Abstract

Within the framework of the thermal-wave model, the quantumlike description of electron optics in terms of the propagator is given. First we briefly review the standard description in configuration space by analogy to quantum mechanics and in connection with recent investigations of charged-particle-beam transport that have used the concept of propagator. Then new insights are given by extension of the analysis of the particle-beam propagator to the phase-space context for which our system is described by the Wigner quasi-distribution function, as well as to the tomography context for which our system is described by the marginal distribution. Furthermore, the integrals of motion of a charged-particle beam and their relation to the propagator concept are discussed. Finally, the perturbation theory for a charged-particle-beam propagator is developed in the above-described two contexts and is applied to some simple optical devices.

© 2000 Optical Society of America

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  1. J.-M. Lagniel, “Halos and chaos in space-charge dominated beams,” in Proceedings of the Fifth European Particle Accelerator Conference (EPAC96), S. Meyer, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, J. Poole, eds. (Institute of Physics, Bristol, UK, 1996), pp. 163–167; S. O’Connel, T. P. Wangler, R. S. Mills, K. R. Krandal, “Beam halo formation from space-charge dominated beams in uniform focusing channels,” in Proceedings of the Particle Accelerator Conference, S. T. Corneliussen, ed. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 3657–3659; C. Chen, R. C. Davidson, “Nonlinear properties of the Kapchinskij–Vladimirskij equilibrium and envelope equation for an intense charged-particle beam in a periodic focusing field,” Phys. Rev. E 49, 5679–5687 (1994); M. Raiser, N. Brown, “Proposed high-current linear accelerators with beams in thermal equilibrium,” Phys. Rev. Lett. 74, 1111–1114 (1995).
    [CrossRef]
  2. A. Piwinski, “Beam losses and lifetime,” in Proceedings of the CERN Accelerator School, P. Briant, S. Turner, eds. (Centre European de Recherches Nucleaires, Geneva, 1985), Vol. 85-19, pp. 415–431; A. H. Sørensen, “Introduction to intrabeam scattering,” in Proceedings of the CERN Accelerator School, S. Turner, ed. (Centre European de Recherches Nucleaires, Geneva, 1987), Vol. 87-10, pp. 135–151.
  3. S. A. Khan, M. Pusterla, “Quantum-mechanical aspects of the halo puzzle,” in Proceedings of the Particle Accelerator Conference (PAC’99), A. Luccio, W. MacKay, eds. (Institute of Electrical and Electronic Engineers, New York, 1999), pp. 3280–3281; S. A. Khan, M. Pusterla, “Quantum-like approaches to the beam halo problem,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000).
  4. R. Fedele, G. Miele, “A thermal-wave model for relativistic charged-particle-beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
    [CrossRef]
  5. M. A. Leontovich, “A method of solving the problem of electromagnetic-wave propagation along the Earth’s surface,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16–22 (1994); M. A. Leontovich, V. A. Fock, “Solution of the problem of propagation of electromagnetic waves along the Earth’s surface by the method of parabolic equation,” Zh. Eksp. Teor. Fiz. 16, 557–573 (1946).
  6. R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
    [CrossRef]
  7. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  8. R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
    [CrossRef]
  9. S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
    [CrossRef]
  10. R. Fedele, V. I. Man’ko, “Phase-space electronic-ray description for charged-particle-beam transport. Quantum-like corrections versus the classical picture,” Phys. Scr. T 75, 283–287 (1998).
    [CrossRef]
  11. R. Fedele, V. I. Man’ko, “Quantum-like corrections and tomography in beam physics,” in Proceedings of the Sixth European Particle Accelerator Conference, S. Meyers, L. Liljeby, Ch. Petit-Jean-Genaz, J. Poole, K.-G. Rensfelt, eds. (Institute of Physics, Bristol, UK, 1998), pp. 1268–1270.
  12. Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).
  13. I. A. Malkin, V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems, in Russian (Nauka, Moscow, 1979).
  14. V. V. Dodonov, V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute Series (Nova Science, New York, 1989), Vol. 183.
  15. R. Fedele, V. I. Man’ko, “Quantum-like corrections and semiclassical description of a charged-particle-beam transport,” Phys. Rev. E 58, 992–1001 (1998).
    [CrossRef]
  16. O. Man’ko, V. I. Man’ko, “Quantum states in probability representation and tomography,” J. Russ. Laser Res. 18, 407–444 (1997).
    [CrossRef]
  17. V. I. Man’ko, R. V. Mendes, “Noncommutative time-frequency tomography,” Phys. Lett. A 263, 53–61 (1999).
    [CrossRef]
  18. J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932).
  19. M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signals, and the Green function of a harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999); M. A. Man’ko, “Optical tomography approach in signal analysis,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000); M. A. Man’ko, “Fractional Fourier analysis and quantum propagators,” in Proceedings of the International Symposium on Quantum Theory and Symmetries, H.-D. Doebner, V. K. Kobrev, J.-D. Hennig, W. Luecke, eds. (World Scientific, Singapore, 2000), pp. 226–231; M. A. Man’ko, “Fractional Fourier transform in signal analysis and information processing as the quantum propagator,” in Proceedings of the International Workshop on Optoelectronics, N. V. Hieu, P. H. Khoi, eds. (World Scientific, Singapore, to be published); M. A. Man’ko, “Quasidistribution, tomography and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
    [CrossRef]
  20. H. M. Ozaktas, D. Mendlovich, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]
  21. P. Ermakov, “Differential equations of the second order and the conditions of integrability in finite form,” Izv. Sant Vladimir Univ. (Kiev) 20(9), Part II, Sec. III, 1–25 (1880).
  22. H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
    [CrossRef]
  23. A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
    [CrossRef]
  24. R. Simon, K.-B. Wolf, “Structure of the set of the paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
    [CrossRef]

