Photon wave mechanics and position eigenvectors
Margaret Hawton"
One and two photon wave functions are derived by projecting the quantum state vector onto
simultaneous eigenvectors of the number operator and a recently constructed photon position oper-
ator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only
the Landau-Peierls wave function defines a positive definite photon density, a similarity transfor-
mation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell s equations
preserves eigenvalues and expectation values. We show that this real space description of photons
is compatible with all of the usual rules of quantum mechanics and provides a framework for under-
standing the relationships amongst different forms of the photon wave function in the literature. It
also gives a quantum picture of the optical angular momentum of beams that applies to both one
photon and coherent states. According to the rules of qunatum mechanics, this wave function gives
the probability to count a photon at any position in space.
I. INTRODUCTION operator with commuting components that transforms as
a vector [10].
Our position operator [11] has commuting components
The current interest in entanglement and its applica-
but is not rotationally invariant and does not transform
tion to quantum information has rekindled the contro-
as a vector [12], and thus it is consistent with the pre-
versy surrounding the photon wave function [1, 2, 3, 4, 5,
vious work. Description of a localized state requires a
6]. It is still unclear what form a real space photon wave
sum over all k, and a localized photon can have definite
function should take, or if one exists. In the standard
spin in the k-direction, that is it can have definite he-
formulation of quantum mechanics, the coordinate space
licity, but it cannot have definite spin along any fixed
wave function is the projection of the state vector onto
axis. It is the total angular momentum (AM) that can
an orthonormal basis of eigenvectors of a Hermitian po-
have a definite value along some specified direction in
sition operator. It has been claimed since the early days
space [13]. The position eigenvectors are not spherically
of quantum mechanics that there is no position opera-
symmetric, instead they have a vortex structure as is ob-
tor for the photon that allows the introduction of a wave
served for twisted light [14]. Compared to the Newton
function in this way. Contrary to these claims, we have
Wigner position operators for which transformation of
recently constructed a photon position operator whose
a particle s spin and position are separable, the photon
transverse eigenvectors form a real space basis. Here we
position operator must incorporate an additional unitary
will use this basis to obtain a photon wave function that
transformation that reorients this vortex.
is compatible with the usual rules of quantum mechanics.
Valid position eigenvectors cannot violate the
We will show that this clarifies a number of previously
Hegerfeldt [15] and Paley-Wiener [16] theorems based
unresolved issues regarding the real space description of
on Fourier transform theory. Hegerfeldt proved that
one photon and multiphoton states.
a positive frequency wave function can be exactly
In 1933 Pauli [7] stated that the nonexistence of a den-
localized at only one instant in time and interpreted
sity for the photon corresponds to the fact that the po-
this to imply a violation of causality. Bialynicki-
sition of a photon cannot be associated with an opera-
Birula [17] noted that the Paley-Wiener theorem limits

"
tor in the usual sense. Based on definitions of center of
g (x) = dkh (k) exp (-ikx) of the form exp (-Axł)
0
mass, Pryce found the k-space photon position operator
to Å‚ < 1. He then applied this to separate outgoing

= i" - ik/2k + k×S/k where Sj are the 3 × 3 spin
rP
and incoming exponentially localized spherical pulses in

1 matrices, k is a unit wave vector, and "j = "/"kj three dimensions. However, their sum is not subject to
the exponential localization limit, as can be seen from
[8]. This operator does not have commuting components
the form of the k-integral. Position eigenvectors require
which suggests that three spatial coordinates cannot si-
a sum over all wave vectors, and thus must be a sum
multaneously have a definite value. In 1949 Newton and
of outgoing and incoming waves that interfere to give
Wigner sought rotationally invariant localized states and
exact localization at a single instant in time, consistent
the corresponding position operators. They were success-
with the Hegerfeldt theorem.
ful for massive particles and zero mass particles with spin
Maxwell s equations (MEs) are analogous to the
0 and 1/2, but found for photons  no localized states in
Dirac equation when written in terms of the Riemann-
the above sense exist [9]. This result is widely quoted
Silberstein (RS) field vector, proportional to EÄ…icB
as a proof of the nonexistence of a photon position oper-
where c is the speed of light in vacuum, E is the elec-
ator. It has been proved that there is no photon position
tric field, and B is the magnetic induction. This sug-
gests that the photon is an elementary particle like any
other, and that MEs provide a first quantized description
"
Electronic address: [e-mail] of the photon. Use of the positive frequency RS vector
arXiv:0705.3196v1 [quant-ph] 22 May 2007
2
as a photon wave function in vacuum and in a medium this section. Subscripts denote eigenvalues and Cartesian
has been thoroughly studied [1, 18, 19, 20]. If a field components of the vectors È and e. Cartesian compo-
¨(1/2) with quantum electrodynamic weighting, k1/2, is nents are used where it is necessary to avoid confusing
used as wave function, a metric factor k-1 is required in vector notation. The parameter Ä… allows for both LP
the scalar product. The real space squared norm then and field based wave functions. The position eigenvec-

tors are electric and/or magnetic fields if Ä… = 1/2, the
goes as d3r d3r2 ¨(1/2)" (r) · ¨(1/2) (r2 ) / |r - r2 |2 and
thus its integrand cannot be interpreted as a local num- vector potential if Ä… = -1/2, or LP wave functions if
ber density [18, 21]. Since the photon has no mass, it has Ä… = 0. This is consistent with the QED based interpre-
tation that a mode with frequency Ék has energy Ék so
been suggested that there is no photon number density,
that the square of the fields gives energy density while
only energy density [20]. Photon number density based
the wave function gives number density. The spherical
on the Landau-Peierls (LP) wave function ¨(0) (without
polar definite helicity unit vectors are
the factor k1/2) was investigated as early as 1930 [22, 23].
Its absolute value squared is positive definite but it has

"

the disadvantage that its relationship to electric current
e(0) = ¸ + iÃĆ / 2 (2)
k,Ã
density and the electromagnetic fields is nonlocal in real
space [18, 22, 23, 24].

where ¸ and Ć are unit vectors in the increasing ¸ and
Returning to field-like ¨(1/2) functions, we will show
Ć directions. Periodic boundary conditions in a finite
here that it is possible to define a biorthonormal basis
volume are used here to simplify the notation, and the
that gives a local density by combining the eigenvectors
limit as V " can be taken to calculate derivatives
of an operator with those of its adjoint. This formalism
and perform sums. If the wave function (1) is a position
has recently been applied to pseudo-Hermitian Hamilto-
eigenvector it should satisfy the eigenvector equation
nians that possess real spectra [25]. Such a basis provides
an interesting alternative to explicit inclusion of a metric
(Ä…)È(Ä…) (k) = rÈ(Ä…) (k) (3)
r
r,Ã,j r,Ã,j
operator when working with electromagnetic fields. The
density ¨(1/2)" (r)·¨(-1/2) (r) is local, but it not positive
where (Ä…) is the k-space representation of the position
r
definite since it is not an absolute value squared. Only
operator and its eigenvalues, r, can be interpreted as pho-
the LP wave function defines a positive definite photon
2
ton position.
¨(0)
density, equal to (r) . However, we will show that
The operator arrived at in [11] using the condition (3)
the biorthogonal field-potential pair gives the same re-
is
sults in most calculations.
In the present paper one and two photon wave func-
(Ä…) = (Ä…) + SkĆ cot ¸/k (4)
r rP
tions and photon density will be obtained by projection
onto a basis of position eigenvectors. In Section II the where
photon position operator will be reviewed and the scalar

