Abstract

A unitary transformation between Cartesian and polar pixellations of finite two-dimensional images is obtained from the su(2) model for discrete and finite signals. This transformation analyzes the original image into its finite Cartesian “Laguerre–Kravchuk” modes (involving Wigner little-d functions) and synthesizes it back using a polar mode basis with the same set of mode coefficients. The polar basis is derived from the quantum angular momentum theory, and its modes are given by Clebsch–Gordan coefficients.

© 2008 Optical Society of America

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  1. L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981), Vol. 8.
  2. N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
    [CrossRef]
  3. N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
    [CrossRef]
  4. N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
    [CrossRef]
  5. T. Alieva and K. B. Wolf, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365-370 (2008).
    [CrossRef]
  6. M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
    [CrossRef]
  7. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.
  8. N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, 1968).
  9. N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055-1062 (1991).
    [CrossRef]
  10. M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” Acad. Sci., Paris, C. R. 189, 620-622 (1929).
  11. N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
    [CrossRef]
  12. K. B. Wolf and G. Krötzsch, “Geometry and dynamics in the fractional discrete Fourier transform,” J. Opt. Soc. Am. A 24, 651-658 (2007).
    [CrossRef]
  13. L. E. Vicent, “Coherent states for the finite su(2)-oscillator model,” Int. J. Mod. Phys. B 20, 1934-1941 (2006).
    [CrossRef]
  14. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
    [CrossRef]
  15. A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998).
  16. R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368-2381 (2000).
    [CrossRef]
  17. K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).
  18. L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
    [CrossRef]
  19. R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications (Wiley, 1974).
  20. L. E. Vicent, “Análisis de Señales Discretas Finitas mediante el Modelo de Oscilador Finito de su(2),” Ph.D. dissertation (Universidad Autónoma del Estado de Morelos, 2007).
  21. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).
  22. N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
    [CrossRef]
  23. W. Miller, Jr., “Symmetry groups and separation of variables,” in Encyclopedia of Mathematics, G.-C.Rota, ed. (Addison-Wesley, 1977), Vol. 4.

2008

2007

K. B. Wolf and G. Krötzsch, “Geometry and dynamics in the fractional discrete Fourier transform,” J. Opt. Soc. Am. A 24, 651-658 (2007).
[CrossRef]

L. E. Vicent, “Análisis de Señales Discretas Finitas mediante el Modelo de Oscilador Finito de su(2),” Ph.D. dissertation (Universidad Autónoma del Estado de Morelos, 2007).

2006

L. E. Vicent, “Coherent states for the finite su(2)-oscillator model,” Int. J. Mod. Phys. B 20, 1934-1941 (2006).
[CrossRef]

2004

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

2003

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
[CrossRef]

2001

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

2000

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368-2381 (2000).
[CrossRef]

1999

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
[CrossRef]

1998

A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998).

1997

1991

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

1988

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).

1981

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981), Vol. 8.

1977

W. Miller, Jr., “Symmetry groups and separation of variables,” in Encyclopedia of Mathematics, G.-C.Rota, ed. (Addison-Wesley, 1977), Vol. 4.

1974

R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications (Wiley, 1974).

1968

N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, 1968).

1953

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

1929

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” Acad. Sci., Paris, C. R. 189, 620-622 (1929).

Alieva, T.

Arik, M.

M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
[CrossRef]

Atakishiyev, N. M.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

Barker, L.

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Biedenharn, L. C.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981), Vol. 8.

Çandan, Ç.

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

Frank, A.

A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998).

Gilmore, R.

R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications (Wiley, 1974).

Hakioglu, T.

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Khersonskii, V. K.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).

Krawtchouk, M.

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” Acad. Sci., Paris, C. R. 189, 620-622 (1929).

Krötzsch, G.

Kutay, M. A.

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Louck, J. D.

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981), Vol. 8.

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

Miller, W.

W. Miller, Jr., “Symmetry groups and separation of variables,” in Encyclopedia of Mathematics, G.-C.Rota, ed. (Addison-Wesley, 1977), Vol. 4.

Moskalev, A. N.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

Ozaktas, H. M.

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Pogosyan, G. S.

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

Simon, R.

Suslov, S. K.

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

Van Isacker, P.

A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998).

Varshalovich, D. A.

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).

Vicent, L. E.

