Distribution of xp in some molecular rotational states

Liu, Q.H.

doi:10.1016/j.physleta.2014.01.025

Cited by 8 publications

(8 citation statements)

References 29 publications

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“…In consequence, we have e r · P ⊥ + P ⊥ · e r = 0. The components in Cartesian coordinates give the standard projections [1][2][3][4][5][6][7][8][9][10],…”

Section: Radial and Transverse Decomposition Of The Gradient Operamentioning

confidence: 99%

Geometric momentum and angular momentum for charge-monopole system

Xiao

Liu

2018

Mod. Phys. Lett. A

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For a charge-monopole pair, though the definition of the orbital angular momentum is different from the usual one, and the transverse part of the momentum that includes the vector potential as an additive term turns out to be the so-called geometric momentum that is under intensive study recently. For the charge is constrained on the spherical surface with monopole at the origin, the commutation relations between all components of geometric momentum and the orbital angular momentum satisfy the so(3, 1) algebra. With construction of the geometrically infinitesimal displacement operator based on the geometric momentum, the so(3, 1) algebra implies the Aharonov-Bohm phase shift. The related problems such as charge and flux quantization are also addressed. PACS numbers: 14.80.Hv Magnetic monopoles, 03.65.-w Quantum mechanics.For a particle in the centrally symmetrical potential field, for instance, an electron interacts with proton that is placed at the origin of the coordinates, the orbital angular momentum is a conservative quantity, which is defined by,where the position of the particle is defined by its radius vector r, and P is the canonical momentum. Once splitting the momentum P into a radial P and a transverse part P ⊥ in the following, with r ≡ |r| and e r ≡ r/r, P ≡ e r (e r · P) , and P ⊥ ≡ P − e r (e r · P) ,we immediately find that in the angular momentum only the transverse part P ⊥ remains and its radial part P has zero effect in it for we have r × P = 0 and an equivalent definition of the orbital angular momentum,In quantum mechanics, P ≡ −i ∇. The splitting of P means the gradient operator ∇ for the centrally symmetrical system is also splittable in the similar fashion. In fact, the transverse momentum with a given radius r is the geometric one which is recently under extensive investigations [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], and its explicit form is given shortly, c.f., (11a)-(11c).In section II, we show how to split the gradient operator ∇ in spherical polar coordinates. In section III, we apply this splitting to the charge-monopole system. In final section IV, a brief conclusion is given. In whole of present paper, only the nonrelativistic motion of the charge is considered and the spin is ignored. I. RADIAL AND TRANSVERSE DECOMPOSITION OF THE GRADIENT OPERATOR IN SPHERICAL POLAR COORDINATESFor a spherically symmetrical potential field, the most useful coordinates are the spherical polar ones, and the transformation relation between the Cartesian coordinates (x, y, z) and the spherical polar coordinates (r, θ, ϕ) is,where θ is the polar angle from the positive z-axis with 0 ≤ θ < π, and ϕ is the azimuthal angle in the xy-plane from the x-axis with 0 ≤ ϕ < 2π. The gradient operator in the Cartesian coordinates, ∇ cart ≡ e x ∂ x + e y ∂ y + e z ∂ z , can be expressed in (r, θ, ϕ) by either of the following two ways, ∇ sp = e r ∂ ∂r + e θ 1 r ∂ ∂θ + e ϕ 1 r sin θ ∂ ∂ϕ , and (5a) ∇ sp = ∂ ∂r e r + 1 r ∂ ∂θ e θ + 1 r sin θ ∂ ∂ϕ e ϕ . (5b) * Electronic address: quanhuiliu@gmail.com

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“…In consequence, we have e r · P ⊥ + P ⊥ · e r = 0. The components in Cartesian coordinates give the standard projections [1][2][3][4][5][6][7][8][9][10],…”

Section: Radial and Transverse Decomposition Of The Gradient Operamentioning

confidence: 99%

Geometric momentum and angular momentum for charge-monopole system

Xiao

Liu

2018

Mod. Phys. Lett. A

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“…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%

“…[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: 99%

On relationship between canonical momentum and geometric momentum

Xiao,

Liu

2016

Preprint

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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.

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“…In Ref. 5, we presented the proper form of the posmom on a two-dimensional spherical surface S 2 and successfully worked out its distribution densities for some molecular rotational states. In this paper, we explore the posmom distribution of quantum states on a circle S 1 which frequently models the planar rigid rotor, molecular rotation constrained on a plane, etc.…”

Section: Introductionmentioning

confidence: 99%

Distribution of x ⋅ p for quantum states on a circle

Liu

Qin

Huang

et al. 2015

Int. J. Geom. Methods Mod. Phys.

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We first give the proper definition of the particle's position-momentum dot product, the so-called posmom x · p, to quantum states on a circular circle, in which the momentum turns out to be the geometric one that is recently intensively studied. Second, we carry out the posmom distributions for eigenstates of the free motion on the circle, i.e. exp(imϕ)/ √ 2π, (m = 0, ±1, ±2, . . .). The results are not only potentially experimentally testable, but also reflect a fact that the embedding of the circle S 1 in two-dimensional flat space R 2 is physically reasonable.

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Distribution of xp in some molecular rotational states

Cited by 8 publications

References 29 publications

Geometric momentum and angular momentum for charge-monopole system

Geometric momentum and angular momentum for charge-monopole system

On relationship between canonical momentum and geometric momentum

Distribution of x ⋅ p for quantum states on a circle

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