2000 (2)

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
[CrossRef]

R. Simon, K.-B. Wolf, “Structure of the set of the paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

1999 (2)

V. I. Man’ko, R. V. Mendes, “Noncommutative time-frequency tomography,” Phys. Lett. A 263, 53–61 (1999).
[CrossRef]

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signals, and the Green function of a harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999); M. A. Man’ko, “Optical tomography approach in signal analysis,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000); M. A. Man’ko, “Fractional Fourier analysis and quantum propagators,” in Proceedings of the International Symposium on Quantum Theory and Symmetries, H.-D. Doebner, V. K. Kobrev, J.-D. Hennig, W. Luecke, eds. (World Scientific, Singapore, 2000), pp. 226–231; M. A. Man’ko, “Fractional Fourier transform in signal analysis and information processing as the quantum propagator,” in Proceedings of the International Workshop on Optoelectronics, N. V. Hieu, P. H. Khoi, eds. (World Scientific, Singapore, to be published); M. A. Man’ko, “Quasidistribution, tomography and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

1998 (2)

R. Fedele, V. I. Man’ko, “Phase-space electronic-ray description for charged-particle-beam transport. Quantum-like corrections versus the classical picture,” Phys. Scr. T 75, 283–287 (1998).
[CrossRef]

R. Fedele, V. I. Man’ko, “Quantum-like corrections and semiclassical description of a charged-particle-beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

1997 (2)

O. Man’ko, V. I. Man’ko, “Quantum states in probability representation and tomography,” J. Russ. Laser Res. 18, 407–444 (1997).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

1996 (1)

S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
[CrossRef]

1995 (2)

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

H. M. Ozaktas, D. Mendlovich, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

1994 (1)

M. A. Leontovich, “A method of solving the problem of electromagnetic-wave propagation along the Earth’s surface,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16–22 (1994); M. A. Leontovich, V. A. Fock, “Solution of the problem of propagation of electromagnetic waves along the Earth’s surface by the method of parabolic equation,” Zh. Eksp. Teor. Fiz. 16, 557–573 (1946).

1991 (1)

R. Fedele, G. Miele, “A thermal-wave model for relativistic charged-particle-beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
[CrossRef]

1969 (1)

H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

1880 (1)

P. Ermakov, “Differential equations of the second order and the conditions of integrability in finite form,” Izv. Sant Vladimir Univ. (Kiev) 20(9), Part II, Sec. III, 1–25 (1880).