L. E. Vicent, “Análisis de Señales Discretas Finitas mediante el Modelo de Oscilador Finito de su(2),” Ph.D. dissertation (Universidad Autónoma del Estado de Morelos, 2007).

L. E. Vicent, “Coherent states for the finite su(2)-oscillator model,” Int. J. Mod. Phys. B 20, 1934-1941 (2006).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

Vilenkin, N. Ya.

N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, 1968).

Wolf, K. B.

T. Alieva and K. B. Wolf, “Rotation and gyration of finite two-dimensional modes,” J. Opt. Soc. Am. A 25, 365-370 (2008).
[CrossRef]

K. B. Wolf and G. Krötzsch, “Geometry and dynamics in the fractional discrete Fourier transform,” J. Opt. Soc. Am. A 24, 651-658 (2007).
[CrossRef]

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

R. Simon and K. B. Wolf, “Fractional Fourier transforms in two dimensions,” J. Opt. Soc. Am. A 17, 2368-2381 (2000).
[CrossRef]

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
[CrossRef]

N. M. Atakishiyev and K. B. Wolf, “Fractional Fourier-Kravchuk transform,” J. Opt. Soc. Am. A 14, 1467-1477 (1997).
[CrossRef]

Acad. Sci., Paris, C. R.

M. Krawtchouk, “Sur une généralization des polinômes d'Hermite,” Acad. Sci., Paris, C. R. 189, 620-622 (1929).

Int. J. Mod. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite one-dimensional oscillator,” Int. J. Mod. Phys. A 18, 317-327 (2003).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, and K. B. Wolf, “Contraction of the finite radial oscillator,” Int. J. Mod. Phys. A 18, 329-341 (2003).
[CrossRef]

Int. J. Mod. Phys. B

L. E. Vicent, “Coherent states for the finite su(2)-oscillator model,” Int. J. Mod. Phys. B 20, 1934-1941 (2006).
[CrossRef]

J. Comput. Appl. Math.

N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” J. Comput. Appl. Math. 107, 73-95 (1999).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: I. The Cartesian model,” J. Phys. A 34, 9381-9398 (2001).
[CrossRef]

N. M. Atakishiyev, G. S. Pogosyan, L. E. Vicent, and K. B. Wolf, “Finite two-dimensional oscillator: II. The radial model,” J. Phys. A 34, 9399-9415 (2001).
[CrossRef]

M. Arik, N. M. Atakishiyev, and K. B. Wolf, “Quantum algebraic structures compatible with the harmonic oscillator Newton equation,” J. Phys. A 32, L371-L376 (1999).
[CrossRef]

L. Barker, Ç. Çandan, T. Hakioğlu, M. A. Kutay, and H. M. Ozaktas, “The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform,” J. Phys. A 33, 2209-2222 (2000).
[CrossRef]

Theor. Math. Phys.

N. M. Atakishiyev and S. K. Suslov, “Difference analogs of the harmonic oscillator,” Theor. Math. Phys. 85, 1055-1062 (1991).
[CrossRef]

Other

L. C. Biedenharn and J. D. Louck, “Angular momentum in quantum physics,” in Encyclopedia of Mathematics and Its Applications, G.-C.Rota, ed. (Addison-Wesley, 1981), Vol. 8.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, 1953), Vol. 2.

N. Ya. Vilenkin, Special Functions and the Theory of Group Representations (American Mathematical Society, 1968).

R. Gilmore, Lie Groups Lie Algebras and Some of Their Applications (Wiley, 1974).

L. E. Vicent, “Análisis de Señales Discretas Finitas mediante el Modelo de Oscilador Finito de su(2),” Ph.D. dissertation (Universidad Autónoma del Estado de Morelos, 2007).

D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskiĭ, Quantum Theory of Angular Momentum (World Scientific, 1988).

W. Miller, Jr., “Symmetry groups and separation of variables,” in Encyclopedia of Mathematics, G.-C.Rota, ed. (Addison-Wesley, 1977), Vol. 4.

A. Frank and P. Van Isacker, Algebraic Methods in Molecular and Nuclear Structure Physics (Wiley, 1998).

K. B. Wolf, Geometric Optics on Phase Space (Springer-Verlag, 2004).

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Figures (11)

Fig. 1
Fig. 1

(a) Cartesian and (b) polar arrangements of 17 2 pixel centers.