(Ä…) = iI" - iIÄ…k/k + k×S/k, (5)
rP
product and Hermiticity will be discussed. In Section III
the orthonormal and biorthonormal eigenkets of the posi-
is a generalization of the Pryce operator, I is a 3 × 3 unit
tion operator will be obtained in the Heisenberg picture
matrix, (Si)jk = -i%EÅ‚ijk, and the component of spin par-
(HP). We will then derive photon wave functions from

quantum electrodynamics (QED) in Section IV by pro- allel to k, Sk = k·S, extracts the helicity Ã. The opera-
jecting the state vector onto simultaneous eigenvectors
tor (Ä…) is essentially the usual k-space position operator,
r
of the photon position, helicity, and number operators.
i", with terms added to compensate for differentiation
We will discuss MEs, photon wave mechanics, and an-
of the unit vectors and kÄ… by ". The term k×S/k gives a
gular momentum and beams in Sections V, VI and VII

transverse vector, while SkĆ cot ¸/k rotates ¸ and Ć back
respectively and then conclude.

to their original orientations, and -iIÄ…k/k corrects for
differentiation of kÄ…. It was proved in [11] that (Ä…) has
r
commuting components and satisfies the other expected
II. POSITION OPERATOR
commutation relations.
The photon s position coordinates must be real, and
We start with a discussion of the photon position op-
this normally implies that the position operator must be
erator. The procedure used in [11] was to construct an
Hermitian. In the LP case the k-space inner-product is
operator with transverse eigenvectors of definite helicity,


à = ą1. In k-space, it is reasonable to expect that the

¨(0)|¨(0) = ¨(0)" (k) ¨(0) (k)
j j
transverse function
k, j
"
È(Ä…) (k) = (Ék)Ä… e(Ç) exp (-ik · r) / V (1)
r,Ã,j k,Ã,j
where ¨(Ä…) (k) and ¨(Ä…) (k) are any two state vectors.
It can be proved by converting the sum to an inte-

describes a photon located at position r, where Ék = kc

gral and integrating by parts that ¨(0)| (0)¨(0) =
r
in vacuum and the parameter Ç will be discussed later in
3


venience and the choice Ç = -mĆ gives [13]
(0)¨(0)|¨(0) which implies that (0) is Hermitian.
r r
The case Ä… = 1/2 with inner-product
z
x - iy
e(-mĆ) = " (cos ¸ - Ã) ei(mÃ+1)Ć - " sin ¸eimÃĆ
k,Ã

2 2 2

¨(1/2)|¨(1/2) = k-1¨(1/2)" (k) ¨(1/2) (k) (6)
j j x + iy

"
+ (cos ¸ + Ã) ei(mÃ-1)Ć. (9)
k, j
2 2
was considered in [21] and [11]. Integration by parts in

For example, Ç = -Ć (m = 1) rotates ¸ and Ć back
this case requires differentiation of k-1,which again gives
to the x and y axes to give unit vectors that approach
"

¨(1/2)| (1/2)¨(1/2) = (1/2)¨(1/2)|¨(1/2) , proving + iÃy) / 2 in the ¸ 0 limit. This is useful in the de-
r r (x
scription of paraxial beams since the unit vectors describe
that (1/2) is Hermitian based on the inner-product (6).
r
spin alone. Their coefficients then describe all of the or-
This leads to the nonlocal real space density discussed in
bital angular momentum so that a factor exp (ilzĆ) im-
the Introduction. Alternatively the inner-product can be
plies a z-component of orbital angular momentum equal
written as
to lz.


The spin and orbital AM of a photon are not separable

¨(1/2)|¨(-1/2) = ¨(1/2)" (k) ¨(-1/2) (k)
j j
beyond the paraxial approximation. For unit vectors of
k, j
the form (9) the z-component of total angular momentum
and photon position operators satisfy [13]

by defining ¨(-1/2) = ¨(1/2)/k, thus avoiding explicit

inclusion of the factor k-1 and the consequent nonlocal
Jz, rk = i %EÅ‚zklrl. (10)

real space density. The expectation value of the posi-


tion operator then satisfies ¨(1/2)| (-1/2)¨(-1/2) = This is just the usual commutation relation satisfied by a
r

vector operator and an angular momentum component.

(1/2)¨(1/2)|¨(-1/2) . If we apply this to the localized
r
Here it implies that photon position transforms as a vec-
tor under rotations about the axis of symmetry of the
state ¨(Ä…) = ¨(Ä…) = È(Ä…) this proves that the eigen-
r2 ,Ã
localized states. A photon on the z-axis satisfies the un-
value r2 is still real. However, the position operators
certainty relation "Jz"rk 0. Unit vectors of the form
(1/2) and (-1/2) = (1/2) are not self-adjoint. Opera-
r r r
(9) contribute a definite z-component of the total AM,
tors such are these are referred to a pseudo-Hermitian by
consistent with {sz, lz} equal to {-1, mà + 1}, {0, mÃ}
Mostafazadeh [25]. Use of pseudo-Hermitian operators
or {1, mà - 1} with jz = mÃ, that is total AM has a
and a biorthonormal basis is discussed in more detail in
definite value, but spin and orbital AM do not.
the next section.
In [12] the position operator was generalized to allow
for rotation about k through the Euler angle Ç (¸, Ć) to
III. POSITION EIGENVECTORS
give the most general transverse basis,
Here we will obtain the eigenvectors of the position
e(Ç) = e-iÃÇe(0) . (7)
k,Ã k,Ã
operators discussed in Section II. The LP form of the po-
sition operator, (0), is self adjoint, has real eigenvalues,
r
It was found that the position operator can be written as
and defines an orthonormal basis as is usual in quantum
mechanics. To obtain QED-like fields as eigenvectors,
(Ä…) = D kÄ…i"k-Ä… D-1 (8)
r
the choice Ä… = 1/2 is required. In this section we will use
the mathematical properties of pseudo-Hermitian opera-
where D = exp (-iSkÇ) exp (-iS3Ć) exp (-iS2¸). Start-
tors to obtained a completeness relation for the field-like
ing from a wave vector parallel to and transverse unit
z
photon wave function and investigate how it is related to

vectors x and y, D rotates k from the z-axis to an ori-
the LP wave function. The operators will be obtained
entation described by the angles ¸ and Ć, while at the
in the HP picture, so time dependence as determined by

same time rotating the transverse vectors first to ¸ and
the Hamiltonian must also be considered.

Ć and then about k through Ç. For example, when (1/2) We will start with an examination of the expectation
r

values to motivate the use of the biorthonormal formal-
acts on a transverse field parallel to Ć it rotates it to
"
ism. Any Hermitian operator o satisfies the eigenvector


y and divides it by Ék, then operates on it with the
equation o = on |fn and its eigenvalues, on, are real.
|fn
usual k-space position operator i". It then reverses the
"
To transform from LP position eigenvectors to fields,
process by multiplying it by Ék and rotates it back to
"
multiplication by Ék is required. Assume that · is an
its original transverse orientation. This allows (1/2) to
r
"
operator with positive square root Á = · which will
extract the position of the photon from the phase of the
"
equal Ék in the present application. We can write
coefficient of the transverse unit vector.