Atakishiyev, N. M.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Chumakov, S. M.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Dodonov, V. V.

V. V. Dodonov, V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute Series (Nova Science, New York, 1989), Vol. 183.

Ermakov, P.

P. Ermakov, “Differential equations of the second order and the conditions of integrability in finite form,” Izv. Sant Vladimir Univ. (Kiev) 20(9), Part II, Sec. III, 1–25 (1880).

Fedele, R.

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
[CrossRef]

R. Fedele, V. I. Man’ko, “Phase-space electronic-ray description for charged-particle-beam transport. Quantum-like corrections versus the classical picture,” Phys. Scr. T 75, 283–287 (1998).
[CrossRef]

R. Fedele, V. I. Man’ko, “Quantum-like corrections and semiclassical description of a charged-particle-beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

R. Fedele, G. Miele, “A thermal-wave model for relativistic charged-particle-beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
[CrossRef]

R. Fedele, V. I. Man’ko, “Quantum-like corrections and tomography in beam physics,” in Proceedings of the Sixth European Particle Accelerator Conference, S. Meyers, L. Liljeby, Ch. Petit-Jean-Genaz, J. Poole, K.-G. Rensfelt, eds. (Institute of Physics, Bristol, UK, 1998), pp. 1268–1270.

Galluccio, F.

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

Khan, S. A.

S. A. Khan, M. Pusterla, “Quantum-mechanical aspects of the halo puzzle,” in Proceedings of the Particle Accelerator Conference (PAC’99), A. Luccio, W. MacKay, eds. (Institute of Electrical and Electronic Engineers, New York, 1999), pp. 3280–3281; S. A. Khan, M. Pusterla, “Quantum-like approaches to the beam halo problem,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000).

Kim, Y. S.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

Lagniel, J.-M.

J.-M. Lagniel, “Halos and chaos in space-charge dominated beams,” in Proceedings of the Fifth European Particle Accelerator Conference (EPAC96), S. Meyer, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, J. Poole, eds. (Institute of Physics, Bristol, UK, 1996), pp. 163–167; S. O’Connel, T. P. Wangler, R. S. Mills, K. R. Krandal, “Beam halo formation from space-charge dominated beams in uniform focusing channels,” in Proceedings of the Particle Accelerator Conference, S. T. Corneliussen, ed. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 3657–3659; C. Chen, R. C. Davidson, “Nonlinear properties of the Kapchinskij–Vladimirskij equilibrium and envelope equation for an intense charged-particle beam in a periodic focusing field,” Phys. Rev. E 49, 5679–5687 (1994); M. Raiser, N. Brown, “Proposed high-current linear accelerators with beams in thermal equilibrium,” Phys. Rev. Lett. 74, 1111–1114 (1995).
[CrossRef]

Leontovich, M. A.

M. A. Leontovich, “A method of solving the problem of electromagnetic-wave propagation along the Earth’s surface,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16–22 (1994); M. A. Leontovich, V. A. Fock, “Solution of the problem of propagation of electromagnetic waves along the Earth’s surface by the method of parabolic equation,” Zh. Eksp. Teor. Fiz. 16, 557–573 (1946).

Lewis, H. R.

H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Malkin, I. A.

I. A. Malkin, V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems, in Russian (Nauka, Moscow, 1979).

Man’ko, M. A.

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
[CrossRef]

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signals, and the Green function of a harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999); M. A. Man’ko, “Optical tomography approach in signal analysis,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000); M. A. Man’ko, “Fractional Fourier analysis and quantum propagators,” in Proceedings of the International Symposium on Quantum Theory and Symmetries, H.-D. Doebner, V. K. Kobrev, J.-D. Hennig, W. Luecke, eds. (World Scientific, Singapore, 2000), pp. 226–231; M. A. Man’ko, “Fractional Fourier transform in signal analysis and information processing as the quantum propagator,” in Proceedings of the International Workshop on Optoelectronics, N. V. Hieu, P. H. Khoi, eds. (World Scientific, Singapore, to be published); M. A. Man’ko, “Quasidistribution, tomography and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

Man’ko, O.