Fig. 2
Fig. 2

Classical sphere with rotation axes j 1 , j 2 , j 3 . Tangent to it, the phase space plane with translation directions q , p . The harmonic oscillator evolution rotates the figure around its vertical axis. We deform q j 1 , p j 2 , and identify the Hamiltonian with h j 3 + r , where r is the radius of the sphere.

Fig. 3
Fig. 3

Kravchuk functions, Ψ n ( N ) ( q ) in Eq. (11), of the finite su ( 2 ) oscillator for N = 65 ( j = 32 ) . Selected values of n show the lower, middle, and higher states. Note that the latter reproduce the former with alternating signs.

Fig. 4
Fig. 4

Rhombus pattern for the two-dimensional finite oscillator states classified by the total mode number n = n x + n y .

Fig. 5
Fig. 5

Rhombus pattern of two-dimensional finite oscillator Kravchuk functions. The upper half of the rhombus reproduces the lower one with a checkerboard of alternating signs.

Fig. 6
Fig. 6

Rhombus pattern of two-dimensional Laguerre–Kravchuk modes with definite angular momentum. Since the modes are complex, we plot the real modes Λ n , m ( N ) , c = 1 2 ( Λ n , m ( N ) + Λ n , m ( N ) ) for m 0 on the right-hand side and Λ n , m ( N ) , s = 1 2 i ( Λ n , m ( N ) Λ n , m ( N ) ) for m > 0 on the left-hand side. The m = 0 modes are real. Again, the upper triangle reproduces the lower one with a checkerboard of alternating signs.

Fig. 7
Fig. 7

States in the rhombus classified by mode number n = j + λ and angular momentum m are related to states classified by radius ρ and the same angular momentum through Clebsch–Gordan coefficients.

Fig. 8
Fig. 8

States classified by radius ρ and angular momentum m are related, at each circle of radius ρ, to 2 ρ + 1 kets associated with equally spaced points through the finite FT.

Fig. 9
Fig. 9

Rhombus pattern of two-dimensional modes ( n , m ) with definite mode number and angular momentum on a screen with pixels that follow polar coordinates. Again, the modes are complex, so we show the real part of the modes for m 0 on the right-hand side and the imaginary part for m > 0 on the left-hand side. The functions are zero for radii ρ < m .

Fig. 10
Fig. 10

Left: unitary map of a high-contrast image of 65 2 pixels from Cartesian (top) to polar (bottom) coordinates. Right: map of a low-contrast version. White corresponds to 1 and black to 0.

Fig. 11
Fig. 11

Top: map of a square grid of one-pixel thick lines (in white). Bottom: map of a partial square grid of three-pixel thick lines (in white). As in Fig. 10, white corresponds to 1 and black to 0.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

