Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects

D. Galetti; E.C. Silva; M. A. Marchiolli; M. Ruzzi

REVIEW

Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects

M. Ruzzi, M. A. Marchiolli, E.C. Silva, D. Galetti

Quantum Physics · Submitted 26 Feb 2006 · 26 references

Pith's verdict: Solid and useful s-parametrized phase-space formalism on the circle, with a real gap around the P function that a referee should push on.

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arXiv IDquant-ph/0602216 (version 2)

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CitationD. Galetti, E.C. Silva, M. A. Marchiolli, M. Ruzzi. "Quasiprobability distribution functions for periodic phase-spaces: I. Theoretical Aspects." arXiv:quant-ph/0602216 (2006). https://pith.science/paper/LUSASIIH

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Editor's note

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Ruzzi et al. build the first s-parametrized quasiprobability family for angle-angular momentum phase space, and the core of the construction is sound. The new objects are the mapping kernel T^(s) (Eq. 7), the s-family of distributions F^(s), and the clean identity |m,theta> W -> H via the smoothing kernel Z^(1) is also new, and Z^(1) itself is well-behaved because it contains K, not K^{-1}; the stress-test note to the contrary has a sign error.

The real problem is the positive-s side. T^(+1) is defined through [K(l,alpha)]^{-1}, and K decays like a Gaussian in l, so the sum over l diverges badly. The paper's convergence remark applies to Eq. (8), but the same issue in the definition of T^(s) itself is not addressed. As a result, the Glauber-Sudarshan P function, the diagonal representation (16), and the P -> W smoothing step (19) are formal. This is not merely a technicality: without a distributional definition of T^(+1), the claimed complete s-parametrized formalism is not established.

Properties (iv) and (vi) are asserted without proof; they look right but are load-bearing. The coherent states themselves closely follow de Bievre-Gonzalez and Kowalski et al., so the novelty is in the parametrized mapping, not the states.

Overall: the central projector identity and the Wigner/Husimi side are solid; the P-function side needs rigorous treatment. A referee should ask the authors to either prove convergence in a suitable distribution space or weaken the claims. The paper is for researchers in phase-space methods, quantum rings, and rotational dynamics, and it deserves peer review rather than desk rejection.

Referee report

Major objections (3)

[Sec. 3, Eqs. (7), (12)–(14), properties (v)–(vi)] The definition of T^(s) in Eq. (7) is inconsistent with the projector identity (14) by a factor of 2π. Inserting Eq. (13) into Eq. (12) gives |m,θ><m,θ| = ∑_l ∫ dα/(2π) exp[−i(lθ−mα)] D(l,α) K(l,α), which is (1/2π) times the s = −1 case of Eq. (7). Thus Eq. (14) should either read T^(−1) = 2π |m,θ><m,θ| or the factor 1/(2π) is missing in Eq. (7). This also breaks property (v): from Eq. (7), Tr[T^(s)] = 2π, not 1, and property (vi) evaluated via Eq. (8) for s' = −s gives 4π^2 δ δ, not 2π δ δ. The normalization must be fixed and propagated through Eqs. (15)–(21).
[Sec. 3.1, Eq. (16) and P function] The positive-s sector is not justified. From the scalar product (iv), K(l,α) decays like exp(−l^2/4) for large l, so [K(l,α)]^(−1) grows super-polynomially. Consequently the kernel T^(+1)(m,θ) in Eq. (7) and the objects F^(+1)(m,θ) = Tr[T^(+1)ρ] are not well-defined as ordinary distributions on the circle for generic ρ, and the diagonal representation (16) is not established. The note in Sec. 3.2 that Eq. (8) is well-defined only for Re(s'−s) ≥ 0 concerns the smoothing kernels Z^(s'−s), not the definition of T^(+1). I note that Z^(1) itself is actually well-behaved: substituting the appropriate parameters into Eq. (8) gives an integrand [K]^(+1), which decays. The unresolved object is T^(+1), and the derivation of Eqs. (18)–(20) via Eq. (17) assumes the inversion formula (9) for s' = 1, which requires the ill-defined T^(1). The paper should restrict the formalism to Re(s) ≤ 0, define the P function for a restricted class of states or by analytic continuation, or prove that the divergent sums define a distribution in an appropriate sense.
[Sec. 3, properties (iv) and (vi), Eq. (8)] The central technical properties (iv) and (vi) are asserted without proof, despite being load-bearing: (iv) gives the large-l behavior of K used in the convergence analysis, and (vi) is the orthogonality relation underlying the inversion formula (9). Moreover, property (vi) as printed is not consistent with Eq. (8) as printed, as detailed in the first comment. The authors should provide explicit derivations or at least a clear statement of the normalization conventions that makes Eqs. (7), (8), (14), and (vi) mutually consistent.