O. Man’ko, V. I. Man’ko, “Quantum states in probability representation and tomography,” J. Russ. Laser Res. 18, 407–444 (1997).
[CrossRef]

Man’ko, V. I.

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
[CrossRef]

V. I. Man’ko, R. V. Mendes, “Noncommutative time-frequency tomography,” Phys. Lett. A 263, 53–61 (1999).
[CrossRef]

R. Fedele, V. I. Man’ko, “Phase-space electronic-ray description for charged-particle-beam transport. Quantum-like corrections versus the classical picture,” Phys. Scr. T 75, 283–287 (1998).
[CrossRef]

R. Fedele, V. I. Man’ko, “Quantum-like corrections and semiclassical description of a charged-particle-beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

O. Man’ko, V. I. Man’ko, “Quantum states in probability representation and tomography,” J. Russ. Laser Res. 18, 407–444 (1997).
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
[CrossRef]

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

V. V. Dodonov, V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute Series (Nova Science, New York, 1989), Vol. 183.

I. A. Malkin, V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems, in Russian (Nauka, Moscow, 1979).

R. Fedele, V. I. Man’ko, “Quantum-like corrections and tomography in beam physics,” in Proceedings of the Sixth European Particle Accelerator Conference, S. Meyers, L. Liljeby, Ch. Petit-Jean-Genaz, J. Poole, K.-G. Rensfelt, eds. (Institute of Physics, Bristol, UK, 1998), pp. 1268–1270.

Mancini, S.

S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
[CrossRef]

Mendes, R. V.

V. I. Man’ko, R. V. Mendes, “Noncommutative time-frequency tomography,” Phys. Lett. A 263, 53–61 (1999).
[CrossRef]

Mendlovich, D.

Miele, G.

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

R. Fedele, G. Miele, “A thermal-wave model for relativistic charged-particle-beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
[CrossRef]

Noz, M. E.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

Ozaktas, H. M.

Piwinski, A.

A. Piwinski, “Beam losses and lifetime,” in Proceedings of the CERN Accelerator School, P. Briant, S. Turner, eds. (Centre European de Recherches Nucleaires, Geneva, 1985), Vol. 85-19, pp. 415–431; A. H. Sørensen, “Introduction to intrabeam scattering,” in Proceedings of the CERN Accelerator School, S. Turner, ed. (Centre European de Recherches Nucleaires, Geneva, 1987), Vol. 87-10, pp. 135–151.

Pusterla, M.

S. A. Khan, M. Pusterla, “Quantum-mechanical aspects of the halo puzzle,” in Proceedings of the Particle Accelerator Conference (PAC’99), A. Luccio, W. MacKay, eds. (Institute of Electrical and Electronic Engineers, New York, 1999), pp. 3280–3281; S. A. Khan, M. Pusterla, “Quantum-like approaches to the beam halo problem,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000).

Riesenfeld, W. B.

H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

Rivera, A. L.

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Simon, R.

Tombesi, P.

S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
[CrossRef]

von Neumann, J.

J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932).

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, K.-B.

R. Simon, K.-B. Wolf, “Structure of the set of the paraxial optical systems,” J. Opt. Soc. Am. A 17, 342–355 (2000).
[CrossRef]

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Izv. Akad. Nauk SSSR Ser. Fiz. (1)

M. A. Leontovich, “A method of solving the problem of electromagnetic-wave propagation along the Earth’s surface,” Izv. Akad. Nauk SSSR Ser. Fiz. 8, 16–22 (1994); M. A. Leontovich, V. A. Fock, “Solution of the problem of propagation of electromagnetic waves along the Earth’s surface by the method of parabolic equation,” Zh. Eksp. Teor. Fiz. 16, 557–573 (1946).

Izv. Sant Vladimir Univ. (Kiev) (1)

P. Ermakov, “Differential equations of the second order and the conditions of integrability in finite form,” Izv. Sant Vladimir Univ. (Kiev) 20(9), Part II, Sec. III, 1–25 (1880).

