Abstract

Using the symmetries of the three-dimensional Paul trap, we derive the solutions of the time-dependent Schrödinger equation for this system, in both Cartesian and cylindrical coordinates. Our symmetry calculations provide insights that are not always obvious from the conventional viewpoint.

© 2001 Optical Society of America

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References

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  1. J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D 7, 359–362 (1973).
    [Crossref]
  2. P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I–IV. Reprinted by (AIP, Woodbury, NY, 1995).
  3. D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
    [Crossref]
  4. W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. 62, 531–540 (1998).
    [Crossref]
  5. M. Combescure, “A quantum particle in a quadrupole radio-frequency trap,” Ann. Inst. Henri Poincare 44, 293–314 (1986).
  6. M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).
  7. M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. 2, 18.1–18.9 (2000). Eprint quant-ph/0002050.
    [Crossref]
  8. G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
    [Crossref]
  9. W. Miller, Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).
  10. M. M. Nieto and D. R. Truax (in preparation).
  11. V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
    [Crossref] [PubMed]
  12. J. R. Klauder, private communication.
  13. The phase factor can also be obtained [14, 15], by solving the eigenvalue equation M3Znz=(nz+12)Znz,, where M3=i{ϕ3∂t+12(z∂z+12)−i4ϕ¨3z2}.. Then, solving the equation Jz -Z0=0 will yield the extremal state function up to a factor of (π)-1/4.
  14. D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {∂xx +2i∂t -2g2(t)x2-2g1(t)x-2g0(t)}Ψ(x, t]=0.” J. Math. Phys. 23, 43–54 (1982).
    [Crossref]
  15. M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. 38, 84–97 (1997).
    [Crossref]
  16. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).
  17. V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
    [Crossref]
  18. M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

2000 (1)

M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. 2, 18.1–18.9 (2000). Eprint quant-ph/0002050.
[Crossref]

1998 (2)

W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. 62, 531–540 (1998).
[Crossref]

M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).

1997 (1)

M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. 38, 84–97 (1997).
[Crossref]

1995 (1)

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

1993 (1)

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

1986 (1)

M. Combescure, “A quantum particle in a quadrupole radio-frequency trap,” Ann. Inst. Henri Poincare 44, 293–314 (1986).

1985 (1)

V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
[Crossref]

1983 (1)

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
[Crossref]

1982 (1)

D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {∂xx +2i∂t -2g2(t)x2-2g1(t)x-2g0(t)}Ψ(x, t]=0.” J. Math. Phys. 23, 43–54 (1982).
[Crossref]

1973 (1)

J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D 7, 359–362 (1973).
[Crossref]

Combescure, M.

M. Combescure, “A quantum particle in a quadrupole radio-frequency trap,” Ann. Inst. Henri Poincare 44, 293–314 (1986).

Dawson, P. H.

P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I–IV. Reprinted by (AIP, Woodbury, NY, 1995).

Eberly, J. H.

J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D 7, 359–362 (1973).
[Crossref]

Feng, M.

M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).

Glauber, R. J.

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

Itano, W. M.

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
[Crossref]

Klauder, J. R.

J. R. Klauder, private communication.

Kosteleck, V. A.

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

Kostelecky, V. A.

V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
[Crossref]

Magnus, W.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).

Man’ko, V. I.

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

Miller, W.

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).

Nieto, M. M.

M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. 2, 18.1–18.9 (2000). Eprint quant-ph/0002050.
[Crossref]

M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. 38, 84–97 (1997).
[Crossref]

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
[Crossref]

M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

M. M. Nieto and D. R. Truax (in preparation).

Oberhettinger, F.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).

Paul, W.

W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. 62, 531–540 (1998).
[Crossref]

Schleich, W. P.

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

Schrade, G.

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

Singh, L. P. S.

J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D 7, 359–362 (1973).
[Crossref]

Soni, R. P.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).

Truax, D. R.

M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. 2, 18.1–18.9 (2000). Eprint quant-ph/0002050.
[Crossref]

M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. 38, 84–97 (1997).
[Crossref]

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
[Crossref]

D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {∂xx +2i∂t -2g2(t)x2-2g1(t)x-2g0(t)}Ψ(x, t]=0.” J. Math. Phys. 23, 43–54 (1982).
[Crossref]

M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

M. M. Nieto and D. R. Truax (in preparation).

Van Dyck, R. S.

