Abstract

A wave function for single- and many-photon states is defined by associating photons with different momenta with different spectral and polarization components of the classical, generally complex, electromagnetic field that propagates in a definite direction. When each spectral component of the classical field is scaled to the square root of the photon energy, the appropriately normalized photon wave function acquires the desired interpretation of probability density amplitude, in contradistinction to the Riemann–Silberstein wave function that can be considered the amplitude of the photon probability energy density.

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  1. T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
    [CrossRef]
  2. A. S. Wightman, "On the localizability of quantum mechanical systems," Rev. Mod. Phys. 34, 845-872 (1962).
    [CrossRef]
  3. P. W. Milonni, The Quantum Vacuum (Academic, 1994).
  4. S. Weinberg, "The cosmological constant problem," Rev. Mod. Phys. 61, 1-23 (1989).
    [CrossRef]
  5. R. H. Good, Jr., "Particle aspect of the electromagnetic field equations," Phys. Rev. 105, 1914-1920 (1957).
    [CrossRef]
  6. D. H. Kobe, "A relativistic Schrödinger-like equation for a photon and its second quantization," Found. Phys. 29, 1203-1231 (1999).
    [CrossRef]
  7. I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
    [CrossRef]
  8. J. E. Sipe, "Photon wave functions," Phys. Rev. A 52, 1857-1883 (1995).
    [CrossRef]
  9. A. Gersten, "Maxwell equations--the one-photon quantum equation," Found. Phys. 31, 1211-1231 (2001).
    [CrossRef]
  10. G. S. Agarwal and R. Simon, "Berry phase, interference of light beams, and the Hannay angle," Phys. Rev. A 42, 6924-6927 (1990).
    [CrossRef] [PubMed]
  11. R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
    [CrossRef]
  12. V. V. Dvoeglazov, "Generalized Maxwell equations from the Einstein postulate," J. Phys. A 33, 5011-5016 (2000).
    [CrossRef]
  13. T. Inagaki, "Quantum-mechanical approach to a free photon," Phys. Rev. A 49, 2839-2843 (1994).
    [CrossRef] [PubMed]
  14. J. R. Oppenheimer, "Note on light quanta and the electromagnetic field," Phys. Rev. 38, 725-746 (1931).
    [CrossRef]
  15. I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. 86, 97-116 (1994).
  16. P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, 1958).
  17. F. J. Dyson, "Feynman's proof of the Maxwell equations," Am. J. Phys. 58, 209-211 (1990).
    [CrossRef]
  18. R. J. Cook, "Lorentz covariance of photon dynamics," Phys. Rev. A 26, 2754-2760 (1982).
    [CrossRef]
  19. R. J. Cook, "Photon dynamics," Phys. Rev. A 25, 2164-2167 (1982).
    [CrossRef]
  20. M. Hawton and T. Melde, "Photon number density operator iÊ∙Â," Phys. Rev. A 51, 4186-4190 (1995).
    [CrossRef] [PubMed]
  21. L. Mandel, "Configuration-space photon number operators in quantum optics," Phys. Rev. 144, 1071-1077 (1966).
    [CrossRef]
  22. C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
    [CrossRef]

2001 (1)

A. Gersten, "Maxwell equations--the one-photon quantum equation," Found. Phys. 31, 1211-1231 (2001).
[CrossRef]

2000 (1)

V. V. Dvoeglazov, "Generalized Maxwell equations from the Einstein postulate," J. Phys. A 33, 5011-5016 (2000).
[CrossRef]

1999 (1)

D. H. Kobe, "A relativistic Schrödinger-like equation for a photon and its second quantization," Found. Phys. 29, 1203-1231 (1999).
[CrossRef]

1997 (1)

C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
[CrossRef]

1996 (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

1995 (2)

J. E. Sipe, "Photon wave functions," Phys. Rev. A 52, 1857-1883 (1995).
[CrossRef]

M. Hawton and T. Melde, "Photon number density operator iÊ∙Â," Phys. Rev. A 51, 4186-4190 (1995).
[CrossRef] [PubMed]

1994 (2)

T. Inagaki, "Quantum-mechanical approach to a free photon," Phys. Rev. A 49, 2839-2843 (1994).
[CrossRef] [PubMed]

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. 86, 97-116 (1994).

1990 (2)

F. J. Dyson, "Feynman's proof of the Maxwell equations," Am. J. Phys. 58, 209-211 (1990).
[CrossRef]

G. S. Agarwal and R. Simon, "Berry phase, interference of light beams, and the Hannay angle," Phys. Rev. A 42, 6924-6927 (1990).
[CrossRef] [PubMed]

1989 (1)

S. Weinberg, "The cosmological constant problem," Rev. Mod. Phys. 61, 1-23 (1989).
[CrossRef]

1982 (2)

R. J. Cook, "Lorentz covariance of photon dynamics," Phys. Rev. A 26, 2754-2760 (1982).
[CrossRef]

R. J. Cook, "Photon dynamics," Phys. Rev. A 25, 2164-2167 (1982).
[CrossRef]

1974 (1)