J. Math. Phys. (1)

H. R. Lewis, W. B. Riesenfeld, “An exact quantum theory of the time-dependent harmonic oscillator and of a charged particle in a time-dependent electromagnetic field,” J. Math. Phys. 10, 1458–1473 (1969).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Russ. Laser Res. (3)

O. Man’ko, V. I. Man’ko, “Quantum states in probability representation and tomography,” J. Russ. Laser Res. 18, 407–444 (1997).
[CrossRef]

R. Fedele, M. A. Man’ko, V. I. Man’ko, “Wave-optics applications in charged-particle-beam transport,” J. Russ. Laser Res. 21, 1–32 (2000).
[CrossRef]

M. A. Man’ko, “Fractional Fourier transform in information processing, tomography of optical signals, and the Green function of a harmonic oscillator,” J. Russ. Laser Res. 20, 226–238 (1999); M. A. Man’ko, “Optical tomography approach in signal analysis,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000); M. A. Man’ko, “Fractional Fourier analysis and quantum propagators,” in Proceedings of the International Symposium on Quantum Theory and Symmetries, H.-D. Doebner, V. K. Kobrev, J.-D. Hennig, W. Luecke, eds. (World Scientific, Singapore, 2000), pp. 226–231; M. A. Man’ko, “Fractional Fourier transform in signal analysis and information processing as the quantum propagator,” in Proceedings of the International Workshop on Optoelectronics, N. V. Hieu, P. H. Khoi, eds. (World Scientific, Singapore, to be published); M. A. Man’ko, “Quasidistribution, tomography and fractional Fourier transform in signal analysis,” J. Russ. Laser Res. 21, 411–437 (2000).
[CrossRef]

Nuovo Cimento D (1)

R. Fedele, G. Miele, “A thermal-wave model for relativistic charged-particle-beam propagation,” Nuovo Cimento D 13, 1527–1544 (1991).
[CrossRef]

Phys. Lett. A (3)

V. I. Man’ko, R. V. Mendes, “Noncommutative time-frequency tomography,” Phys. Lett. A 263, 53–61 (1999).
[CrossRef]

R. Fedele, F. Galluccio, V. I. Man’ko, G. Miele, “Full phase-space analysis of particle-beam transport in the thermal wave model,” Phys. Lett. A 209, 263–276 (1995).
[CrossRef]

S. Mancini, V. I. Man’ko, P. Tombesi, “Symplectic tomography as a classical approach to quantum systems,” Phys. Lett. A 213, 1–6 (1996).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Phys. Rev. A (1)

A. L. Rivera, N. M. Atakishiyev, S. M. Chumakov, K.-B. Wolf, “Evolution under polynomial Hamiltonians in quantum and optical phase space,” Phys. Rev. A 55, 876–889 (1997).
[CrossRef]

Phys. Rev. E (1)

R. Fedele, V. I. Man’ko, “Quantum-like corrections and semiclassical description of a charged-particle-beam transport,” Phys. Rev. E 58, 992–1001 (1998).
[CrossRef]

Phys. Scr. T (1)

R. Fedele, V. I. Man’ko, “Phase-space electronic-ray description for charged-particle-beam transport. Quantum-like corrections versus the classical picture,” Phys. Scr. T 75, 283–287 (1998).
[CrossRef]

Other (8)

R. Fedele, V. I. Man’ko, “Quantum-like corrections and tomography in beam physics,” in Proceedings of the Sixth European Particle Accelerator Conference, S. Meyers, L. Liljeby, Ch. Petit-Jean-Genaz, J. Poole, K.-G. Rensfelt, eds. (Institute of Physics, Bristol, UK, 1998), pp. 1268–1270.

Y. S. Kim, M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, Singapore, 1991).

I. A. Malkin, V. I. Man’ko, Dynamic Symmetries and Coherent States of Quantum Systems, in Russian (Nauka, Moscow, 1979).

V. V. Dodonov, V. I. Man’ko, Invariants and Evolution of Nonstationary Quantum Systems, Proceedings of the P. N. Lebedev Physical Institute Series (Nova Science, New York, 1989), Vol. 183.