position : Q = J 1 ,
momentum : P = J 2 ,
3 - projection : L = J 3 ,
mode number : N = J 3 + j 1 ,
energy : H = J 3 + ( j + 1 2 ) 1 .
[ H , Q ] = i P , [ P , H ] = i Q ,
[ Q , P ] = i L = i ( H ( j + 1 2 ) 1 ) .
J 2 Q 2 + P 2 + L 2 = j ( j + 1 ) 1 .
f ( q ) j , q f 1 , { J 2 j , q 1 = j ( j + 1 ) j , q 1 , Q j , q 1 = q j , q 1 ,
Ψ n ( N ) ( q ) j , q j , Λ 3 1 = d n j , q j ( 1 2 π ) , { J 2 j , λ 3 = j ( j + 1 ) j , λ 3 , L j , λ 3 = λ j , λ 3 .
d m , m j ( β ) = d m , m j ( β ) = j , m exp ( i β P ) j , m 3 3 ,
d m , m j ( β ) = ( 1 ) m m ( m m ) ! ( j m ) ! ( j + m ) ! ( j + m ) ! ( j m ) ! ( cos 1 2 β ) 2 j + m m ( sin 1 2 β ) m m F 1 2 ( m j , m j ; m m + 1 ; tan 2 1 2 β ) .
Ψ n ( N ) ( q ) = d n j , q j ( 1 2 π ) = ( 1 ) n 2 j ( 2 j n ) ( 2 j j + q ) K n ( j + q ; 1 2 , 2 j ) ,
Q x , P x , L x = N x j 1 = H x ( j + 1 2 ) 1 su ( 2 ) x ,
Q y , P y , L y = N y j 1 = H y ( j + 1 2 ) 1 su ( 2 ) y .
Ψ n x , n y ( N ) ( q x , q y ) = Ψ n x ( N ) ( q x ) Ψ n y ( N ) ( q y ) = d n x j , q x j ( 1 2 π ) d n y j , q y j ( 1 2 π ) = j , q x j , n x j 3 x 1 x × j , q y j , n y j 3 y 1 y j , q x ; j , q y j , n x j ; j , n y j 3 1 ,
F ( q x , q y ) = j , q x ; j , q y F 1 .
antisymmetric FT , F ¯ 1 1 4 ( P ¯ x 2 P ¯ y 2 + Q ¯ x 2 Q ¯ y 2 ) ,
gyration , F ¯ 2 1 2 ( P ¯ x P ¯ y + Q ¯ x Q ¯ y ) ,
rotation , F ¯ 3 1 2 ( Q ¯ x P ¯ y Q ¯ y P ¯ x ) 1 2 M ¯ .
symmetric FT , F ¯ 0 1 4 ( P ¯ x 2 + P ¯ y 2 + Q ¯ x 2 + Q ¯ y 2 2 1 ) .
Λ n , m ( N ) ( q x , q y ) n x + n y = n e i π ( n x n y ) 4 d 1 2 m , 1 2 ( n x n y ) n 2 ( 1 2 π ) Ψ n x , n y ( N ) ( q x , q y ) ,
n { 0 , 1 , , 4 j } ,
{ n 2 j m { n , n + 2 , , n } , n 2 j m { 4 j + n , 4 j + n + 2 , , 4 j n } .
j ; n , m ) LK n x + n y = n e i π ( n x n y ) 4 d 1 2 m , 1 2 ( n x n y ) n 2 ( 1 2 π ) j , n x j ; j , n y j 3 .
Ψ 1 2 ( n + m ) ( n + 1 ) ( 1 2 ( n x n y ) ) = d 1 2 m , 1 2 ( n x n y ) n 2 ( 1 2 π ) .
J x × J x = i J x J x × J y = 0 J y × J y = i J y } J + J x + J y J J x J y { J + × J + = i J + J + × J = i J J × J = i J + .
J + ( R + S + L ) , J ( S R 1 2 M ) ,
[ M , R ± ] = ± i R , [ M , S ± ] = ± i S .
[ L , R ± ] = i S ± , [ L , S ± ] = i R .
R 2 R + 2 + R 2 + M 2 commutes with M .
j ; λ , m L ,
eigenkets of : J x 2 J y 2 J x 3 J y 3 ,
eigenvalues : j ( j + 1 ) j ( j + 1 ) 1 2 ( λ + m ) 1 2 ( λ m ) ,
j ; ρ , m R ,
eigenkets of : ( J x + J y ) 2 J x J y R 2 M ,
eigenvalues : 2 j ( j + 1 ) 0 ρ ( ρ + 1 ) m .
j ; ρ , m j ; λ , m L R = φ ( j , ρ , λ , m ) C 1 2 ( m + λ ) , 1 2 ( m λ ) , m j , j , ρ ,
φ ( j , ρ , λ , m ) = ( 1 ) j + ρ + 1 2 ( m m ) e i π λ 2 .
j ; ρ , θ k ) A 1 2 ρ + 1 m = ρ ρ e i m θ k j ; ρ , m R ,
Φ n , m ( N ) ( ρ , θ k ) ( j ; ρ , θ k j ; n j , m L A = 1 2 ρ + 1 m = ρ ρ e i m θ k φ ( j , ρ , n j , m ) C 1 2 ( m + n j ) , 1 2 ( m n + j ) , m j , j , ρ .
F ( ρ , θ k ) ( j ; ρ , θ k F A
= q x , q y = j j ( j ; ρ , θ k j , q x ; j , q y 1 A × j , q x ; j , q y F 1 .
U ( N ) ( ρ , θ k ; q x , q y ) ( j ; ρ , θ k j , q x ; j , q y 1 A = n , m ( j ; ρ , θ k j ; n j , m L A ( j ; n , m j , q x ; j , q y 1 LK
= n , m Φ n , m ( N ) ( ρ , θ k ) Λ n , m ( N ) ( q x , q y ) * ,
F ( ρ , θ k ) = q x , q y = j j U ( N ) ( ρ , θ k ; q x , q y ) F ( q x , q y ) ,
F ( q x , q y ) = ρ , θ k U ( N ) ( ρ , θ k ; q x , q y ) * F ( ρ , θ k ) ,

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