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
[Crossref]

Wang, K. L.

M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).

Wineland, D. J.

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
[Crossref]

Wu, J. H.

M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).

Adv. Atomic Mol. Phys. (1)

D. J. Wineland, W. M. Itano, and R. S. Van Dyck, “High-resolution spectroscopy of stored ions,” Adv. Atomic Mol. Phys. 19, 135–186 (1983).
[Crossref]

Ann. Inst. Henri Poincare (1)

M. Combescure, “A quantum particle in a quadrupole radio-frequency trap,” Ann. Inst. Henri Poincare 44, 293–314 (1986).

Commun. Theoret. Phys. (1)

M. Feng, J. H. Wu, and K. L. Wang, “A Study of the characteristics of the wave packets of a Paul-trapped ion,” Commun. Theoret. Phys. 29, 497–502 (1998).

J. Math. Phys. (2)

D. R. Truax, “Symmetry of time-dependent Schrödinger equations. II. Exact solutions for the equation {∂xx +2i∂t -2g2(t)x2-2g1(t)x-2g0(t)}Ψ(x, t]=0.” J. Math. Phys. 23, 43–54 (1982).
[Crossref]

M. M. Nieto and D. R. Truax, “Displacement operator squeezed states. I. Time-dependent systems having isomorphic symmetry algebras,” J. Math. Phys. 38, 84–97 (1997).
[Crossref]

New J. Phys. (1)

M. M. Nieto and D. R. Truax, “Coherent states sometimes look like squeezed states, and visa versa: The Paul trap,” New J. Phys. 2, 18.1–18.9 (2000). Eprint quant-ph/0002050.
[Crossref]

Phys. Rev. A (1)

V. A. Kosteleck, V. I. Man’ko, M. M. Nieto, and D. R. Truax, “Supersymmetry and a time-dependent Landau system,” Phys. Rev. A 48, 951–963 (1993).
[Crossref] [PubMed]

Phys. Rev. D (2)

V. A. Kosteleckŷ, M. M. Nieto, and D. R. Truax, “Supersymmetry and the relationship between the Coulomb and oscillator problems in arbitrary dimensions,” Phys. Rev. D 32, 2627–2633 (1985).
[Crossref]

J. H. Eberly and L. P. S. Singh, “Time operators, partial stationarity, and the energy-time uncertainty relation,” Phys. Rev. D 7, 359–362 (1973).
[Crossref]

Quantum Semiclass. Opt. (1)

G. Schrade, V. I. Man’ko, W. P. Schleich, and R. J. Glauber, “Wigner functions in the Paul trap,” Quantum Semiclass. Opt. 7, 307–325 (1995).
[Crossref]

Rev. Mod. Phys. (1)

W. Paul, “Electromagnetic traps for charged and neutral particles,” Rev. Mod. Phys. 62, 531–540 (1998).
[Crossref]

Other (7)

P. H. Dawson, Quadrupole Mass Spectrometry and its Applications (Elsevier, Amsterdam, 1976), Chaps. I–IV. Reprinted by (AIP, Woodbury, NY, 1995).

W. Miller, Symmetry and Separation of Variables (Addison-Wesley, Reading, MA, 1977).

M. M. Nieto and D. R. Truax (in preparation).

M. M. Nieto and D. R. truax, eprint quant-ph/0011062, expands the contents of this manuscript. It contains further information on Ref. [1] and, in an appendix, on J. H. Eberly.

J. R. Klauder, private communication.

The phase factor can also be obtained [14, 15], by solving the eigenvalue equation M3Znz=(nz+12)Znz,, where M3=i{ϕ3∂t+12(z∂z+12)−i4ϕ¨3z2}.. Then, solving the equation Jz -Z0=0 will yield the extremal state function up to a factor of (π)-1/4.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd Edition (Springer, New York, 1966).

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Figures (1)

Fig. 1.
Fig. 1.

A plot of the allowed quantum numbers for the polar coordinates of the Paul trap. Shown are the (n,m) quantum numbers of Eqs. (54) and (55) as well as the (nr,ℓz ) quantum numbers of Eqs. (57) and (60).