R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
[CrossRef]

1966 (1)

L. Mandel, "Configuration-space photon number operators in quantum optics," Phys. Rev. 144, 1071-1077 (1966).
[CrossRef]

1962 (1)

A. S. Wightman, "On the localizability of quantum mechanical systems," Rev. Mod. Phys. 34, 845-872 (1962).
[CrossRef]

1957 (1)

R. H. Good, Jr., "Particle aspect of the electromagnetic field equations," Phys. Rev. 105, 1914-1920 (1957).
[CrossRef]

1949 (1)

T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
[CrossRef]

1931 (1)

J. R. Oppenheimer, "Note on light quanta and the electromagnetic field," Phys. Rev. 38, 725-746 (1931).
[CrossRef]

Adlard, C.

C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and R. Simon, "Berry phase, interference of light beams, and the Hannay angle," Phys. Rev. A 42, 6924-6927 (1990).
[CrossRef] [PubMed]

Baldo, M.

R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. 86, 97-116 (1994).

Cook, R. J.

R. J. Cook, "Photon dynamics," Phys. Rev. A 25, 2164-2167 (1982).
[CrossRef]

R. J. Cook, "Lorentz covariance of photon dynamics," Phys. Rev. A 26, 2754-2760 (1982).
[CrossRef]

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, 1958).

Dvoeglazov, V. V.

V. V. Dvoeglazov, "Generalized Maxwell equations from the Einstein postulate," J. Phys. A 33, 5011-5016 (2000).
[CrossRef]

Dyson, F. J.

F. J. Dyson, "Feynman's proof of the Maxwell equations," Am. J. Phys. 58, 209-211 (1990).
[CrossRef]

Gersten, A.

A. Gersten, "Maxwell equations--the one-photon quantum equation," Found. Phys. 31, 1211-1231 (2001).
[CrossRef]

Good, R. H.

R. H. Good, Jr., "Particle aspect of the electromagnetic field equations," Phys. Rev. 105, 1914-1920 (1957).
[CrossRef]

Hawton, M.

M. Hawton and T. Melde, "Photon number density operator iÊ∙Â," Phys. Rev. A 51, 4186-4190 (1995).
[CrossRef] [PubMed]

Inagaki, T.

T. Inagaki, "Quantum-mechanical approach to a free photon," Phys. Rev. A 49, 2839-2843 (1994).
[CrossRef] [PubMed]

Kobe, D. H.

D. H. Kobe, "A relativistic Schrödinger-like equation for a photon and its second quantization," Found. Phys. 29, 1203-1231 (1999).
[CrossRef]

Mandel, L.

L. Mandel, "Configuration-space photon number operators in quantum optics," Phys. Rev. 144, 1071-1077 (1966).
[CrossRef]

Melde, T.

M. Hawton and T. Melde, "Photon number density operator iÊ∙Â," Phys. Rev. A 51, 4186-4190 (1995).
[CrossRef] [PubMed]

Mignani, R.

R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
[CrossRef]

Milonni, P. W.

P. W. Milonni, The Quantum Vacuum (Academic, 1994).

Newton, T. D.

T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
[CrossRef]

Oppenheimer, J. R.

J. R. Oppenheimer, "Note on light quanta and the electromagnetic field," Phys. Rev. 38, 725-746 (1931).
[CrossRef]

Pike, E. R.

C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
[CrossRef]

Recami, E.

R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
[CrossRef]

Sarkar, S.

C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
[CrossRef]

Simon, R.

G. S. Agarwal and R. Simon, "Berry phase, interference of light beams, and the Hannay angle," Phys. Rev. A 42, 6924-6927 (1990).
[CrossRef] [PubMed]

Sipe, J. E.

J. E. Sipe, "Photon wave functions," Phys. Rev. A 52, 1857-1883 (1995).
[CrossRef]

Weinberg, S.

S. Weinberg, "The cosmological constant problem," Rev. Mod. Phys. 61, 1-23 (1989).
[CrossRef]

Wightman, A. S.

A. S. Wightman, "On the localizability of quantum mechanical systems," Rev. Mod. Phys. 34, 845-872 (1962).
[CrossRef]

Wigner, E. P.

T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
[CrossRef]

Acta Phys. Pol. (1)

I. Bialynicki-Birula, "On the wave function of the photon," Acta Phys. Pol. 86, 97-116 (1994).

Am. J. Phys. (1)

F. J. Dyson, "Feynman's proof of the Maxwell equations," Am. J. Phys. 58, 209-211 (1990).
[CrossRef]

Found. Phys. (2)

D. H. Kobe, "A relativistic Schrödinger-like equation for a photon and its second quantization," Found. Phys. 29, 1203-1231 (1999).
[CrossRef]

A. Gersten, "Maxwell equations--the one-photon quantum equation," Found. Phys. 31, 1211-1231 (2001).
[CrossRef]

J. Phys. A (1)

V. V. Dvoeglazov, "Generalized Maxwell equations from the Einstein postulate," J. Phys. A 33, 5011-5016 (2000).
[CrossRef]

