2.1. Introduction

Classical expression for free photon energy is:

where c is the light velocity in vacuum and p_x, p_y and p_z are the components of photon momentum. If instead of classical momentum components we use quantum-mechanical operatorspν→p^ν=−iℏ∂∂xν;ν= (x,y,z), where ℏ≡h2π= 1,05456×10–34Js is Dirac's constant, we obtain quantum-mechanical single photon Hamiltonian:

This Hamiltonian is not a linear operator that contradicts the principle of superposition (Gottifried, 2003; Kadin, 2005). Klein and Gordon (Sapaznjikov, 1983) skirted this problem solving the eigen-problem of square of Hamiltonian (2):

since the square of Hamiltonian is a linear operator. This approach has given satisfactory description of single photon behaving. Up to now it is considered that this approach gives real picture of photon. Here will be demonstrated that Kline–Gordon picture of photon is incomplete.

Here we shall try to examine single photon behavior by means of linearized Hamiltonian (2). Linearization procedure is analogous to the procedure that was used by Dirac’s in the analysis of relativistic electron Hamiltonian (Dirac, 1958). We shall take that

i.e. we shall transform the sum of squares into the square of the sum using α^,β^ and χ^ matrices. In accordance with (4) these matrices fulfill the following relations:

It is easy to show (Tošić, et al., 2008; Delić, et al., 2008) that (5) conditions are fulfilled by Pauli’s matrices

Combining (6), (4) and (2), we obtain linearized photon Hamiltonian which completely reproduces the quantum nature of light (Holbrow, et al., 2001; Torn, et al., 2004) in the form:

Since linearized Hamiltonian is a 2×2 matrix, photon eigen-states must be columns and rows which two components. Operators of other physical quantities must be represented in the form of diagonal 2×2 matrices.

At the end of this presentation, it is important to underline the orbital momentum operator(L→^00L→^); L→^=r→^×p→^does not commute with Hamiltonian (7). It means that it is not integral of motion as in Klein-Gordon theory (Davidov, 1963). It can be shown that integral of motion represents total momentum(J→^00J→^), where J→^ is the sum of orbital momentum L→^ and rotation momentum S→^ which corresponds to 1/2 spin.

In further the eigen-problem of linearized single photon Hamiltonian will be analyzed in Cartesian’s, cylindrical and spherical coordinates.

2.2. Photons in Cartesian's picture

The eigen-problem of single photon Hamiltonian in Cartesian coordinates (we shall take it with plus sign) has the following form:

wherefrom we obtain the following system of equations from:

wherek=Eℏc. It follows from (9) that:

Since the operators ∂∂z±ik and ∂∂x±i∂∂y commute, through (11) we come to the following relation:

In the same manner, from (9) and (11), we come to the relation:

The two last relations are of identical form and can be substituted by one unique relation:

If we take in (14) that k2=kx2+ky2+kz2 andΨ(x,y,z)=A(x)B(y)C(z), we come to the following equation:

which is fulfilled if:

Equations (16) can be easily solved and each of them has two linearly independent particular integrals:

Based on these expressions, we conclude that eigen-vector of single photon (Ψ1Ψ2) has the following form:

Since k→ is a continuous variable, the normalization of (18) mast be done to δ–function, wherefrom follows:

Solving these integrals, we come to: 2 D² (2π)³ = 1, wherefrom we get the normalized single photon eigen-vector as:

As it can be seen from (20), the components of single photon eigen-vector are progressive plane wave ~eik→r→ and the regressive one ~e−ik→r→. Since we consider a free single photon, the obtained conclusion is physically acceptable.

2.3. Photons in cylindrical picture

In this section of first part of the paper we are going to analyze the same problem in cylindrical coordinates. Since solving of partial equation of (Δ+k2)Ψ=0 type in cylindrical coordinates requires more general approach than that which was used in Cartesian's coordinates, it is necessary to examine single photon eigen-problem in cylindrical system.