Equations (67)

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V ( x , y , z , t ) = V x ( x , t ) + V y ( y , t ) + V z ( z , t ) ,
V x ( x , t ) = + e 2 r 0 2 V ( t ) x 2 g ( t ) x 2 ,
V y ( y , t ) = + e 2 r 0 2 V ( t ) y 2 g ( t ) y 2 ,
V z ( z , t ) = e r 0 2 V ( t ) z 2 g 3 ( t ) z 2 .
x ̈ c l = F x ( x , t ) = d V x ( x , t ) d x = 2 g ( t ) x c l ,
y ̈ cl = F y ( y , t ) = d V y ( y , t ) dy = 2 g ( t ) y c l ,
z ̈ cl = F z ( z , t ) = d V z ( z , t ) dz = 2 g 3 ( t ) z c l .
S Ψ ( x , y , z , t ) = { x x + y y + z z + 2 i t 2 g ( t ) ( x 2 + y 2 ) 2 g 3 ( t ) z 2 } Ψ ( x , y , z , t ) = 0 .
g ( t ) = + e 2 r 0 2 V ( t ) , g 3 ( t ) = e r 0 2 V ( t ) .
V ( t ) = V d c V a c cos ω ( t t 0 )
x = r cos θ , y = r sin θ , z = z ,
S c y l Φ ( r , θ , z , t ) = { r r + 1 r r + 1 r 2 θ θ + z z + 2 i t 2 g ( t ) r 2 2 g 3 ( t ) z 2 } Φ ( r , θ , z , t )
= 0 .
Ψ ( x , y , z , t ) = X ( x , t ) Y ( y , t ) Z ( z , t ) ,
S x X ( x , t ) = { x x + 2 i t 2 g ( t ) x 2 } X ( x , t ) = 0 ,
S y Y ( y , t ) = { yy + 2 i t 2 g ( t ) y 2 } Y ( y , t ) = 0 ,
S z Z ( z , t ) = { zz + 2 i t 2 g 3 ( t ) z 2 } Z ( z , t ) = 0 ,
Φ ( x , y , z , t ) = Ω ( r , θ , t ) Z ( z , t ) ,
S r θ Ω ( r , θ , t ) = { r r + 1 r r + 1 r 2 θ θ 2 g ( t ) r 2 } Ω ( r , θ , t ) = 0 .
[ S , L ] = ( x , y , z , t ) S .
L = C 0 t + C 1 x + C 2 y + C 3 z + C ,
x = x ϕ 1 / 2 ( t ) , y = y ϕ 1 / 2 ( t ) , z = z ϕ 3 1 / 2 ( t ) , t = t ,
ρ = x 2 + y 2 ϕ 1 / 2 ( t ) , θ = sin 1 ( y x 2 + y 2 ) , z = z ϕ 3 1 / 2 ( t ) , t = t .
ϕ = 2 ξ ξ ¯ ϕ 3 = 2 ξ 3 ξ ¯ 3 ,
γ ̈ + 2 g ( t ) γ = 0 , γ ̈ 3 + 2 g 3 ( t ) γ 3 = 0 ,
W ( ξ , ξ ¯ ) = W ( ξ 3 , ξ ¯ 3 ) = i .
J z = ξ 3 z i ξ ˙ 3 z = + 1 2 ( ξ ¯ 3 ξ 3 ) 1 / 2 [ z + z ( 1 i 2 ϕ ˙ 3 ) ] ,
J z + = ξ ¯ 3 z + i ξ ¯ ˙ 3 z = 1 2 ( ξ ¯ 3 ξ 3 ) 1 / 2 [ z z ( 1 + i 2 ϕ ˙ 3 ) ] .
[ J z , J z + ] = I ,
J z Z n z ( z , t ) = n z Z n z 1 ( z , t ) ,
J z + Z n z ( z , t ) = n z + 1 Z n z + 1 .
i t Z n z = H Z n z = i [ i t Z n z Z n z ] Z n z [ Const ] Z n z .
J z Z 0 ( z , t ) = 0 .
Z 0 ( z , t ) = f ( t ) exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } ,
Z 0 ( z , t ) = ( π ϕ 3 ) 1 / 4 ( ξ ¯ 3 ξ 3 ) 1 / 4 exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } .
