θ) but in an ambiguous manner. In line with Franke-Arnold et al. [New J. Phys. 6, 103 (2004)], it is shown that experimental data of the uncertainties Δθ and ΔL can be sufficient to reveal quantum-mechanical features of such states ψ(θ) without necessarily making use of quasi-probability densities.
In this work we study various notions of uncertainty for angular momentum in the spin- s representation of SU(2). We characterize the ‘uncertainty regions’ given by all vectors, whose components are specified by the variances of the three angular momentum components. A basic feature of this set is a lower bound for the sum of the three variances. We give a method for obtaining optimal lower bounds for uncertainty regions for general operator triples, and evaluate these for small s . Further lower bounds are derived by generalizing the technique by which Robertson obtained his state-dependent lower bound. These are optimal for large s , since they are saturated by states taken from the Holstein–Primakoff approximation. We show that, for all s , all variances are consistent with the so-called vector model, i.e., they can also be realized by a classical probability measure on a sphere of radius s ( s + 1 ) . Entropic uncertainty relations can be discussed similarly, but are minimized by different states than those minimizing the variances for small s . For large s the Maassen–Uffink bound becomes sharp and we explicitly describe the extremalizing states. Measurement uncertainty, as recently discussed by Busch, Lahti and Werner for position and momentum, is introduced and a generalized observable (POVM) which minimizes the worst case measurement uncertainty of all angular momentum components is explicitly determined, along with the minimal uncertainty. The output vectors for the optimal measurement all have the same length r ( s ) , where r ( s ) / s → 1 as s → ∞ .
The quantum mechanical description of the azimuthal rotation angle and the orbital angular momentum around the polar axis differs greatly from the description of position and momentum. This has led to a controversy over the existence of a self-adjoint angle operator in the literature. It is possible to circumvent the problems of defining a self-adjoint operator for a periodic variable by using trigonometric functions of operators as a basis for the quantum mechanical description. This approach, however, does not allow us to study the properties of the angle operator itself. By using a state space of an arbitrarily large yet finite number of dimensions, it is possible to introduce angle and orbital angular momentum as a conjugate pair of variables both represented by Hermitian matrices. After physical and measurable quantities have been calculated in this state space the number of dimensions is allowed to tend to infinity in a limiting procedure. The first part of this chapter reviews the difficulties associated with the quantum mechanical formulation of angle and orbital angular momentum and presents a way to overcome these problems using a finite-dimensional state space. In the second part this formulation is generalised to include the phenomenon of non-integer orbital angular momentum. This part is based on a chapter of the PhD thesis of Götte [1], where further details may be found. Introduction The correct description of a periodic variable such as a rotation angle or the optical phase has been a long standing problem in quantum mechanics. At the root of the problem is the question of whether the variable itself is restricted in a 2 π radian range or whether it evolves continuously without bound.