The quantum numbers {r, Ã} index the basis states


fn |o| fm = fn Á Á-1oÁ Á-1 fm = Ćn O Èm

for a given Ç (¸, Ć). The z-axis can be selected for con-
4

where O = Á-1oÁ is a similarity transformation, |Èm = and the completeness relation


Á-1 |fm and |Ćn = ( fn| Á) = Á Á |Èn . The eigenvec-


È(Ä…)

d3r È(-Ä…) = 1. (16)
tor equation becomes O |Èn = on |Èn and the eigenval-
r,Ã,j r,Ã,j
Ã,j
ues and inner-products are preserved. If Á is a unitary

operator, that is if Á-1 = Á , then O = O is Hermi-
Here ´3 is the 3-dimensional Dirac ´-function and we can
tian and |Ćn = |Èn . The |Èn and |Ćn eigenvectors
interchange Ä… and -Ä…. The field and the LP operators,
are the same, and the usual quantum mechanical for-
o, are related as

malism is obtained. On the other hand, if Á is a Her-
mitian operator satisfying Á = Á then |Ćn = |Èn and


O = É1/2oÉ-1/2. (17)

there are two distinct sets of eigenvectors. We can deal k k
with this is one of two ways: (1) The metric operator
consistent with (8). This transforms the LP position
· = Á2 can introduced to give the new inner-product
be
operator (0) to (1/2), introducing an addition term
r r
Ćn|·-1Ćm and work only with the |Ćn basis. (2) We

-iIk/2k. The momentum and angular momentum op-

can use the eigenvectors of O and the eigenvectors of
erators k and (-k × i" + S) are unaffected by the

O = ·O·-1 are |Èn and |Ćn respectively. Since

which similarity transformation (17). In the angular momen-

fn|fm = fn ÁÁ-1 fm transforms to Ćn|Èm , the

tum case this is because k × k = 0.
eigenvectors |Èm and |Ćn are biorthonormal [26]. If
Time dependence is determined by the Hamiltonian
there is degeneracy, a biorthonormal basis can be ob-

H + H0 with
tained by defining a complete set of commuting operators

(CSCO).

H = Éka ak,Ã (18)
k,Ã
The properties of pseudo-Hermitian operators and
k,Ã
biorthonormal bases have recently been investigated by

Mostafazadeh and can be summarized as [25]

where the zero point terms H0 = Ék/2 which are
k,Ã
unaffected by the photon state will be omitted here. The
"

O |Èn = On |Èn , O |Ćn = On |Ćn ,
operator ak,Ã annihilates a photon with wave vector k

O = ·O·-1, Ćn|Èm = ´n,m,
and helicity à and satisfies the commutation relations

2 2
ak,Ã, a = ´Ã,Ã ´k,k . The operators and their eigen-
k2 ,Ã2
with the completeness relation
kets are time dependent in the HP [27]. Using the unitary

time evolution operator
|Èn Ćn| = |Ćn Èn| = 1,
n n

U (t) = exp(-iHt), (19)

where · is a metric operator and 1 is the unit operator.
"
the HP position operator becomes
If Á = · exists,
(Ä…) = U (t) (Ä…)U (t) = (Ä…) + "Ékt (20)
rHP r r

o = ÁOÁ-1 = Á-1O Á (11)


È(Ä…) È(Ä…)
is self-adjoint and the eigenvectors On are real. Expec- with eigenvectors U (t) with given by Eq.

r,Ã r,Ã
tation values are preserved by the similarity transforma-
(1) in k-space. The coefficient of t in the last term of
tion, ·.
(20) is the photon group velocity.
To apply this formalism to the photon we take · = Ék We can define 1-photon HP annihilation and creation
and work in k-space. Then o = (0) is self-adjoint and
r
operators for a photon with helicity à at position r and

the operators O = (1/2) and O = (-1/2) have the
r r
time t as
(-1/2)
biorthonormal eigenvectors È(1/2) (k) and
r,Ã
" "Èr,Ã (k) k
eik·r-iÉ t
given by Eq. (1) that go as Ék and 1/ Ék respec- (Ä…)
Èr,Ã,j (t) a" (Ék)Ä… e(Ç) ak,Ã " , (21)
k,Ã,j
tively as required by QED for the electromagnetic fields V
k
and the vector potential. The position operators and
k
e-ik·r+iÉ t
(Ä…)
their eigenvectors satisfy
Èr,Ã,j (t) a" (Ék)Ä… e(Ç)" a " . (22)
k,Ã,j k,Ã
V
k
(-Ä…) = (Ä…), (12)
r r
For Ä… = 1/2, Eq. (21) implies that the biorthonormal
È(-1/2) (k) = É-1/2È(0) (k) , (13)
r,Ã r,Ã
k
pairs are related through
È(1/2) (k) = É1/2È(0) (k) , (14)
r,Ã k r,Ã
(-1/2)
(1/2) "Èr,Ã (t)

Èr,Ã (t) = i (23)
the biorthonormality condition
"t


analogous to the relationship between the vector poten-
2
È(-Ä…) |È(Ä…) = ´3 (r - r2 ) ´Ã,Ã , (15)
r2 ,Ã2 ,j r,Ã,j
tial and the electric field in the Coulomb gauge. The
j
5
1-photon position eigenkets normalized according to (15) IV. WAVE FUNCTION
are

(Ä…)
In this section we will obtain one and two photon wave
È(Ä…) (t) = Èr,Ã (t) |0 ,

(24)
r,Ã
functions and photon density by projection onto the basis
of position eigenvectors found in Section III. This density
where |0 is the electromagnetic vacuum state. The
is a 2-point correlation function that is based on the LP
projection of (24) onto the momentum-helicity basis,
or biorthonormal basis, rather than electric fields alone
{|k, Ã } , gives back Eq. (1) in the Schrödinger picture.
as in Glauber photodetection theory [28].
The free space operators for a photon with helicity Ã
The QED state vector describing a pure state in which
satisfy the r-space dynamical equation
the number of photons and their wave vectors are uncer-
tain can be expanded as
(Ä…)
"Èr,Ã (t) (Ä…)


i = Ãc" × Èr,Ã (t) . (25)
"t
|¨ = c0 |0 + ck,Ãa |0 (32)
k,Ã
k,Ã
The annihilation and creation operators satisfy the equal


time commutation relations 1
2 2
+ Nk,Ã;k ,Ã2 ck,Ã;k ,Ã2 a a |0 + ..

k,Ã k2 ,Ã2

2!
k,Ã;k2 ,Ã2
(-Ä…) (Ä…)
2
Èr,Ã,j (t) , Èr ,Ã2 ,j (t) = ´Ã,Ã ´3 (r - r2 ) . (26)
2
j
2
where c0 = 0|¨ , ck,Ã a" 0 |ak,Ã| ¨ , ck,Ã;k ,Ã2 a"
2 2 2
ck = 0 |ak,Ãak ,Ã2 | ¨ , and Nk,Ã;k ,Ã2 = 1 +
The Hermitian operator describing the density of photons ,Ã2 ;k,Ã
2 2
´k,k ´Ã,Ã . Division by 2! corrects for identical states ob-
with helicity Ã, obtained by averaging over the Ä… and -Ä…
tained when the {k, Ã} subscripts are permuted while
forms, is
"
N /2 normalizes doubly occupied states. A more gen-
(Ä…)
1
(-Ä…)
eral state requires a formulation in terms of density ma-
n(Ä…) (r, t) = Èr,Ã (t) · Èr,Ã (t) + H.c.. (27)

Ã
2
trices that will not be attempted here.
The total number operator is The 1-photon real space wave function in the helicity

basis, equal to the projection of state vector onto

this

N = d3rn(Ä…) (r, t) = a ak,Ã. (28)

eigenvectors of (Ä…) as È(Ä…) |¨ , is
rHP r,Ã,j
k,Ã
k,Ã
k
eik·r-iÉ t
An alternative linear polarization basis can be ob-
¨(Ä…) (r, t) = ck,Ãe(Ç) (Ék)Ä… " (33)
Ã
k,Ã
tained if we define operators that annihilate a photon
V
k

state with polarization in the ¸ and Ć directions as
where we have used Eqs. (22), (24) and (32). The ex-
"
(Ä…) (Ä…)
(Ä…) pansion coefficients depend on the choice of basis, for
Èr (t) = Èr,1 (t) + Èr,-1 (t) / 2, (29)
example when Ç Ç + "Ç the coefficients ck,Ã