J.-M. Lagniel, “Halos and chaos in space-charge dominated beams,” in Proceedings of the Fifth European Particle Accelerator Conference (EPAC96), S. Meyer, A. Pacheco, R. Pascual, Ch. Petit-Jean-Genaz, J. Poole, eds. (Institute of Physics, Bristol, UK, 1996), pp. 163–167; S. O’Connel, T. P. Wangler, R. S. Mills, K. R. Krandal, “Beam halo formation from space-charge dominated beams in uniform focusing channels,” in Proceedings of the Particle Accelerator Conference, S. T. Corneliussen, ed. (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 3657–3659; C. Chen, R. C. Davidson, “Nonlinear properties of the Kapchinskij–Vladimirskij equilibrium and envelope equation for an intense charged-particle beam in a periodic focusing field,” Phys. Rev. E 49, 5679–5687 (1994); M. Raiser, N. Brown, “Proposed high-current linear accelerators with beams in thermal equilibrium,” Phys. Rev. Lett. 74, 1111–1114 (1995).
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A. Piwinski, “Beam losses and lifetime,” in Proceedings of the CERN Accelerator School, P. Briant, S. Turner, eds. (Centre European de Recherches Nucleaires, Geneva, 1985), Vol. 85-19, pp. 415–431; A. H. Sørensen, “Introduction to intrabeam scattering,” in Proceedings of the CERN Accelerator School, S. Turner, ed. (Centre European de Recherches Nucleaires, Geneva, 1987), Vol. 87-10, pp. 135–151.

S. A. Khan, M. Pusterla, “Quantum-mechanical aspects of the halo puzzle,” in Proceedings of the Particle Accelerator Conference (PAC’99), A. Luccio, W. MacKay, eds. (Institute of Electrical and Electronic Engineers, New York, 1999), pp. 3280–3281; S. A. Khan, M. Pusterla, “Quantum-like approaches to the beam halo problem,” in Proceedings of the Sixth International Conference on Squeezed States and Uncertainty Relations, D. Han, Y. S. Kim, S. Solimeno, eds. (NASA Goddard Space Flight Center, Greenbelt, Md., 2000).

J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer-Verlag, Berlin, 1932).

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Equations (59)