Minor objections (4)

[Sec. 2, Eq. (1) and Sec. 2.1] The parameter a is fixed to a = (2π)^(−1) in Eq. (1) but is varied in Fig. 1; please clarify whether the subsequent formalism, especially the mapping kernel and hierarchy, is intended to hold for all a or only for the fixed value.
[Sec. 3.2, text after Eq. (18)] The sentence about Eq. (8) being well-defined only for Re(s'−s) ≥ 0 is confusing because Eq. (8) itself contains the exponent −(s'+s). The condition should be stated in terms of the actual exponent of [K] after the substitution used for Z^(s'−s).
[Sec. 2, Eq. (1)] The fermion case (half-integer m) is introduced but never used; please state explicitly that only the boson case (integer m) is developed in this paper.
[Sec. 3.2, Eq. (21)] For readability, explicitly connect |⟨m',θ'|m,θ⟩|² = |K(m'−m,θ'−θ)|² to the Fourier representation (8) of Z^(1), since this clarifies why the hierarchy is a convolution in the phase-space variables.

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The paper

The paper claims

On a circular phase space, the full family of quasiprobability functions is anchored by the identity $|m,\theta\rangle\langle m,\theta| = T^{(-1)}(m,\theta)$.

The reading

This paper aims to give circular phase spaces — the angle and angular-momentum variables of a particle on a ring or a rotor — the same complete family of quasiprobability distributions that the Cahill–Glauber formalism provides for position and momentum. It constructs coherent states adapted to periodicity, then proves that the projector onto any such state is exactly the mapping kernel at $s = -1$; from that identity the Husimi function becomes a genuine coherent-state expectation value, and the Wigner and Glauber–Sudarshan functions appear as the $s = 0$ and $s = +1$ members of the same family. The three are ordered by a smoothing chain $P \to W \to H$, so any one of them determines the others. The result matters because only the Wigner function previously existed on a circle, leaving rotational degrees of freedom without the phase-space toolkit routinely used for Cartesian systems.

What carries the argument

The load-bearing object is the $s$-parametrized mapping kernel $T^{(s)}(m,\theta)$ of Eq. (7): a sum over angular-momentum labels and an integral over the angle variable of displacement operators $D(l,\alpha)$, weighted by powers $[K(l,\alpha)]^{-s}$ of the coherent-state overlap $K(l,\alpha) = \langle 0,0|l,\alpha\rangle$, all built from Jacobi $\theta$ functions. Its trace properties (v)–(vi) make the one-to-one map between bounded operators and functions on the circle phase space (Eq. 9), and its coincidence with the coherent-state projector at $s = -1$ (Eq. 14) is what turns a formal expansion into a hierarchy of genuine quasiprobabilities. The inter-conversion kernel $Z^{(1)}$ that links the three functions is itself the Wigner function evaluated on coherent states. The $\theta$ functions play the role the Gaussian plays in the Cartesian case: they keep the vacuum state periodic, near minimum-uncertainty, and give the overlaps the algebraic structure the hierarchy needs.

Extended reading notes

Core claim

The paper's central claim is the identity (Eq. 14) $|m,\theta\rangle\langle m,\theta| = T^{(-1)}(m,\theta)$: the projector onto an angle–angular-momentum coherent state coincides with the mapping kernel evaluated at $s = -1$. From it, the Husimi function is the diagonal coherent-state expectation value $H(m,\theta) = \langle m,\theta|\rho|m,\theta\rangle$, the kernel at $s = 0$ retrieves the known circular Wigner function, and the $s = +1$ function $P(m,\theta)$ appears as the coefficient of that same projector in the diagonal representation $\rho = \sum_m \int \frac{d\theta}{2\pi}\, P(m,\theta)\,|m,\theta\rangle\langle m,\theta|$. The trace relation (8) then yields convolution formulas (Eqs. 19–20) showing that the Wigner function is a smoothed P function and the Husimi function a smoothed Wigner function, with a direct Husimi-from-P formula (Eq. 21) that bypasses the intermediate step. The paper notes that the double Fourier transform in (8) is well defined only for $\mathrm{Re}(s' - s) \geq 0$, leaving the P function and the inversion formula (9) with an unproven distributional status.