"
(Ä…) ck,Ã exp(-iÃ"Ç) analogous to gauge changes of the vec-
(Ä…) (Ä…)
Ćr (t) = -i Èr,1 (t) - Èr,-1 (t) / 2,
tor potential describing a magnetic monopole in real
space [12]. In any basis the inner-product ¨|¨ =

respectively. While the direction of these eigenvectors
|ck,Ã|2 a" |c1|2 where |c1|2 is the net probability for
k,Ã
depends on k, they do not rotate in space and time, and
1-photon in state |¨ . The free space 1-photon dynam-
in that sense they are linearly polarized. The inverse
ical equations mirror the operator Eqs. (23) and (25).
transformation is
They are

"
(Ä…) (Ä…)
(Ä…)
Èr,Ã (t) = Èr (t) + iÃĆr (t) / 2. (30)
"¨(-1/2) (r, t)
Ã
i = ¨(1/2) (r,t) , (34)
Ã
"t
In free space
"¨(Ä…) (r, t)
Ã
i = Ãc" × ¨(Ä…) (r, t) .
Ã
(Ä…)
"t
"Èr (t)
(Ä…)
= c" × Ä†r (t) , (31)
"t
To obtain the 2-photon wave function we can project
(Ä…) |¨ onto the 2-photon real space basis
(Ä…)
"Ćr (t)

= -c" × Èr (t) ,
"t
(Ä…) (Ä…)
Èr,Ã,j (t) , Èr ,Ã2 ,j2 (t2 ) = Èr,Ã,j (t) Èr ,Ã2 ,j2 (t2 ) |0
2 2
If Ä… = 0 these are the operators introduced by Cook [24],
while if Ä… = 1/2 their dynamics is ME-like.
giving
The localized definite helicity basis states are eigen-


vectors of a CSCO, so it is the helicity basis that will

È(Ä…) (Ä…)
¨(Ä…) (r, r2 ; t, t2 ) = 0 (t) Èr ,Ã2 ,j2 (t2 ) ¨ .
2
Ã,Ã2 ;j,j2 r,Ã,j

be used here. Linearly polarized fields can be found by
taking the sum and difference as in (29). (35)
6

The bases obtained here provide a real space descrip-
Use of Eq. (22) and ak,Ã, a = ´k,k ´Ã,Ã to evaluate
k2 ,Ã2 2 2
tion of the multiphoton state that  encodes the maximum
(Ä…) (Ä…)
0| Èr,Ã,j (t) Èr ,Ã2 ,j2 (t2 ) a a |0 then gives
2
total knowledge describing the system as discussed in
k,Ã2 2 k2 ,Ã2 2 2
Ref. [3]. The electric field wave function used in [2, 5]


1
2
¨(Ä…) (r, r2 ; t, t2 ) = Nk,Ã;k ,Ã2 (36)or RS vectors in [1, 18] by themselves do not provide a
Ã,Ã2 ;j,j2
2!V
basis, and this is the root of the criticism of [2] made
k,Ã;k2 ,Ã2
in [3]. The 2-photon wave function (36) is symmetric in
2 2
×ck,Ã;k ,Ã2 (ÉkÉk )Ä…
agreement with [1, 2].

2
2
k
× e(Ç) e(Ç) eik·r-iÉ teik ·r2 -iÉk t2
k,Ã,j k2 ,Ã2 ,j2

2 2
2
+e(Ç) e(Ç) eik·r -iÉkt2 eik ·r-iÉk t
k2 ,Ã2 ,j k,Ã,j2
V. MAXWELL S EQUATIONS
which becomes a 2-photon wave function if we set t2 = t.
In this section we will show that MEs can be obtained
A separate symmetrization step is not required since its
from QED in two distinct ways. The first is the conven-
symmetric form is a direct consequence of the commu-
tional approach of calculating the expectation value of
tation relations satisfied by the photon annihilation and
operators with all modes populated as coherent states.
creation operators.
The fields obtained in this way are real and they cannot
To obtain an n-photon basis the creation operator can
be interpreted as wave functions. The second approach is
be applied to the vacuum n times with each occurrence
to project the state vector onto the position eigenvectors
having different parameters r, Ã, and j. The state vector
obtained when a field operator acts on the vacuum state
can then be projected onto this basis to give the n-photon
to give fields proportional to the 1-photon wave function
term. The result is the symmetric n-photon real space
components in Section IV.
function
If the multipolar Hamiltonian is used, the displace-
n-1


ment is purely photonic, while the vector potential will
¨(Ä…) r, .., r[n-1]; t, .., t[n-1] = È(Ä…) |¨ (37)

m
{m}
include photon and matter contributions [31]. The vec-
m=0
tor potential operator is a sum over positive and negative

frequencies, photon and matter parts, and both helicities.
È is a short hand for È(Ä…)
(Ä…)
where t[m]

m
r[m],Ã[m],j[m]
We can define
and m represents the mth set of variables, quantum num-

bers and components r[m], t[m], Ã[m], j[m] . Generally

A (r, t) = A(+) (r, t) + A(-) (r, t) , (40)
the n-photon states provide more information than can

A(+) (r, t) = A(+) (r, t) + A(+) (r, t)
p m
be measured. Instead the real space helicity à photon
density, equal to the expectation value of the number

A(+) (r, t) = A(+) (r, t) + A(+) (r, t) ,
p 1 -1
density operator, (27), can be defined as

where A(-) = A(+) and the subscripts m and p denote
n(Ä…) (r, t) = ¨ | (r, t)| ¨ (38)
nÃ
Ã

matter and photon parts respectively. The electric field


1

È(Ä…) (-Ä…)
= ¨ (t) Èr,Ã,j (t) ¨ + c.c. and magnetic induction are then given by
r,Ã,j

2
j

E = -"A/"t - "Ć, (41)
The 0-photon contribution to n is 0, while the 1-photon

contribution is B = " × A.
1

n(Ä…) (r, t) = ¨(Ä…)" (r, t) · ¨(-Ä…) (r, t) + c.c.. (39)
In the presence matter of with polarization operator P
à à Ã
2

and magnetization M the displacement and magnetic
For the 2-photon state (35), substitution of (26) gives
field operators are


1

n(Ä…) (r, t) = d3r2 ¨(Ä…)" (r, r2 ; t, t) D = %EÅ‚0E + P, (42)
à Ã,Ã2 ;j,j2
2
Ã2 ;j,j2

H = B/µ0 - M,
ר(-ą) (r, r2 ; t, t) + c.c.,
Ã,Ã2 ;j,j2
where SI units are used, %EÅ‚0 is the permittivity and µ0 the
"
implying that unobserved photons are summed over. A
permeability of vacuum, and c = 1/ %EÅ‚0µ0.
similar argument can be applied to each n-photon term.
The momentum conjugate to the vector potential is
Photons are noninteracting particles and the existence of

-DÄ„" where DÄ„" is the transverse part of the displace-
a photon density is consistent with Feynman s conclu-
ment operator [20, 31]. These operators satisfy canoni-
sion the photon probability density can be interpreted as
(-Ä…) (Ä…)

particle density [29, 30]. cal commutation relations. Since Èr,Ã and Èr,Ã satisfy
7
(26) we can choose that there is no position basis for the photon, this re-
sult is new. We can define the 1-particle states |Vr,Ã =


(-) (+)

Vr,Ã |g, 0 with V(-) = V(+) and V(+) = VÃ for
(-1/2)
Ã
A(+) (r, t) = Èr,Ã (t) , (43)
Ã
2%EÅ‚0
any field operator V such that

(1/2)
%EÅ‚0


D(+) (r, t) = i Èr,Ã (t) .
(+) (+)
Ä„",Ã
VÃ (r, t) = g, 0|Vr,Ã |¨ . (47)
2

This is equivalent to the usual QED expansion of A and
This can be viewed as the projection of the photon-

D and thus is consistent with the operator MEs
matter state vector state onto the n = 1 term of number-
position-helicity basis. In the ground state |0 both the

