On schwinger’s angular-momentum calculus and the dirac bracket

Györgyi, G.; Kövesi‐Domokos, S.

doi:10.1007/bf02711789

Cited by 12 publications

(19 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following is a list of various forms of the GM in the current literature. Because (6)- (8) give the most general GM of p (α,β)i satisfying the fundamental commutators (20 ) (v) The fifth choice is made by many groups based on different theoretical grounds and it should be α = 1 and β = 0 13,17,[19][20][21][22][23]30 .…”

Section: Geometric Momenta For a Particle On The Sphere: A Reviewmentioning

confidence: 99%

See 1 more Smart Citation

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

2011

View full text Add to dashboard Cite

In Dirac's canonical quantization theory on systems with second-class constraints, the commutators between the position, momentum and Hamiltonian form a set of algebraic relations that are fundamental in construction of both the quantum momentum and the Hamiltonian. For a free particle on a two-dimensional sphere or a spherical top, results show that the well-known canonical momentum p θ breaks one of the relations, while three components of the momentum expressed in the three-dimensional Cartesian system of axes as p i (i = 1, 2, 3) are satisfactory all around. This momentum is not only geometrically invariant but also self-adjoint, and we call it geometric momentum. The nontrivial commutators between p i generate three components of the orbital angular momentum; thus the geometric momentum is fundamental to the angular one. We note that there are five different forms of the geometric momentum proposed in the current literature, but only one of them turns out to be meaningful.

show abstract

Section: Geometric Momenta For a Particle On The Sphere: A Reviewmentioning

confidence: 99%

“…where ϕ α (α = 1, 2) are, respectively, the primary and second-class constraint (12) and (13) and the matrix elements C αβ is defined by…”

Section: Complete Determination Of Geometric Momenta and The Hammentioning

confidence: 99%

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

2011

View full text Add to dashboard Cite

show abstract

“…Exploration of the proper form and the physical meaning of momentum in curvilinear coordinates attracts constant attention since the birth of quantum mechanics. [1][2][3][4][5][6][7][8][9][10][11][12][13] In this paper, we show that canonical momenta P ξ associated with its conjugate canonical positions, or coordinates, ξ, are closely related to mean curvatures of the surface ξ = const. So, the geometric momenta [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] that are under extensive studies and applications are closely related to natural decompositions of the momentum operator in gaussian normal, curvilinear in general, coordinates.…”

Section: Introductionmentioning

confidence: 82%

“…where, Π r , Π θ and Π ϕ are so-called the geometric momenta [9][10][11][12][13][14][15][16][17][18][19][20] for the corresponding surfaces, though the last one M ϕ = 0 is trivial,…”

Section: Natesmentioning

confidence: 99%

On relationship between canonical momentum and geometric momentum

Xiao,

Liu

2016

Preprint

View full text Add to dashboard Cite

Decompositing of N + 1-dimensional gradient operator in terms of Gaussian normal coordinates (ξ 0 , ξ µ ), (µ = 1, 2, 3, ..., N ) and making the canonical momentum P 0 along the normal direction n to be hermitian, we obtain nP 0 = −i (n∂ 0 − M 0 ) with M 0 denoting the mean curvature vector on the surface ξ 0 = const. The remaining part of the momentum operator lies on the surface, which is identical to the geometric one.

show abstract

“…where the occupation numbers are respectively n 1 = a † 1 a 1 and n 2 = a † 2 a 2 , and the eigenvalues of the occupation number operator are n α = 0, 1, 2, .... The unit direction vector N and the geometric momentum Π are respectively [12,28,29],…”

mentioning

confidence: 99%

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

Liu

Shen

Xun

et al. 2013

Int. J. Geom. Methods Mod. Phys.

View full text Add to dashboard Cite

With a recently introduced geometric momentum that depends on the extrinsic curvature and offers a proper description of momentum on two-dimensional sphere, we show that the annihilation operators whose eigenstates are coherent states on the sphere take the expected form αx + iβp, where α and β are two operators that depend on the angular momentum and x and p are the position and the geometric momentum, respectively. Since the geometric momentum is manifestly a consequence of embedding the two-dimensional sphere in the three-dimensional flat space, the coherent states reflects some aspects beyond the intrinsic geometry of the surfaces. PACS numbers: 03.65.-w Quantum mechanics; 02.40.-k Differential geometry; 92.60.hc Mesosphere; 73.20.Fz Quantum localization on surfaces and interfacesThe coherent states on two-dimensional sphere, generated by the annihilation operators, were discovered around the turn of present century, independently by Hall [1][2][3][4][5][6] in the Bargmann representation, and by in the position representation, respectively. Once each group of them got to know the work of the other, both soon realized that their coherent states are essentially the same [6,9], and the equivalence was also noted by other group [11]. However, it is puzzling that in the annihilation operators they introduced, there is a fundamental quantity that is represented by a non-hermitian operator and has the same dimension of linear momentum, but it does not bear a transparent physical nor geometric meaning. This article points out that the physical and geometric interpretation of the fundamental quantity is easily available, based on the geometric momentum that is recently introduced to offer a proper description of momentum on sphere [12], whose general form for an arbitrary twodimensional curved surface with M denoting the mean curvature, is given by [12][13][14], *

show abstract

On schwinger’s angular-momentum calculus and the dirac bracket

Cited by 12 publications

References 8 publications

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

Geometric momentum: The proper momentum for a free particle on a two-dimensional sphere

On relationship between canonical momentum and geometric momentum

On Relation Between Geometric Momentum and Annihilation Operators on a Two-Dimensional Sphere

Contact Info

Product

Resources

About