Load-bearing premise

The load-bearing premise is that the mapping kernel (7) and the trace relation (8) are well-defined distributions for every $|s| \leq 1$; the paper itself concedes in Section 3.2 that the transform in (8) converges only for $\mathrm{Re}(s' - s) \geq 0$, so the P function and the inversion formula (9) stand on ground the paper does not secure.

Editorial extensions

If this is right

A Husimi function on a circle is simply $\langle m,\theta|\rho|m,\theta\rangle$, hence everywhere nonnegative and normalized, giving a directly plottable picture of angle–momentum states.
Any one of the three functions determines the others: $P$ and $H$ are each linked to $W$ by a convolution with $Z^{(1)}$ (Eqs. 19–20), and $H$ connects to $P$ directly through the overlap probability $|K|^2$ in Eq. (21).
Rotational degrees of freedom — deformed rotors, molecular systems with rigid deformation, and quantum rings — become accessible to phase-space analysis, including semiclassical limits, through the von Neumann–Liouville formalism.
The trace identity (vi) provides a one-to-one correspondence between bounded operators and functions on the circular phase space, opening the way to tomographic reconstruction for angle–momentum systems.
The formalism supplies an operator-ordering structure for circle variables, which the authors offer as a route toward settling the creation–annihilation algebra of the angle–angular-momentum pair.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The convergence caveat on Eq. (8) suggests the Glauber–Sudarshan function on a circle may exist only for states whose angular-momentum distribution decays fast enough, just as P functions on the plane are typically more singular than Wigner or Husimi; a rigorous distributional treatment would settle how far the hierarchy extends.
Because $Z^{(1)}$ plays the Gaussian's smoothing role, the chain $P \to W \to H$ can be read as coarse-graining over a theta-function wavepacket of fixed width — a quantitative measure of 'quantumness' of angular states through Wigner negativity.
The same theta-function construction should transfer to other periodic or compact manifolds — tori, spheres, or discrete rotors — wherever the overlaps are available; testing it on a torus would show whether the mechanism is specific to the circle.
The states' near-minimum-uncertainty property (deviation of about 4% at the adopted width) puts them within reach of orbital-angular-momentum photon experiments that directly measure angular uncertainty relations, where the new family could serve as a tomographic tool.

Original abstract

An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed. For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	An approach featuring $s$-parametrized quasiprobability distribution functions is developed for situations where a circular topology is observed.	contribution 0.97	0.86	0.19	asserted 1.00
s1	For such an approach, a suitable set of angle-angular momentum coherent states must be constructed in appropriate fashion.	method 0.93	0.35	0.08	asserted 0.96

What kind of paper the classifier reports this is: method or tool 0.64 · evidence from proof or derivation 0.99.

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arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

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S sections
T body sentences
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Plots of δU(θ) versus θ ∈ [−π, π) for (a) a = 1/20π, (b) a = 1/2π, and (c) a = 10/2π. These pictures show that 0 ≤ δU(θ) ≤ 1, and for δU(θ) = 0 the coherent states {|m, θi} can be considered minimum uncertainty states. It is worth mentioning that the main differences between intelligent and minimum uncertainty states have been discussed in [23] for the experimental context. 2.1. Uncertainty relations To avoid some mathematical inconsistencies in the uncertainty relations following the commutation relation between the J and Θ operators, Carruthers and Nieto [21], following Louisell…

Figures reproduced from arXiv:quant-ph/0602216 by the authors.