"B EM field and any matter present are in their lowest en-

" · B = 0, " × E = - , (44)

"t ergy configurations. The operator V(-) creates 1-particle
that can be a photon or a material excitation. Since the
"D

" · D = Á, " × H = j+ ,
space and time dependence originates entirely in the field
"t
operators, these functions satisfy ME dynamics. The 1-
where Á and j are the free charge and current den- photon MEs are, using (44),
" "

sities. In free space D(+) / %EÅ‚0 = iÃB(+)/ µ0 =
Ã
Ä„",Ã
" "B(+)
(1/2)

" · B(+) = 0, " × E(+) = - , (48)
F(+)/ 2 = i /2Èr,Ã where the RS operator is
Ã
"t
" "

F(+) = D(+)/ 2%EÅ‚0 + B(+)/ 2µ0 as defined in [18].
à à Ã
"D(+)
" · D(+) = Á(+), " × H(+) = j(+)+ .
Coherent states are the most classical, and they can
"t
be used to establish a connection between QED and the
real Maxwell fields. Following Cohen-Tannoudji et. al. Projection of the state vector onto a basis of 1-photon
[31] the complex Fourier transforms of the classical field position eigenvectors results in intrinsically positive fre-
vectors, quency electric and magnetic fields defined by (47) that
satisfy MEs. They can be manipulated to give any of the

exp (-ik · r)
commonly used forms of MEs.
Vk (t) = d3rV (r, t) " ,
A wave equation can be obtained from (48) in the usual
V
way to give
and the normal variables,

1 "2E(+) " "P(+)

+ " × " × E(+) = -µ0 (49)
%EÅ‚0 Ä„"
c2 "t2 "t "t
Å‚k (t) = -i Ek (t) - ck × Bk (t) ,

2 Ék
+"×M(+) + j(+) .
can be defined. For a coherent state with the pho-
ton occupancy of mode {k, Ã} described by the complex
The terms on the right hand side are the polarization,
parameter Å‚k,Ã, the average photon number is nk,Ã =
magnetic and external contributions to the time deriva-

Å‚k,Ã 2 the probability amplitude for n-photons is tive of the current density [31]. If there is no magne-
and

"
tization or external current and the polarization is lin-
Å‚k,Ã 2
exp - /2 Å‚n / n!. The quasi-classical coher-
k,Ã
ear and isotropic we can write P = %EÅ‚0Ç (k) E which
ent state is a Gaussian wave packet that oscillates with-
can be combined with the "2E(+)/"t2 term. Writing
out deformation and with relative number uncertainty
%EÅ‚ (k) = %EÅ‚0 [1 + Ç (k)] the angular frequency in (18) is

Å‚k,Ã
"nk,Ã/nk,Ã = 1/ . In the limit of infinite pho-
Ék = kc 1 + Ç (k). Analogous to the creation of an ex-
ton number the electric and magnetic fields oscillate in a

citation of the electromagnetic field (a photon) by D(-),
well defined way as do the solutions to the classical MEs.

the polarization operator P(-) creates a matter excita-
Thus
tion. Energy can be transferred between matter and the


electromagnetic fields, so the matter and EM modes are

AÄ„"(+) (r, t) = Å‚k,Ã AÄ„"(+) (r, t) Å‚k,Ã (45)

p
coh
coupled. Self-consistent solution of the matter-photon

k
eik·r-iÉ t dynamical equations gives the polariton frequencies Ék
= Å‚k,Ãe(Ç) " ,
that determine time dependence.
2%EÅ‚0Ék k,Ã V
k,Ã
The energy, linear momentum and angular momentum
of the free electromagnetic field are conserved. Their den-
AÄ„" (r, t) = AÄ„"(+) (r, t) + c.c. (46)
coh coh
sities and associated currents satisfy continuity equations
and the fields derived from it behave classically in the of the form "Á/"t + " · j = 0. This can be verified using
large photon number limit. MEs, and the steps in this derivation are still valid if we
It is also possible to derive one photon positive fre- replace the products of classical real fields with Hermi-
quency MEs from QED. Since it is still widely believed tian linear combinations of products of operators. For
8
example, the current describing the flow energy den- density and has important consequences. It is responsi-
of
ble for the spin term in the AM (53). This can be seen
sity D(-) · D(+)/2%EÅ‚0 + B(-) · B(+)/2µ0 is c2 times the
by writing
linear momentum density


-r× D(-) · " A(+) = D(-) × A(+)
1
D(-)

P (r, t) = ¨ × B(+) - B(-) × D(+) ¨ . (50)


2
- D(-) · " r × A(+)
Together with their associated current densities the com-
ponents of also satisfy continuity equations which im- where a × b = -i (a · S) b gives
P
plies that d3rP (r, t) is a constant of the motion. If |¨
3


is a 1-photon state ¨| D(-) × B(+) |¨ = ¨| D(-) |0 × 1
D(-) × A(+) = ¨(1/2)" S¨(-1/2).
j j

0| B(+) |¨ so that 2
j=1

1
Since ".D(-) = Á(-), the last term contributes
P (r, t) = D(-) (r, t) × B(+) (r, t) + c.c. (51)

2
d3rr × ÁA(+) to d3rJ (r, t) after integration by parts
which is zero in the absence of free charge.
with the fields derived using (41), (42), and (47). For
a

coherent state, the quasi-classical expectation value

Å‚k,Ã = Dcoh × Bcoh for small Å‚k,Ã
Å‚k,Ã D × B .

VI. PHOTON WAVE MECHANICS
However (50) can be evaluated exactly using ak,Ã Å‚k,Ã
to give
In this section we will discuss first quantized photon

quantum mechanics. For definiteness we will refer to the
1
P (r, t) = D(-) (r, t) × B(+) (r, t) + c.c. (52)
coh coh Barut-Marlin rules for Schrödinger and Dirac particles
2
stated in [32] as: (a) A basis for the space of wave func-
tions, which describe all the possible states of a particle,
with AÄ„"(+) given by (45). In either case the angular
coh
is defined by a wave equation. (b) A inner-product is
momentum density is
defined in the space of the wave functions. (c) Expres-
sions for the probability density and probability current
J (r, t) = r × P (r, t) . (53)
are found. They should form a 4-vector whose diver-
gence vanishes. The expression for the probability den-
We are now in a position to compare the classical and
sity should be positive definite. (d) Operators which cor-
quantum fields and densities. Eq. (46) describes real
respond to measurements are defined, in particular, mo-
fields that are the expectation values for coherent quan-
mentum and position operators. (e) The eigenfunction
tum states. Expectation values do not describe 1-photon
of the operators, normalized to 1 (in the case of discrete
states since in this case the expectation values of the field
spectrum) or a ´-function (in the case of a continuous
operators are zero. Instead, it is projection onto a basis
function), are found. (f) The position operator, defined
of position eigenvectors that gives 1-photon positive fre-
in (d), and the inner-product, defined in (b), uniquely
quency fields, proportional to components of the wave
determine an expression for the probability density. The
function. For 1-photon and coherent states momentum
theory is consistent only if this uniquely determined ex-
density can be written as a cross product of fields as in
pression is identical with the one defined in (c) to satisfy
(51) and (52). Eq. (50) can be used to interpolate be-
a continuity equation. This is a consistency test.
tween these two extreme cases.
In brief, these rules apply to the r-space wave mechan-
The density D(-) × B(+) can be rewritten as [31]
ics of a single free photon in free space in the following

sense: (a) Solutions to (34),
D(-) × B(+) = D(-) × " × A(+)
3 i"¨(Ä…) (r, t) /"t = Ãc" × ¨(Ä…) (r, t) , (54)