In order to examine this problem, we shall start from the equation (14) in which Laplacian ∂2∂x2+∂2∂y2+∂2∂z2≡Δ will be given in cylindrical coordinates (ρ,φ,z) whereρэ [0,∞], φэ[0,2π]andzэ [–∞,+∞]. The Laplacian in cylindrical coordinates has the following form:

Δ=∂2∂ρ2+1ρ∂∂ρ+1ρ2∂2∂ϕ2+∂2∂z2

and therefore (14) withΨ(x,y,z) => Φ(ρ,φ,z), reduces to:

The square of wave vector k will be separated into two partskx2+ky2+kz2=q2+kz2. On the basis of this the equation (21) can be written as follows:

By the substitution:

the equation (22) reduces to:

This equation is fulfilled if:

Now we separate the variables by substitution:

after which, the (25) goes over to:

Introduction of the variables separation constant m² represents generalization with respect to approach used in previous section. Since the function S(φ)must be single-sign S(φ) =S(φ+2π) we must that m is integer, i.e. m= 0,±1, ±2,...

Relation (27) is separated into two differential equations:

The equation (25) has two particular integrals:

while the solution of the equation (28) is:

By the substitution of argumentρ=bξ, the equation (28) reduces to

and taking thatb=1q, we translate (31) into Bessel's equation with integer index m:

It means that the solution of (28) is the m–order Bessel’s function: J_m, i.e.

Taking into account (29), (30) and (33), we obtain the components of single photon eigen-vector:

Since q and k_z are continuous variables, while m is a discrete one the normalization of eigen-vector must be done partially to δ–functions and partially to Kronecker’s symbol. It means that normalization condition is the following:

Using formula for normalization of Bessel functions with integer index (Korn & Korn, 1961):

the normalization condition reduces into:|D1|2+|D2|2=14π2. It means that normalized single photon eigen-vector in cylindrical coordinates is given by:

The first component Φ₁ corresponds to photon (velocity +c), while second component Φ₂ corresponds to anti-photon (velocity –c). From this formula we conclude that single photon eigen-vector components are progressive and regressive plane waves along z-axis. In the (x,y) planes components change periodically with polar angle φ and decrease by the rule ρ^-1/2 with distance between the axis and envelope of cylinder. The last is concluded on the basis of asymptotic behaving of Bessel’s functions (Korn & Korn, 1961):Jm(ρ)≈sinρρ, whenρ→ ∞. We have seen during the analysis of a photon in Cartesian’s coordinates that only zero values of parameters of variables separation are physically imposed. In cylindrical coordinates, due to physical reasons again, one parameter of variable separation had zero value, while the other has to be a square of integer. The last is necessary since the solution must be single-sign function.

2.4. Photon in spherical picture

The analysis of single photon eigen-problem in spherical coordinates, as it well be shown later, requires introduction of two variable separation parameters. We start from the equation (14), where the Laplace’s operator will be written down in spherical coordinates (r,θ,φ), whererÎ[0,∞], θÎ[0,π]andφÎ[0,2π]. In these coordinates it is of the form:

It means that (14), withΨ(x,y,z) → Ω(r,θ,φ), becomes:

In the first stage of variables separation, we shall take that:

after which substitution into (37), it goes over to:

where Λ² is the variable separation parameter. Double equality in (39) gives two equations:

It should be noted that equation (40) represents eigen-problem of L^2ℏ2 operator. It means that Λ² determines orbital quantum numbers. In this equation we shall take that:

after this substitution, which goes over to:

In this double equality the variable separation parameter m must be integer since the solution S(φ)must be single-signed function. The same requirement appeared in the previous section where single photon vas analyzed in cylindrical coordinates. The equation (42) gives two second order differential equations:

When the solution (43) is:

the equation (43) is associated Legendre’s equation (Gottifried, 2003; Davidov, 1963). The complete procedure of solving of this equation cannot be found in literature. Instead of the general solving procedure of the equation (43) is solved for m = 0. Its solutions are Legendre’s polynomials (Korn & Korn, 1961; Janke, et al., 1960). Differentiating these polynomials m-th times it was possible to conclude that solution (43) can be expressed through m-th Legendre’s polynomials derivations.