References

Works cited (26)

[1]

Kim Y S and Noz M E 1991 Phase Space Picture of Quantum Mechanics: Group Theoretica l Approach - Lecture Notes in Physics Vol. 40 (Singapore: W orld Scientiﬁc Publishing)

View on Pith 1991
[2]

For example, see the following textbooks: Scully M O and Z ubairy M S 1997 Quantum Optics (New York: Cambridge University Press); Orszag M 2000 Quantum Optics (Berlin: Springer); Schleich W P 2001 Quantum Optics in Phase Space (Berlin: Wiley-VCH)

View on Pith 1997
[3]

Carmichael H J 1993 Lecture Notes in Physics m 18: An Open Systems Approach to Quantum Optics (Berlin: Springer Verlag); Carmichael H J 1999 Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Berlin: Springer); Gardiner C W and Zoller P 2000 Quantum Noise: A Handbook of Markovian and Non-Markovian Qu antum Stochastic Me...

View on Pith 1993
[4]

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer- Verlag); Almeida A M O 1992 Hamiltonian Systems: Chaos and Quantization - Cambridge Monographs on Mathematical Physics (New York: Cambridge Un iversity Press); Haake F 2001 Quantum Signatures of Chaos (Berlin: Springer)

View on Pith 1990
[5]

Hillery M, O’Connell R F, Scully M O and Wigner E P 1984 Dist ribution functions in physics: fundamentals Phys. Rep. 106 121

View on Pith 1984
[6]

Lee H-W 1995 Theory and application of the quantum phase- space distribution functions Phys. Rep. 259 147

View on Pith 1995
[7]

Mukunda N 1979 Wigner distribution for angle coordinate s in quantum mechanics Am. J. Phys. 47 182

View on Pith 1979
[8]

Bizarro JP 1994 W eyl-Wigner formalism for rotation-ang le and angular-momentum variables in quantum mechanics Phys. Rev. A 49 3255

View on Pith 1994
[9]

Ruzzi M and Galetti D 2002 Schwinger and Pegg-Barnett app roaches and a relationship between angular and Cartesian quantum descriptions: II. Phase spac es J. Phys. A: Math. Gen. 35 4633

View on Pith 2002
[10]

de Bi` evre S and Gonz´ alez J A 1993 Semiclassical behavi our of coherent states on the circle Quantization and Coherent States Methods (Proceedings of the XI Workshop on Geometric Methods in Physics, Bialowieza, Poland 7-14 July 1992) ed S T Ali, I M Ladanov and A Odzijewicz (Singapore: W orld Scientiﬁc); Gonz´ alez J A and del Olmo M A 1998 Coherent st...

View on Pith 1993
[11]

Brzezi´ nski T, Rembieli´ nski J and Smoli´ nski K A 1993 Q uantum particle on a quantum circle Mod. Phys. Lett. A 8 409

View on Pith 1993
[12]

Ohnuki Y and Kitakado S 1993 Fundamental algebra for qua ntum mechanics on SD and gauge potentials J. Math. Phys. 34 2827

View on Pith 1993
[13]

Kowalski K, Rembieli´ nski J and Papaloucas L C 1996 Cohe rent states for a quantum particle on a circle J. Phys. A: Math. Gen. 29 4149; Kowalski K and Rembieli´ nski J 2002 Exotic behaviour of a quantum particle on a circle Phys. Lett. A 293 109

View on Pith 1996
[14]

Hall B C and Mitchell J J 2002 Coherent states on spheres J. Math. Phys. 43 1211

View on Pith 2002
[15]

Gazeau J-P and Piechocki W 2004 Coherent state quantiza tion of a particle in de Sitter space J. Phys. A: Math. Gen. 37 6977

View on Pith 2004
[16]

Kastrup H A 2006 Quantization of the canonically conjug ate pair angle and orbital angular momentum Phys. Rev. A 73 052104

View on Pith 2006
[17]

Cahill K E and Glauber R J 1969 Ordered Expansions in boso n amplitude operators Phys. Rev. 177 1857; Cahill K E and Glauber R J 1969 Density operators and qua siprobability distributions Phys. Rev. 177 1882

View on Pith 1969
[18]

Whittaker E T and W atson G N 2000 Cambridge Mathematical Library: A Course of Modern Analysis (Cambridge: Cambridge University Press)

View on Pith 2000
[19]

Bellman R 1961 A Brief Introduction to Theta Functions (New York: Holt, Rinehart and Winston)

View on Pith 1961
[20]

Klauder J R and Skagerstam B S 1985 Coherent States. Applications in Physics and Mathematical Physics (Singapore: W orld Scientiﬁc) and references therein

View on Pith 1985
[21]