à Ã
(-)
= Dj "A(+) - D(-) · " A(+).
j
include positive and negative frequencies. The negative
j=1
frequency solution can be eliminated on physical grounds
[19, 29], thus cutting the Hilbert space in half as is done
Its first term, equal to
for solutions to the Dirac equation [32]. (b) The inner-

3 3 ¨

1 product of the wave functions describing states and

(-)
Dj "A(+) = ¨(1/2)" (i ") ¨(-1/2),
j j j
2
|¨ ,
j=1 j=1



is the integrand in the expectation value of the real
¨(Ä…)|¨(-Ä…) = d3r¨(Ä…) (r, t) · ¨(-Ä…) (r, t) ,
à Ã
space momentum operator -i ". The last term,
Ã

D(-) · " A(+), also contributes to the flow of energy (55)
9
exists and is invariant under similarity transformations in the linear polarization basis. If Ä… = 1/2, (60) is of the
between Ä… = 1/2 and Ä… = 0. (c) The real number and form considered by Bialynicki-Birula and Sipe [18, 20],
current densities obtained by averaging the Ä… and -Ä… while if Ä… = 0 (61) is the form used by Inagaki [29].
densities However, (60) and (61) themselves imply that either the
helicity or the linear polarization basis can be used in
1
n(Ä…) (r, t) = ¨(Ä…)" · ¨(-Ä…) + c.c., (56)
à à combination with field-like ą = 1/2 wave functions or
2
Ã
LP Ä… = 0 wave functions. The operator on the right

iÃc
hand sides of (60) and (61) is the real space 1-photon
j(Ä…) (r, t) = - ¨(Ä…)" × ¨(-Ä…) + c.c.,
à Ã
2 Hamiltonian.
Ã
The density i%EÅ‚0E · A/ has appeared before in the
satisfy the continuity equation
classical context and in applications to beams. Cohen-
Tannoudji et. al. [31] transform the classical electromag-
"n(Ä…) (r, t)
+ " · j(Ä…) (r, t) = 0. (57)
netic angular momentum as
"t
This can be verified using the wave equation. The



¨(0) 2
J = %EÅ‚0 d3rr× (E × B) (62)
density n(0) = is positive definite, while

Ã
Ã



n(1/2), j(1/2) is a 4-vector that can be written as the 3

contraction of second rank EM field tensors with 4-
= %EÅ‚0 d3r Ei (r × ") Ai + E × A
potentials. (d) The momentum operator is k and the
i=1
position operator is given by Eq. (4). (e) The eigenvec-
tors of these operators are ´-function normalized accord-
by requiring that the fields go to zero sufficiently quickly
ing to (15). (f) The position
"operator and inner-product at infinity. Although this looks like an expectation value,

1
the fields are classical. In a discussion of optical beams,
give the density È(Ä…)|¨ È(-Ä…)|¨ + c.c.. Some of
r,Ã r,Ã
2
van Enk and Nienhuis [34] separate monochromatic fields
these points will now be discussed in more detail.
into their positive and negative frequency parts using
Both positive and negative frequency solutions of the
wave equation are mathematically allowed. The classi-

"
cal solutions are real, and real waves do not satisfy a
V = V(+) exp (-iÉt) + V(-) exp (iÉt) / 2
continuity equation or allow a probability interpretation
[33]. It has been argued by Inagaki for LP wave functions
that the negative frequency solutions with momentum in and obtain for total field linear momentum and AM
opposite direction to the wave propagation should be dis-


carded from the physical photon state [29]. A similar case 3

(+)"
is made by Bialynicki-Birula for elimination of the nega-
P = -i d3r Di (i") A(+) , (63)
i
tive frequency fields in field-like wave functions [18, 19].
i=1

As with MEs the photon wave equations can be written
3

(+)"
in a number of equivalent ways, and this will be consid-
J = -i d3r Di (-r×i" + S) A(+) .(64)
i
ered next to allow comparison with the existing photon
i=1
wave function literature. The six component wave func-
tion
Here we have assumed the absence of matter in writ-

ing D = %EÅ‚0E, substituted A(+) = iÉD(+), and changed
¨(Ä…)
1
¨(Ä…) = (58)
the notation a bit for consistency with the present work.
hel
¨(Ä…)
-1
These are classical expressions, but terms at frequency
2É do not contribute to the total momentum and an-
in the helicity basis and
gular momentum, P and J [35]. They look like the ex-

¨(Ä…)
pectations values of the linear and angular momentum
¨(Ä…) = (59)
lin
Åš(Ä…)
operators that would be
obtained using the biorthonor-
"
mal wave function pair %EÅ‚0/ A(+)Ä„" and -iD(+)/ %EÅ‚0 .
in the linear polarization basis can be defined. The photon

Schrödinger equation is then, using (54) and (31) with The number operator iD(-) · A(+)/2 + H.c. was shown
" × a = -i (S · ") a, previously to be the zeroth component of a four-vector

obtained by contraction of the second rank EM field ten-

"
¨(Ä…) -iS·" 0 ¨(Ä…)
sor with the four-potential (Ć, A) [36]. This demonstates
1 1
i = c (60)
0 iS·"
"t that the biorthonormal basis is of value for comparison
¨(Ä…) ¨(Ä…)
-1 -1
with the existing literature.
in the helicity basis and
It was noted in Section III that the biorthonormal

" inner-product is equivalent to the use of a metric opera-
¨(Ä…) 0 S·" ¨(Ä…)
i = c (61)

-S·" 0 tor. Using (23) and H = kc in k-space and substituting
"t Åš(Ä…) Åš(Ä…)
10

H for i"/"t the inner-product (55) can be written as are

(Ä…) (-Ä…)
1

(Ä…) (-Ä…)
n(Ä…) (r, t) = Èr · Èr + Ćr · Ćr + H.c. (65)
,
d3k
2
¨|¨ = ¨(1/2)" (k, t) ¨(1/2) (k, t)
Ã,j Ã,j

kc
Ã,j (Ä…) (Ä…)
(-Ä…) (-Ä…)
(Ä…) (r, t) = 1 Èr × Ä†r - Ćr × Èr + H.c. ,

j

2

= d3r¨(1/2)" (r, t) H-1¨(1/2) (r, t)
Ã,j Ã,j
Ã,j
(Ä…)
(Ä…)
with Èr and Ćr given by (29). Mandel and Wolf noted
the convenience of a photon number operator, equal to
as in [21, 38].
(0) (0)

Èr · Èr in the present notation, to the theory of pho-
The number density is the expectation value of the
ton counting for an arbitrary quantum state [39]. Cook
number density operator (27) as discussed in Section IV.
sought detector independent photon density and current
The Ä… = Ä…1/2 wave function pair gives a real local 1-
operators that satisfy a continuity equation. His opera-
photon density n(1/2), but this density is not positive
tors are just (65) if we take Ä… = 0. Inagaki reformulated
definite. This can be seen from the following example: If
Cook s theory in terms of conventional quantum mechan-
|¨ is a 1-photon state that includes only wave vectors
ics [29]. These authors discuss the restrictions imposed
k1 and k2 = k1 + "k both with helicity à where ck ,à =
1
"
by photon nonlocalizability, but the existence of a ba-
ck ,Ã = 1/ 2 then
2
sis of position eigenvectors makes this unnecessary here.
Our operators describe microscopic densities, and there


