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The Journal of Chemical Physics

Dirac states for unit position and momentum: Phase consistency of their angular momentum representations

Robert F. Snider

Published Apr 1, 1982• 7 citations
Publisher

Abstract

It is shown that the position and momentum directional representations of angular momentum states must satisfy Σλs〈r̂‖λs〉〈λs‖p̂〉 = Σλs(i)λYλs(r̂)Yλs(p̂)*. This imposes phase constraints on the relation between 〈r̂‖λs〉, 〈p̂‖λs〉, Yλs(r̂), and Yλs(p̂). In the accompanying paper, it is shown that this resolves a problem in the centrifugal sudden approximation of molecular collision theory.

DOI
10.1063/1.443456

Related papers

Reply to ‘‘Comment on ‘Dirac states for unit position and momentum: Phase consistency of their angular momentum representations’ ’’a)

Robert F. Snider

It is stressed that the defined states and their relationship deal exclusively with the quantum mechanics of directions and angular momentum. In particular, they have nothing to do with the possibility of factoring three dimensional motion into magnitude and directional components. A magnitude independent derivation of the previously obtained result is given.

The Journal of Chemical Physics•Mar 1, 1983•0 citations

Comment on ‘‘Dirac states for unit position and momentum: Phase consistency of their angular momentum representations’’

G. E. Hahne

The argument presented makes two points about Snider’s [J. Chem. Phys. 76, 3543 (1982)] transformation between ‘‘unit position’’ states ‖r̂〉 and ‘‘unit momentum’’ states ‖p〉 for the motion of a structureless particle in three space: Snider’s transformation (1) implicitly defines an operator p̂ that has a classical limit a unit vector that is perpendicular to both r̂ and l=r×p, while on a classical scattering trajectory for r→∞, the physical unit vector p/p≊±r̂, and (2) attempts, inconsistent with the kinematics of states and observables of the system, to pass from r̂ to p̂ amplitudes of a state vector without any reference to the scalar r or p dependence of the wave functions.

The Journal of Chemical Physics•Mar 1, 1983•1 citation

ANGULAR MOMENTUM-PHASE WIGNER OPERATOR FOR DESCRIBING ELECTRONS IN UNIFORM MAGNETIC FIELDS

Hong-Yi Fan

By introducing the angular momentum-phase Wigner operator both in <λ| representation and <l,r| representation, which are respectively the eigenstates of position and angular momentum of an electron in a uniform magnetic field, we establish the angular momentum-phase Wigner function whose marginal distributions are physically meaningful.

Modern Physics Letters B•Jun 10, 2001•2 citations

Minimum uncertainty states of angular momentum and angular position

Roberta Zambrini, D.T. Pegg, S.M. Barnett +2 more

In this paper, the much less known uncertainty relation for angular position and momentum to find the system at the edge of the phase interval is considered. States that minimize the uncertainty product with a constraint given either by the angular variance or by the angular momentum variance is examined.

European Quantum Electronics Conference (EQEC)•Jan 18, 2006•3 citations

MINIMUM UNCERTAINTY STATES FOR ANGULAR-MOMENTUM–PHASE UNCERTAINTY RELATION IN RADIUS–PHASE DESCRIPTION OF AN ELECTRON IN UNIFORM MAGNETIC FIELD

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Based on the |λ> representation and the radius–phase description of an electron's orbit track in a uniform magnetic field, we introduce the phase operator and derive the minimum uncertainty states for the angular-momentum–phase uncertainty relation. The derivation makes full use of the newly constructed |l, r> representation which is the common eigenvector of the angular momentum Lz and the radius operator (K- - iΠ+)(K+ + iΠ-).

Modern Physics Letters B•Apr 10, 2000•2 citations

The Future of Quantum Theory of Angular Momentum

James D. Louck

THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.

Birkhäuser Boston eBooks•Jan 1, 2000•0 citations
It is shown that the position and momentum directional representations of angular momentum states must satisfy Σλs〈r̂‖λs〉〈λs‖p̂〉 = Σλs(i)λYλs(r̂)Yλs(p̂)*. This imposes phase constraints on the relation between 〈r̂‖λs〉, 〈p̂‖λs〉, Yλs(r̂), and Yλs(p̂). In the accompanying paper, it is shown that this resolves a problem in the centrifugal sudden approximation of molecular collision theory.
The argument presented makes two points about Snider’s [J. Chem. Phys. 76, 3543 (1982)] transformation between ‘‘unit position’’ states ‖r̂〉 and ‘‘unit momentum’’ states ‖p〉 for the motion of a structureless particle in three space: Snider’s transformation (1) implicitly defines an operator p̂ that has a classical limit a unit vector that is perpendicular to both r̂ and l=r×p, while on a classical scattering trajectory for r→∞, the physical unit vector p/p≊±r̂, and (2) attempts, inconsistent with the kinematics of states and observables of the system, to pass from r̂ to p̂ amplitudes of a state vector without any reference to the scalar r or p dependence of the wave functions.
THE QUANTUM THEORY OF ANGULAR MOMENTUM began with Bohr’s enunciation of the rules for the quantization of orbital angular momentum in 1913, and the refinement of these rules by Sommerfeld [ 1 ] and Landé [ 2 ] within the framework of the old quantum theory. A basic result for angular momentum in modern quantum theory, which we call The First Fundamental Result, was given in the third foundational paper in 1926 by Born, Heisenberg, and Jordan [ 3 ], who not only gave the basic commutation relations for the components of the total angular momentum J, but all finite-dimensional Hermitian matrix representations of these components. Dirac [ 4 ] simultaneously deduced these basic commutation relations using his algebraic approach to quantum mechanics, and later gave a marvelously brief derivation of their matrix representations in his book [ 5 ]. Further developments, including the famous Pauli matrices and results relating to angular momentum selection rules for transitions in atomic systems, are given in the papers by Heisenberg and Jordan [ 6 ], Pauli [ 7 ], and Güttinger and Pauli [ 8 ]. These works, and the books by Born and Jordan [ 9 ] and Condon and Shortley [ 10 ], fixed the framework for much of the developments in angular momentum theory, which was hardly distinguishable from the general quantum theory in the context of which it was developed for the purpose of giving a consistent description of the spectra of atoms. Little emphasis was given to the universal and distinctive role of symmetry.