Carruthers P and Nieto M M 1968 Phase and Angle Variables in Quantum Mechanics Rev. Mod. Phys. 40 411

View on Pith 1968
[22]

Louisell W H 1963 Amplitude and Phase Uncertainty Relat ions Phys. Lett. 7 60

View on Pith 1963
[23]

Leach J, Padgett M J, Barnett S M, Franke-Arnold S and Cou rtial J 2002 Measuring the Orbital Angular Momentum of a Single Photon Phys. Rev. Lett. 88 257901; Franke-Arnold S, Barnett S M, Yao E, Leach J, Courtial J and Padgett M 2004 Uncertainty p rinciple for angular position Quasiprobability distribution functions for periodic pha se-spaces 11 and angul...

View on Pith 2002
[24]

Ruzzi M, Marchiolli M A and Galetti D 2005 Extended Cahil l-Glauber formalism for ﬁnite- dimensional spaces: I. Fundamentals J. Phys. A: Math. Gen. 38 6239; Marchiolli M A, Ruzzi M and Galetti D 2005 Extended Cahill-Glauber formalism for ﬁ nite-dimensional spaces: II. Applications in quantum tomography and quantum teleportat ion Phys. Rev. A 72 042308

View on Pith 2005
[25]

Reimers J R and Heller E J 1985 The exact eigenfunctions a nd eigenvalues of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics J. Chem. Phys. 83 511; Balzer B, Dilthey S, Stock G and Thoss M 2003 Quasiclassical and semicl assical wave-packet dynamics in periodic potentials J. Chem. Phys. 119 5795

View on Pith 1985
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Lorke A, Luyken R J, Govorov A O, Kotthaus J P, Garcia J M an d Petroﬀ P M 2000 Spectroscopy of Nanoscopic Semiconductor Rings Phys. Rev. Lett. 84 2223

View on Pith 2000

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

Kim Y S and Noz M E 1991 Phase Space Picture of Quantum Mechanics: Group Theoretica l Approach - Lecture Notes in Physics Vol. 40 (Singapore: W orld Scientiﬁc Publishing)

View on Pith 1991

[2] [2]

For example, see the following textbooks: Scully M O and Z ubairy M S 1997 Quantum Optics (New York: Cambridge University Press); Orszag M 2000 Quantum Optics (Berlin: Springer); Schleich W P 2001 Quantum Optics in Phase Space (Berlin: Wiley-VCH)

View on Pith 1997

[3] [3]

Carmichael H J 1993 Lecture Notes in Physics m 18: An Open Systems Approach to Quantum Optics (Berlin: Springer Verlag); Carmichael H J 1999 Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Berlin: Springer); Gardiner C W and Zoller P 2000 Quantum Noise: A Handbook of Markovian and Non-Markovian Qu antum Stochastic Me...

View on Pith 1993

[4] [4]

Gutzwiller M C 1990 Chaos in Classical and Quantum Mechanics (New York: Springer- Verlag); Almeida A M O 1992 Hamiltonian Systems: Chaos and Quantization - Cambridge Monographs on Mathematical Physics (New York: Cambridge Un iversity Press); Haake F 2001 Quantum Signatures of Chaos (Berlin: Springer)

View on Pith 1990

[5] [5]

Hillery M, O’Connell R F, Scully M O and Wigner E P 1984 Dist ribution functions in physics: fundamentals Phys. Rep. 106 121

View on Pith 1984

[6] [6]

Lee H-W 1995 Theory and application of the quantum phase- space distribution functions Phys. Rep. 259 147

View on Pith 1995

[7] [7]

Mukunda N 1979 Wigner distribution for angle coordinate s in quantum mechanics Am. J. Phys. 47 182

View on Pith 1979

[8] [8]

Bizarro JP 1994 W eyl-Wigner formalism for rotation-ang le and angular-momentum variables in quantum mechanics Phys. Rev. A 49 3255

View on Pith 1994

[9] [9]

Ruzzi M and Galetti D 2002 Schwinger and Pegg-Barnett app roaches and a relationship between angular and Cartesian quantum descriptions: II. Phase spac es J. Phys. A: Math. Gen. 35 4633

View on Pith 2002

[10] [10]

de Bi` evre S and Gonz´ alez J A 1993 Semiclassical behavi our of coherent states on the circle Quantization and Coherent States Methods (Proceedings of the XI Workshop on Geometric Methods in Physics, Bialowieza, Poland 7-14 July 1992) ed S T Ali, I M Ladanov and A Odzijewicz (Singapore: W orld Scientiﬁc); Gonz´ alez J A and del Olmo M A 1998 Coherent st...

