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Cambridge University Press eBooks

Quantum formulation of angle and orbital angular momentum

Jörg B. Götte, Stephen Mark Barnett

Published Nov 8, 2012• 2 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511795213.007

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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.
The question how to quantize a classical system where an angle $\ensuremath{\varphi}$ is one of the basic canonical variables has been controversial since the early days of quantum mechanics. The problem is that the angle is a multivalued or discontinuous variable on the corresponding phase space. The remedy is to replace $\ensuremath{\varphi}$ by the smooth periodic functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$ and $\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$. In the case of the canonical pair $(\ensuremath{\varphi},{p}_{\ensuremath{\varphi}})$, where ${p}_{\ensuremath{\varphi}}$ is the orbital angular momentum (OAM), the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}={\ensuremath{\varphi}∊\mathbb{R}\phantom{\rule{0.2em}{0ex}}\mathrm{mod}\phantom{\rule{0.2em}{0ex}}2\ensuremath{\pi},{p}_{\ensuremath{\varphi}}∊\mathbb{R}}$ has the global topological structure ${S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$ of a cylinder on which the Poisson brackets of the three functions $\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi},\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varphi}$, and ${p}_{\ensuremath{\varphi}}$ obey the Lie algebra of the Euclidean group E(2)\mathrm{E}(2)E(2) in the plane. This property provides the basis for the quantization of the system in terms of irreducible unitary representations of the group E(2)\mathrm{E}(2)E(2) or of its covering groups. A crucial point is that, due to the fact that the subgroup $\mathrm{SO}(2)\ensuremath{\cong}{S}^{1}$ is multiply connected, these representations allow for fractional OAM $l=\ensuremath{\hbar}(n+\ensuremath{\delta}),n∊\mathbb{Z},\ensuremath{\delta}∊[0,1)$. Such $\ensuremath{\delta}\ensuremath{\ne}0$ have already been observed in cases like the Aharonov-Bohm and fractional quantum Hall effects, and they correspond to the quasimomenta of Bloch waves in ideal crystals. The proposal of the present paper is to look for fractional OAM in connection with the quantum optics of Laguerre-Gaussian laser modes in external magnetic fields. The quantum theory of the phase space ${\mathcal{S}}_{\ensuremath{\varphi},{p}_{\ensuremath{\varphi}}}$ in terms of unitary representations of E(2)\mathrm{E}(2)E(2) allows for two types of ``coherent'' states, the properties of which are discussed in detail: nonholomorphic minimal-uncertainty states and holomorphic ones associated with Bargmann-Segal Hilbert spaces.
The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.
The theory of angular momentum performance a significant position in the classical and quantum mechanical study of physical properties, such as studies into nuclear, atomic, and molecular processes, as well as other quantum problems, including spherical symmetry. In this analysis, angular momentum operators are described in multiple ways, based on the angular momentum operator's commutator, matrix, and geometric representation, The eigenvalue and eigenvector were also known for operatorsJ ̂_±,J ⃑ ̂^2, J ̂_x,J ̂_y and J ̂_zwithin the |j,┤ ├ m⟩ basis. Furthermore, in quantum mechanics, angular momentum is called quantized variable, meaning that it comes in discrete quantities. In contrast to the macroscopic system case where a continuous variable is angular momentum.
The uncertainty relation between angle and orbital angular momentum had not been formulated in a similar form as the uncertainty relation between position and linear momentum because the angle variable is not represented by a quantum mechanical self-adjoint operator.Instead of the angle variable operator, we introduce the complex position operator Ẑ = x + iŷ and interpret the order parameter µ = Ẑ / Ẑ † Ẑ as a measure of certainty of the angle distribution.We prove the relation between the uncertainty of angular momentum and the angle order parameter.We also prove its generalizations and discuss experimental methods for testing these relations.