Z n z ( z , t ) = [ n z ! ] 1 / 2 [ J z + ] n z Z 0 ( z , t ) .
Z n z ( z , t ) = ( 2 n z n z ! ) 1 / 2 ( π ϕ 3 ) 1 / 4 ( ξ ¯ 3 ξ 3 ) 1 / 2 ( n z + 1 2 )
H n z ( z ) exp { z 2 2 [ 1 i ϕ ˙ 3 2 ] } .
X n x ( x , t ) = Z n z n x ( z x , ξ 3 ξ , ϕ 3 ϕ , t ) ,
Y n y ( y , t ) = Z n z n y ( z y , ξ 3 ξ , ϕ 3 ϕ , t ) ,
Ψ n x , n y , n z ( x , y , z , t ) = X n x ( x , t ) Y n y ( y , t ) Z n z ( z , t ) .
a = a + = 1 2 ( J x + i J y ) = 1 2 [ ξ ( x + i y ) i ξ ˙ ( x + i y ) ]
= 1 2 ( ξ ξ ¯ ) 1 / 2 e i θ [ ρ + i ρ θ + ρ ( 1 i 2 ϕ ˙ ) ] ,
c = c + = 1 2 ( J x i J y ) = 1 2 [ ξ ( x i y ) i ξ ˙ ( x i y ) ]
= 1 2 ( ξ ξ ¯ ) 1 / 2 e i θ [ ρ i ρ θ + ρ ( 1 i 2 ϕ ˙ ) ] .
κ = J x + J x + J y + J y + 1 = a + a + c + c + 1 ,
z = i ( y x x y ) = i θ ,
f 1 2 ( κ z ) = a + a + 1 2 ,
d 1 2 ( κ + z ) = c + c + 1 2 ,
f Ω n , m = ( n + 1 2 ) Ω n , m , d Ω n , m = ( m + 1 2 ) Ω n , m ,
a Ω n , m = n Ω n 1 , m , c Ω n , m = m Ω n , m 1 ,
a + Ω n , m = n + 1 Ω n + 1 , m , c + Ω n , m = m + 1 Ω n , m + 1 ,
z Ω n , m = ( f d ) Ω n , m = ( m n ) Ω n , m ,
κ Ω n , m = ( d + f ) Ω n , m = ( n + m + 1 ) Ω n , m ,
a Ω 0 , 0 = 0 , c Ω 0 , 0 = 0 .
Ω n , m , ( r , θ , t ) = 𝓡 n , m ( r , t ) Θ n , m ( θ ) = 𝓡 n , m ( r , t ) exp [ i ( m n ) θ ] 2 π .
Ω 0 , 0 ( r , θ , t ) = 1 2 π [ 2 ϕ ] 1 / 2 ( ξ ¯ ξ ) 1 2 exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } = X 0 ( x , t ) Y 0 ( y , t ) .
Ω n , m ( ρ , θ , t ) = ( n ! m ! ) 1 / 2 a + n c + m Ω 0 , 0 ( ρ , θ , t ) .
Ω n , m ( r , θ , t ) = exp [ i ( m n ) θ ] 2 π ( ) k k ! ( n ! m ! ) 1 / 2 ( 2 ϕ ( t ) ) 1 / 2 ( ξ ¯ ξ ) 1 2 ( n + m + 1 )
ρ n m L k ( n m ) ( ρ 2 ) exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } ,
k 1 2 ( n + m m n ) .
= z , z = m n , n r = m + n 0 .
Ω n , m ( r , t ) R n r , ( r , t ) Θ z ( θ ) ,
Θ z ( θ ) = exp [ i z θ ] 2 π ,
R n r , ( r , t ) = ( 1 ) ( n r ) / 2 [ 2 [ n r 2 ] ! ϕ ( t ) [ n r + 2 ] ! ] 1 / 2 ( ξ ¯ ξ ) 1 2 ( n + m + 1 )
ρ L 1 2 ( n r ) ( ) ( ρ 2 ) exp { ρ 2 2 [ 1 i ϕ ˙ 2 ] } .
Φ n r , z , n z ( x , y , z , t ) = R n r , ( r , t ) Θ z ( θ ) Z n z ( z , t ) .

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