In order to avoid such an artificial solving of the equation (43), we shall expose, briefly, its solving by means of potential series. This solving procedure may be comprehended as methodological contribution of this part of the paper. At the first stage, we translate the equation (43) into algebraic form by means of substitution of argumentcosθ=ζ:

The term m21−ζ2 in (45) does not allow the solving of this equation by means of potential series. Consequently this term must be eliminated from the equation. The strategy of elimination is the following: by the substitution ofT=UּV, where U is an arbitrary function, the equation (45) reduces to the same form but with arbitrary constant in linear function with is multiplied by first derivative of V function. This arbitrary coefficient will be taken in the form –2(2s+1)where s is arbitrary. Arbitrary constant s will be determined in a way which eliminates the term m21−ζ2 from equation for V function. By the described strategy the (45) becomes:

This equation is suitable for solving by means of potential series. Arbitrary function U is given byU=(1−ζ2)S, wheres= ±m/2. This means that function T has the form:

Since ζ[–1,+1]the exponent s must not be negative since T would then have singularities in ζ= ±1 not allowing the normalization. Fortunate circumstance is that the exponent of the function 1 –ζ2has ± sign. This means that for m> 0can be takens= +m/2=|m|/2. Ifm< 0, we takes= –m/2 =|m|/2. Based on this reasoning the equation (46) becomes:

The solution of this equation will looked for in the form of potential series:

after which substitution in (48) we obtain recurrent formula for series coefficients:

Here arises a dilemma whether to leave the whole series or to cut it and retain a polynomial instead of series. In order to solve this dilemma, we shall analyze a special case of formula (50) whenm= Λ = 0. In this case formula (50) becomes:

wherefrom it turns out thatvn=v12n+1, and this means that series solution (49) becomes:

From this formula is obvious that the series has singularities for ζ = ±1. This resolves above mentioned dilemma: the series must be cut and the polynomial obtained in this way must be taken as solution. From the formula (50) it is clear that the series will be cut if:

Now is clear that the series is cut whenl= |m|+n, wherefrom it follows that the degree of polynomial is l=|m|–nand that quantum number m per module must not exceed l:|m|≤l. The obtained polynomials of l–|m|degree are called the associated Legendre’s polynomials (Korn & Korn, 1961; Janke, et al., 1960) and by means of them T function is expressed as:

The product of functions (44) and (54) normalized per angles gives spherical harmonics (Gottifried, 2003; Davidov, 1963):

Finally we shall solve the equation (40) in which Λ² is substituted byl(l+1). It means that it goes over to:

Substituting the function R with r–1/2J(r)and substituting the argument r by ρ/k, we translate last equation into Bessel’s equation (Korn & Korn, 1961; Janke, et al., 1960) with l+1/2index having two linearly independent particular solution Jl+1/2(kr)andJ–l–1/2(kr). Consequently the solutions of (40) are:

It is necessary for further to quote behaving of Bessel’s functions with half integer indices. It can be easily shown that:

As well as using recurrent formula for Bessel’s functions (Janke, et al., 1960):

and taking that p= +1/2 andp= –1/2, we obtain respectively:

Due to the factor x–3/2functions J±3/2have strong singularities in zero so that they cannot be normalized in the interval0 ≤r< ∞. Due to the same reasons neitherJ±5/2,J±7/2, etc. cannot be normalized. It can be concluded that only solutions for A which are proportional to J±1/2have chances to be normalized. Those solutions are:

The very important conclusion of this analysis is: only free photons with zero orbital momentum have chances to be normalized exist. For l> 0photon eigen-vector cannot be normalized.

We shall now examine whether the components of photon eigen-vector proportional to R₁ and R₂ can be normalized. Those components are:

The normalization condition is the following:

It is not difficult to show that:∫0∞drcos(k−k′)r=0, so that the condition (63) becomes meaningless. This means that even for l = 0 photon eigen-vector cannot be normalized.

The last possibility for normalization free photons eigen-vector is so called box quantization method. In this method the sphere is substituted by cube enveloping it and cyclic boundary conditions are required:eikr=eik(r+L), wherefrom it follows that wave vector is quantized:

Sincek=2π/λ, it gives that:

It is seen that the first harmonic of electromagnetic waves has the wave length equal to the cube edge.