View on Pith 1993

[11] [11]

Brzezi´ nski T, Rembieli´ nski J and Smoli´ nski K A 1993 Q uantum particle on a quantum circle Mod. Phys. Lett. A 8 409

View on Pith 1993

[12] [12]

Ohnuki Y and Kitakado S 1993 Fundamental algebra for qua ntum mechanics on SD and gauge potentials J. Math. Phys. 34 2827

View on Pith 1993

[13] [13]

Kowalski K, Rembieli´ nski J and Papaloucas L C 1996 Cohe rent states for a quantum particle on a circle J. Phys. A: Math. Gen. 29 4149; Kowalski K and Rembieli´ nski J 2002 Exotic behaviour of a quantum particle on a circle Phys. Lett. A 293 109

View on Pith 1996

[14] [14]

Hall B C and Mitchell J J 2002 Coherent states on spheres J. Math. Phys. 43 1211

View on Pith 2002

[15] [15]

Gazeau J-P and Piechocki W 2004 Coherent state quantiza tion of a particle in de Sitter space J. Phys. A: Math. Gen. 37 6977

View on Pith 2004

[16] [16]

Kastrup H A 2006 Quantization of the canonically conjug ate pair angle and orbital angular momentum Phys. Rev. A 73 052104

View on Pith 2006

[17] [17]

Cahill K E and Glauber R J 1969 Ordered Expansions in boso n amplitude operators Phys. Rev. 177 1857; Cahill K E and Glauber R J 1969 Density operators and qua siprobability distributions Phys. Rev. 177 1882

View on Pith 1969

[18] [18]

Whittaker E T and W atson G N 2000 Cambridge Mathematical Library: A Course of Modern Analysis (Cambridge: Cambridge University Press)

View on Pith 2000

[19] [19]

Bellman R 1961 A Brief Introduction to Theta Functions (New York: Holt, Rinehart and Winston)

View on Pith 1961

[20] [20]

Klauder J R and Skagerstam B S 1985 Coherent States. Applications in Physics and Mathematical Physics (Singapore: W orld Scientiﬁc) and references therein

View on Pith 1985

[21] [21]

Carruthers P and Nieto M M 1968 Phase and Angle Variables in Quantum Mechanics Rev. Mod. Phys. 40 411

View on Pith 1968

[22] [22]

Louisell W H 1963 Amplitude and Phase Uncertainty Relat ions Phys. Lett. 7 60

View on Pith 1963

[23] [23]

Leach J, Padgett M J, Barnett S M, Franke-Arnold S and Cou rtial J 2002 Measuring the Orbital Angular Momentum of a Single Photon Phys. Rev. Lett. 88 257901; Franke-Arnold S, Barnett S M, Yao E, Leach J, Courtial J and Padgett M 2004 Uncertainty p rinciple for angular position Quasiprobability distribution functions for periodic pha se-spaces 11 and angul...

View on Pith 2002

[24] [24]

Ruzzi M, Marchiolli M A and Galetti D 2005 Extended Cahil l-Glauber formalism for ﬁnite- dimensional spaces: I. Fundamentals J. Phys. A: Math. Gen. 38 6239; Marchiolli M A, Ruzzi M and Galetti D 2005 Extended Cahill-Glauber formalism for ﬁ nite-dimensional spaces: II. Applications in quantum tomography and quantum teleportat ion Phys. Rev. A 72 042308

View on Pith 2005

[25] [25]

Reimers J R and Heller E J 1985 The exact eigenfunctions a nd eigenvalues of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics J. Chem. Phys. 83 511; Balzer B, Dilthey S, Stock G and Thoss M 2003 Quasiclassical and semicl assical wave-packet dynamics in periodic potentials J. Chem. Phys. 119 5795

View on Pith 1985

[26] [26]

Lorke A, Luyken R J, Govorov A O, Kotthaus J P, Garcia J M an d Petroﬀ P M 2000 Spectroscopy of Nanoscopic Semiconductor Rings Phys. Rev. Lett. 84 2223

View on Pith 2000