Lett. Nuovo Cimento (1)

R. Mignani, E. Recami, and M. Baldo, "About a Dirac-like equation for the photon according to Ettore Majorana," Lett. Nuovo Cimento 11, 568-572 (1974).
[CrossRef]

Phys. Rev. (3)

R. H. Good, Jr., "Particle aspect of the electromagnetic field equations," Phys. Rev. 105, 1914-1920 (1957).
[CrossRef]

L. Mandel, "Configuration-space photon number operators in quantum optics," Phys. Rev. 144, 1071-1077 (1966).
[CrossRef]

J. R. Oppenheimer, "Note on light quanta and the electromagnetic field," Phys. Rev. 38, 725-746 (1931).
[CrossRef]

Phys. Rev. A (6)

G. S. Agarwal and R. Simon, "Berry phase, interference of light beams, and the Hannay angle," Phys. Rev. A 42, 6924-6927 (1990).
[CrossRef] [PubMed]

J. E. Sipe, "Photon wave functions," Phys. Rev. A 52, 1857-1883 (1995).
[CrossRef]

T. Inagaki, "Quantum-mechanical approach to a free photon," Phys. Rev. A 49, 2839-2843 (1994).
[CrossRef] [PubMed]

R. J. Cook, "Lorentz covariance of photon dynamics," Phys. Rev. A 26, 2754-2760 (1982).
[CrossRef]

R. J. Cook, "Photon dynamics," Phys. Rev. A 25, 2164-2167 (1982).
[CrossRef]

M. Hawton and T. Melde, "Photon number density operator iÊ∙Â," Phys. Rev. A 51, 4186-4190 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

C. Adlard, E. R. Pike, and S. Sarkar, "Localization of one-photon states," Phys. Rev. Lett. 79, 1585-1587 (1997).
[CrossRef]

Prog. Opt. (1)

I. Bialynicki-Birula, "Photon wave function," Prog. Opt. 36, 245-294 (1996).
[CrossRef]

Rev. Mod. Phys. (3)

S. Weinberg, "The cosmological constant problem," Rev. Mod. Phys. 61, 1-23 (1989).
[CrossRef]

T. D. Newton and E. P. Wigner, "Localized states for elementary systems," Rev. Mod. Phys. 21, 400-406 (1949).
[CrossRef]

A. S. Wightman, "On the localizability of quantum mechanical systems," Rev. Mod. Phys. 34, 845-872 (1962).
[CrossRef]

Other (2)

P. W. Milonni, The Quantum Vacuum (Academic, 1994).

P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed. (Clarendon, 1958).

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Equations (21)

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ϵ 0 ( ϵ 0 E ) = 0 , × ( ϵ 0 E ) = c 1 ( μ 0 H ) t ,
μ 0 ( μ 0 H ) = 0 , × ( μ 0 H ) = c 1 ( ϵ 0 E ) t ,
Ψ em = ( ϵ 0 E μ 0 H ) ,
Ψ em t = c × ( J Ψ em ) = c ( S i ) ( J Ψ em ) ,
Ψ em = 0 ,
J = [ 0 I I 0 ] ,
S x = [ 0 0 0 0 0 i 0 i 0 ] , S y = [ 0 0 i 0 0 0 i 0 0 ] , S z = [ 0 i 0 i 0 0 0 0 0 ] .
Φ em t = c ( S k ) J Φ em ,
k Φ em = 0 ,
Φ em ( k , t ) = ( 2 π ) 3 2 Ψ em ( r , t ) exp ( i r k ) d r ,
a = ( a 1 a 2 ) , b = ( b 1 b 2 )
Ψ ± ( r , t ; k ) = [ 2 ( 2 π ) 3 ] 1 2 ψ ± ( k ) exp ( i k r i ω t ) ,
ψ ± ( k ) = ( f ± ( k ) i f ± ( k ) ) = ( f ± ( k ) k k × f ± ( k ) ) ,
f ± ( k ) = 1 2 k 2 ( k x 2 + k y 2 ) ( k x k z ± i k y k k y k z i k x k k x 2 + k y 2 ) ,
i t Ψ = i c ( S p ) ( J Ψ ) = H Ψ .
i Φ t = i c ( S p ) ( J Φ )
Φ ± ( k , t ) = [ 2 ( 2 π ) 3 ] 1 2 ψ ± ( k ) exp ( i ω t ) .
Φ ( k , t ) = [ 2 ( 2 π ) 3 ] 1 2 [ c + , k ψ + ( k ) + c , k ψ ( k ) ] exp ( i ω t ) ,
Ψ ( r , t ) = [ 2 ( 2 π ) 3 ] 1 2 d k [ c + , k ψ + ( k ) + c , k ψ ( k ) ] exp ( i k r i ω t ) ,
Ψ ( r , t ) = [ 2 ( 2 π ) 3 ] 1 2 d k [ ψ + ( k ) a + , k + ψ ( k ) a , k ] exp ( i k r i ω t ) ,
H = d k c k ( a + , k + a + , k + a , k + a , k ) ,

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