1
is no restriction based on wave length.
n(1/2) = {1 + k1/k2 + k2/k1
2 The Lorentz transformation properties of the Ä… = 0
× cos ["k · r - (k1 - k2)ct)]}/V. photon annihilation operators in the linear polarization
basis were also considered by Cook [41]. He concluded
that their continuity equation is covariant in the sense
The cosine term can exceed the spatially uniform time
"
that is is related to the field vectors in the same way in
independent term due to the k factors, leading to neg-
all reference frames. The Hamiltonian, momentum, an-
ative values. If k2 H" k1, n(1/2) is approximately equal to
gular momentum, and Lorentz transformation operators
the positive definite density, n(0), however only the LP
must conform to the Poincaré algebra. Since the position
wave function satisfies the positive definite requirement
operator generates a change in particle momentum, the
exactly.
boost operator is closely related to the position opera-
It thus appears that LP wave functions are essential
tor. For a free photon in k-space the Lorentz operator
to a probability interpretation. Field-like wave functions
corresponding to the Ä… = 1/2 case is [38]
can be obtained from the LP wave function by a similar-
ity transformation, and thus are equivalent to it for the

K(1/2) = k (i") + k × S.
calculation of expectation values. The operators given
by Eq. (29) in the Ä… = 0 case are identical to the opera-

where K(-1/2) = K(1/2) . Using (17) this gives
tors examined by Cook. The equations that they satisfy
differ from those for D and B only in that their relation-

K(0) = k-1/2K(1/2)k1/2
ship to charge and current sources is nonlocal. The LP
number density has been criticized [18, 20, 23], but its
for the LP boost operator which incorporates the sim-
scalar analog, obtained by taking Fourier transforms of
ilarity transformation. In k-space this is simple, but in
the Schmidt modes, has recently been applied to sponta-
r-space it is non-local as discussed by Cook [41].
neous emission of a photon by an atom and spontaneous
It is stated, in [42] for example, that  the non-
parametric down-conversion [6, 37]. For narrowband su-
Hermitian formulation is in most cases a mere change
perpositions of plane wave states the distinction between
of metric of a well posed Hermitian problem. Nonethe-
the LP and field-like form of the wave function has no
less, .. , it has been successfully argued that the non-
observable consequences [6].
Hermitian formalism is often more natural and simplifies
calculations. These comments apply here. The choice of
The operator (21) creates basis states that lead
to the orthogonal transverse 1-photon wave function Ä… does not affect expectation values, the inner-product,

and the existence of a wave equation and a continuity
¨(Ä…) (r, t) = ¨(Ä…), ¨(Ä…) in the helicity basis. The
hel 1 -1
equation. Only the number and current densities them-
wave function components ¨(-1/2) are proportional to
selves are affected. The field and LP bases can be viewed
the vector potential, while ¨(1/2) is related to EM fields.
as alternative descriptions of the photon state. For most
µ½
Contraction of the second rank field tensor F =
purposes fields are more closely related to the physics,
µ½"
"½Aµ - "µA½ with the 4-potential as F A½ gives a 4-
but the LP basis is needed if the band width is large and

vector [36]. Thus n(1/2), j(1/2) is a 4-vector and photon
photon number density is required.
density is its zeroth component.
According to the general rules of quantum mechan-
In the linear polarization basis the density operators ics, for a 1 photon state the probability that a photon
11
with helicity à will be found at position r at time t is of Ç. Taking helicity à = 1 to give a concrete example,
2
¨ (r, t) . More generally the photon number density we can first take m = 0 in (9) to give the spherical polar
(0)


à "

vectors ¸ + iÃĆ / 2 with total AM 0. At ¸ = 0 there
is the expectation value of the number density operator,
is spin AM and the orbital AM is - , while at ¸ = Ä„
n(0) (r, t) H" n(1/2) (r, t) given by (38). Glauber [28] de-
à Ã
the spin and orbital AM are - and respectively. If
fined an ideal photodector to be of negligible size with a
instead we choose m = 1, the ¸ = 0 orbital AM is 0, but
frequency-independent photoabsorption probability. An
at ¸ = Ä„ it is 2 . For a localized state the vortex has
ideal photon counting detector also has a quantum effi-
not been eliminated, it has just been moved. Thus an
ciency of · = 1, that is any photon reaching the detector
understanding of optical AM is essential to the physical
is counted. A detector with all of these characteristics
picture of the localized basis states that are used here to
measures photon position. Consider a 1-photon pulse
obtain the photon wave function.
travelling in the z-direction that is normally incident on
Theoretically, the simplest beams with orbital AM are
a detector of thickness "z and area "A. The probability
the nondiffracting Bessel beams (BBs), and these beams
that a photon is present in this detector, and hence that
are closely related to our localized states. They satisfy
it is counted, is n(Ä…) (r, t) "A"z. In Glauber theory the
Ã

MEs and have definite frequency, ck0, and a definite wave
E(-)
count rate is dnG/dt " ¨ (r, t) · E(+) (r, t) ¨

vector, kz, along the propagation direction. It then fol-
where (dnG/dt) "z/c is the probability the photon is
lows that the k-space transverse wave vector magnitude

2 2
counted during the time that it takes to traverse the
kÄ„" = k0 - kz, and the angle ¸ = tan-1 (kÄ„"/kz) also
detector. Since n(1/2) = i%EÅ‚0E(-) · A(+)/ + c.c where have definite values for BBs. Cylindrical symmetry is
à à Ã
(+)
achieved by weighting all Ć equally with a phase fac-
A(+) H" -iEÃ /É for most beams available in the labora-
Ã
tor exp (imĆ) . When Fourier transformed to r-space the
tory, the predictions of the present photon number based
modes go as exp (-ik0ct + ilzÕ + ikzz) Jl (kÄ„"r) where
theory and Glauber photodetection theory are usually z
lz = m and m Ä… 1 in (9), Jl are Bessel functions, Õ the
indistinguishable. z
real space azimuthal angle, and r is the perpendicular
The number based theory has the advantage that the
distance from the beam axis [43]. If we select Ç = 0 so
probability is normalizable, for example the probability

that the k-space unit vectors are Ć and ¸ in the linear
to count one photon in a 1-photon state in the whole

of space using an array of detectors with · = 1. The polarization basis, B is transverse to for the ¸ mode
z

Glauber form of the count rate is based on the transi-
and E is transverse for the Ć mode and the linearly po-
tion probability, however there are advocates for a photon
larized modes can be called transverse magnetic (TM)
number density approach, even within conventional pho-
and transverse electric (TE) respectively.
ton counting theory. Mandel noted that  there are many
The Bessel functions have a sinusoidal dependence on
problems in quantum optics, particularly those concerned
kĄ"r, and this implies that the BBs are standing waves
with photoelectric measurements of the field, which are
that are a sum of incoming and outgoing waves. If inte-
most conveniently treated with the help of an operator
grated over kĄ" the resulting wave is localized on the z-
representing the number of photons [39]. Mandel and
axis at some instant in time that can be defined as t = 0.
Wolf based their general photon counting theory on a
Localization of beams in this way is discussed in [44, 45].
photon number operator [40]. Cook observed that there
If the BBs are then integrated over kz, the result is equiv-
is no universal proportionality constant that relates pho-
alent to a sum over all wave vectors, and states localized
ton flux to pG, and thus the prevailing theory of pho-
in three dimensions are obtained. But note that this kz
toelectron counting fails to provide a complete descrip-
sum includes waves travelling in the positive and nega-
tion of photon transport [24] . He proposed a modified
tive
z-directions. According to the Paley-Weiner theo-