Photon energy is determined in the standard way:

This expression for energy is in full accordance with Plank’s hypothesis (Planck, 1901).

In the normalization condition (63) the following translations has to be used:

Combining this and (63) we obtain that the normalization constant isW=12πn. On the basis of this the normalized photon eigen-vector is given by:

As it can be seen the analysis of single photon eigen-problem in spherical coordinates has shown it orbital momentum of photon is equal to zero and that the spin S= 1/2is its unique rotational characteristics (Yao, et al., 2005). Physically it is fully understandable that orbital momentum of a free photon is equal to zero since it moves along the straight line. On straight line photon radius-vector r→ and its momentum p→=mfr→˙ are parallel and this gives thatl→=r→×p→=0.

3.1. Introduction

The fact that photon Hamiltonian is not a linear operator has a set of consequences that have not been studied sufficiently so far. The main reason is that photon characteristics have been mainly examined by means of Klein-Gordon’s equation (Gottifried, 2003; Davidov, 1963; Messiah, 1970; Davydov, 1976), which represents eigen-problem of photon Hamiltonian square. In this part of our paper we shall linearized photon Hamiltonian and examine some of photon characteristics witch follow from linearized Hamiltonian. The analogy with Dirac’s approach to the problem of electrons will be used (Gottifried, 2003; Dirac, 1958). Firstly will be examined integrals of motion of free photon and will be shown that the photon integral of motion is not orbital momentum. It will be shown that the integral of motion is total momentum being the sun of orbital one and spin momentum.

The evaluated Green’s function has given possibility for interpretation of photon reflection as a transformation of photon to anti-photon with energy change equal to double energy of photon and for spin change equal to Dirac’s constant (Dirac, 1958; Messiah, 1970). Since photon is relativistic quantum object the exact determining of its characteristics is impossible.

The discussion of obtained results and their comparison to the experimental data will be done at the last part.

3.2. Linearized photon Hamiltonian

We shall not deal with this eigen-problem in further of this paper. Instead of this we shall look for integrals of motion, i.e. those operators that commute with free-photon Hamiltonian (7). It is obvious that any function depending on momentum components represents an integral of motion, but this fact is not of physical interest.

It is of particular importance whether orbital momentum:

is photon integral of motion, since in non-relativistic quantum mechanics operator L→^ is integral of motion of electron (Messiah, 1970; Davydov, 1976). The components of orbital momentum are given as follows:

If we use commutation relations for components of radius vector and the components of momentum:[xi,pj]=iħδij, i,jÎ(x,y,z)and look for commutators of (69) with Hamiltonian (7), we come to the following relations:

based on which it follows that orbital momentum is not a free photon integral of motion.

It should be pointed out that signs in (70) are obtained on the basis of obvious symmetry properties H^(−r→)=H^(r→) andL→(−r→)=L→(r→), where r→ is radius-vector.

In order to find some rotation characteristics that commute with a free photon Hamiltonian, we shall first show that commutation relations for matrices α^,β^ andχ^, given in section 2.1 by expression (6), are:

while commutation relations for spin components (Dirac, 1958; Messiah, 1970):

are very similar to (71). Comparing (71) to (72) we can establish the correspondence between spin operator components and matrices α^,β^ andχ^:

Commutators of matrices α^,β^ and χ^ with Hamiltonian are given by:

We shall now look for a commutator of component J^x of total momentum, with photon Hamiltonian i.e. withH^(r→). Using upper signs in formulas (70) and (74) we obtain:

For lower signs in formulas (70) and (74)[1] - , we have:

It can be proved, in the same manner, that both y and z components of total momentum J→^=L→^+S→^ commute with photon Hamiltonian (the expression (7) with sign +, i.e. H^(r→)will be called photon Hamiltonian). The expression (7) with sign –, i.e. H^(−r→)will be called anti-photon Hamiltonian. In the same manner can be proved that y and z components of total momentum J→^=L→^−S→^ commute with anti-photon Hamiltonian.