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iΨz=-222x2Ψ+U(x, z)Ψ,
G(x, x, z)=x|Uˆ(z)|x,
Ψ(x, z)=Uˆ(z)Ψ(x, 0),
Ψ(x, z)=G(x, x, z)Ψ(x, 0)dx.
iz+222x2-U(x, z)G(x, x, z)=iδ(z)δ(x-x).
Iˆ(z)z+i[Hˆ, Iˆ]=0
Hˆ=-222x2+U(x, z)
Iˆ(z)=Uˆ(z)Iˆ(0)Uˆ(z)
pˆ0(z)=Uˆ(z)pˆUˆ(z)
xˆ0(z)=Uˆ(z)xˆUˆ(z),
pˆ=-ix,xˆ=x.
pˆ0(z)G(x, x, z)=ixG(x, x, z),
xˆ0(z)G(x, x, z)=xG(x, x, z),
ρw(x, p, z; )=12π-+Ψ*x+y2, z×Ψx-y2, zexpipydy,
ρw(x, p, z; )dx dp=1.
12πρw2(x, p, z; )dx dp=1.
ρwz+pρwx+iUx+i2p-Ux-i2pρw=0,
ρw(x, p, z; )=π(x, p, x, p, z; )×ρw(x, p, 0; )dx dp.
πz+pπx+iUx+i2p-Ux-i2pπ=δ(z)δ(x-x)δ(p-p).
Lˆz+px-Uxp,
Lˆρw=k=1(-1)k(2k+1)!22k2k+1Ux2k+12k+1ρwp2k+1,
π(q, p, q, p, z; )=12πGq+u2, q+s2, z×G*q-u2, q-s2, z×exp(-ipu+ips)du ds.
w(X, μ, ν, z; )=exp[-ik(X-μx-νp)]×ρw(x, p, z; )dk dx dp(2π)2,
w(X, μ, ν, z; )0,w(X, μ, ν, z; )dX=1.
wz-μνw+iU-1/Xμ+iν2X-U-1/Xμ-iν2Xw=0.
Lˆw=2n=1(-1)n(2n+1)!U(2n+1)(qˆ)ν22n+12n+1wX2n+1,
Lˆz-μν-νU(1)(qˆ)X,
U(2n+1)(qˆ)2n+1x2n+1U(x=qˆ),qˆ=-1/Xμ.
w(X, μ, ν, z; )=12π|ν|Ψ(y, z)expiμ2νy2-iXνy2.
w(X, μ, ν, z)=Π(X, μ, ν, X, μ, ν, z)×w(X, μ, ν, 0)dXdμdν,
ρ(X, X, z)=Ψ(X, z)Ψ*(X, z),
ρ(X, X, z)=K(X, X, Y, Y, z)ρ(Y, Y, 0)dY dY
K(X, X, Y, Y, z)=G(X, Y, z)G*(X, Y, z).
K(X, X, Z, Z, z)=1(2π)21|ν|expiY-μX+X2-iZ-ZνY+iZ2-Z22νμ×Π(Y, μ, X-X, Y, μ, ν, t)dμdμdYdYdν.
Π(X, μ, ν, X, μ, ν, z)=1(4π)2k2Ga+kν2, y, z×G*a-kν2, z˜, zδ(y-z˜-kν)×expikX-X+μa-μy+z˜2dkdydz˜da.
H=H0+V(x, z),
Ψ(x, z)Ψ0(x, z)+1i0zdzdx×G0(x, x, z, z)V(x, z)Ψ0(x, z)dx,
Ψ0(x, z)=G0(x, x, z, z)Ψ0(x, z)dx.
w(X, μ, ν, z)=w0(X, μ, ν, z)+0zdzdXdμdν×Π0(X, μ, ν, X, μ, ν, z, z)×iV-1/Xμ-iν2X, z-V-1/Xμ+iν2X, z×w0(X, μ, ν, z),
w0(X, μ, ν, z)=Π0(X, μ, ν, X, μ, ν, z, z)×w0(X, μ, ν, z)dXdμdν.
ρω(x, p, z)=ρω(0)(x, p, z)-i0zdzπ0(x, p, x, p, z, z)×Vx+i2p, z-Vx-i2p, z×ρω(0)(x, p, z)dxdp,
H=-122x2+x22,
G(x, x, z)=1(2πi sin z)1/2×expi2cot z(x2+x2)-2xxsin z.
A(z)=exp(iz)2x+x,
A(z)=exp(-iz)2x-x,
pˆ0(z)=A(z)-A(z)i2,
xˆ0(z)=A(z)+A(z)2,
U(x, z)=k1(z)2x2,
A(z)=i2ξ(z)-ix-ξ˙(z)x,
A(z)=-i2ξ*(z)-ix-ξ˙*(z)x,
ξ¨(z)+k1(z)ξ(z)=0,
ξ(0)=1,ξ˙(0)=i.
J(z)=A(z)A(z),
Π(X, μ, ν, X, μ, ν, z)=δ(X-X)δ(ν-ν cos z+μ sin z)×δ(μ-ν sin z-μ cos z).
Π1(X, μ, ν, X, μ, ν, z)=δ(X-X)δν-12i[ξ˙-ξ˙*]ν-12i[ξ-ξ*]μ×δμ-12[ξ˙+ξ˙*]ν-12[ξ+ξ*]μ.
V(x)=λx4,
w(X, μ, ν, z)=w0(X, μ, ν, z)+λ0zdzdXdμdνδ(X-X)×δ(ν-ν cos[z-z]+μ sin[z-z])δ(μ-ν sin[z-z]-μ cos[z-z])×νμν22X2-1(/X)22μ2×w0(X, μ, ν, z).
(a+b)4-(a-b)4=8ab(a2+b2),
a=-1/Xμ,b=-iν2X.

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