"
photodetection theory based on photon number. A pho-
rem, dkz does not allow exact localization, but this
0
ton density n(0) (r, t) , equal to the probability density to
Ã
restriction does not apply to an integral over all positive
count a photon at r at time t, is consistent with Cook s
and negative values. Position is not a constant of the
arguments and with the rules of quantum mechanics.
motion, and localized states can exist only for an instant
in time. Exactly localized states in free space are not
physically possible because they require infinite energy.
VII. ANGULAR MOMENTUM AND BEAMS
However, our primary concern here is with the use of
localized basis states for calculation of the photon wave
function, and we do not require that these basis states
The physical interpretation of the position eigenvectors
have a physical realization.
in [13] involving AM was motivated by the recent exper-
imental and theoretical work on optical vortices. These The real space mathematical description of beams used
vortices are spiral phase ramps described by fields that go to interpret the AM experiments is usually based on the
as exp (ilzÕ) and in experiments appear as annular rings classical energy, linear momentum, and angular momen-
around a dark center. It can be seen by inspection of (9) tum densities. Here, with a basis of position eigenvectors
that the localized states must have orbital AM, and this in hand that leads to a wave function for a photon in an
implies a vortex structure that is affected by the choice arbitrary state, we are in a position to consider the real
12
space description of the AM of beams from a quantum separable, the choice Ç = -Ć in Eq. (7) gives unit
"
mechanical perspective. The Ä… = 1/2 wave function is a

vectors (x + iÃy) / 2 in the paraxial limit which im-
solution to MEs, and any derivation based on MEs can
plies spin quantum number sz = Ä…1. The wave func-
be adapted to the 1-photon case. The expansion of vec-
tion (66) is an eigenvector of Sz with eigenvalue sz and
tor potential in [44] that leads to paraxial fields to a first

of Lz = -i "/"Õ with eigenvalue lz where Õ the real
approximation can be applied to allow application of our
space azimuthal angle. The latter orbital AM is equiv-
formalism to the paraxial beams that are used in most
alent to linear momentum lz/r. For this definite he-
optical experiments. Localized states do not exist within
licity state only one term in the photon density (56)
the paraxial approximation, and the paraxial approxima-
contributes. The probability to detect this photon is
tion cannot be applied to the position eigenvectors.
<"
n(0) (r, t) n(1/2) (r, t) , where these field-potential and
=
A paraxial beam propagating in the
z-direction with
LP densities are essentially equal for a paraxial beam.
frequency É, helicity Ã, and z-component of orbital AM
For a coherent state the expansion coefficients ck,Ã in the
lz can be described in cylindrical polar coordinates by
1-photon wave function (33) are replaced with the am-
the vector potential [46]
plitudes Å‚k,Ã. Small absorbing particles placed in these
beams are essentially photodetectors that conserve AM
1
by spinning about their centers of mass and rotating

A(+) (r, t) = (x + iÃy) u (r) exp [ilzÕ + ikz (z - ct)] .
2
around the beam axis while they absorb photons [49].
(66)
The photon number density gives the probability to ab-
This vector potential is equivalent to the wave function

sorb a photon which carries with it spin AM sz and
2 2
¨Ã (r, t) = ´Ã,à 2%EÅ‚0/ A(+) (r, t). The z-component of
orbital AM lz. For transparent particles the situation is
the time average of the classical AM density, equal to
more complicated, since re-emission should also be con-
1
r× (D"×B + D × B") , is then found to be
2
sidered.


u2
" (r)
1
Jz (r) = %EÅ‚0 Élz |u (r)|2 - ÉÃr . (67)
2 "r VIII. CONCLUSION
It equals the z-component of the AM density (53) with
We have derived one and two photon wave functions
momentum density given (51) or (52) without the need
from QED by projecting the state vector onto the eigen-
for time averaging. Thus (67) can be interpreted as a
vectors of a photon position operator. Largely because
quantum mechanical AM density that is valid for coher- it is still widely believed that there is no position opera-
ent and 1-photon states, while (50) interpolates between
tor, this is the first time that a photon wave function has
these two cases.
been obtained in this way. The two photon wave func-
The first term of (67) is consistent with orbital AM lz tion is symmetric, in agreement with [1] and [2]. While
per photon since the photon density given by (39) reduces only the LP wave function gives a positive definite photon
density, field-like wave functions are widely used and are
to n(1/2) (r, t) = %EÅ‚0É |u (r)|2 / . The last term of (67) does
more convenient in many applications. Also, they given
not look like photon spin density. The most paradoxical
energy momentum and angular momentum density as in
case is a plane wave, as discussed in [47]. For example a
(51) for example. In the field-like helicity basis the wave

wave function proportional to (x + iÃy) exp (ikz - iÉt)
function pair is
implies linear momentum k per photon and hence no
z

z-component of AM. But we know that such a beam de-
2%EÅ‚0
scribes a stream of photons each with spin AM Ã. It was
¨(-1/2) (r, t) = A(+) (r, t) , (68)
à Ã

observed in 1936 by Beth [48] that a circularly polarized

beam can cause a disk to rotate, so the beam really does 2
¨(1/2) (r, t) = -i D(+) (r, t) .
Ã
carry AM that it can transfer to the disk. The AM of
%EÅ‚0 Ã
this beam resides in its edges, as can be seen from Eq.
The wave function components ¨(Ä…) are given by Eq.
(67). A new edge is created if the disk intercepts part Ã
(33). For definite helicity fields in free space, B(+) and
of the beam and this reduces the AM of the beam, al-
the Reimann-Silberstein field vector are just proportional
lowing the conservation of total AM [35]. This is analo-
to D(+), and thus are equivalent to it. The linear polar-
gous to the continuum description of a dielectric where
ization basis of TM and TE fields can be obtained by tak-
it is know that the medium is composed of atoms, but a
ing the sum and difference of the definite helicity modes
continuum description of a uniformly polarized dielectric
as in (29). The photon density is (56)
results only is a surface charge. An even closer anal-
ogy exists between spin AM and a continuous magnetic
1
n(Ä…) (r, t) = ¨(Ä…)" · ¨(-Ä…) + c.c., (69)
medium where a current in individual molecules reduces
à à Ã
2
to a macroscopic current at the edges of the medium.
In quantum mechanics operators describe observables where n(1/2) is essentially equal to n(0)except for very
à Ã
and their eigenvalues are the possible results of a mea- broad band signals. The 1-photon density can be gener-
surement. While spin and orbital AM are in general not alized to describe the photon density in an arbitrary pure
13
state using the expectation value of the number operator, field-potential wave function pair obtained by solution of
(38). MEs are ideally suited to the interpretation of photon
Systematic investigation of photon position operators counting experiments using a detector that is small in
and their eigenvectors clarifies the role of the photon wave comparison with the spatial variations of photon density.
function in classical and quantum optics. The LP wave It is not subject to limitations based on nonlocalizability,
function defines a positive definite photon number den- and coarse graining or restriction to length scales smaller
sity and results in photon wave mechanics equivalent to than a wave length is not required. Exact localization
Inagaki s single photon wave mechanics [29]. It is related in vacuum requires infinite energy and is not physically
to field based wave functions through a similarity trans- possible, but position eigenvectors provide a useful math-
formation that preserves eigenvalues and scalar products. ematical description of photon density . Photon number
In free space the field (68) is proportional to the RS density is equivalent to integration over undetected pho-
wave function investigated in [1, 18, 19, 20]. The field tons in a multiphoton beam. In an experiment where ab-
D(+) (r, t) is proportional to the Glauber wave function sorbing particles are placed in a beam, the particles act
[2, 5, 28] which gives the photodetection amplitude for as photodetectors which can sense the spin and orbital
a detector that responds to the electric field [1]. While angular momentum of the photons. Our formalism justi-
only fields and potentials are locally related to charge and fies the use of positive frequency Laguerre-Gaussian fields
current sources, Fourier transformation of k-space prob- as photon wave functions and gives a rigorous theoreti-
ability amplitudes naturally leads to the LP form [6, 37]. cal basis for extrapolation of their range of applicability
The similarity transformation between the field-potential from the many photon to the 1-photon regime.
and LP wave functions makes the choice a matter of con-
Acknowledgement 1 The author acknowledges the fi-
venience for most purposes.
nancial support of the Natural Science and Engineering
By the general rules of quantum mechanics the LP
Research Council of Canada.
wave function is the probability amplitude to detect a
photon at a point in space. It and the closely related
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