Photon State-vector Function And Its Application

From ancient times, the people started to the research for the light, and had gradually formed the special discipline-optics. Today, modern optics may divide into four levels on the whole, namely the ray optics (i.e. geometrical optics), the wave optics, the electromagnetic optics and the photon optics. These four levels are already not each other separations, also is not interweaves mutually, but is the relations which the cascade contains, in other words, wave optics to contain the ray optics, electromagnetic optics has contained ray optics and wave optics, photon optics has contained ray optics, wave optics and electromagnetic optics.From the historical development viewpoint, ray optics, wave optics and electromagnetic optics all are the successful optics, explained the multitudinous optical phenomenon, obtained the understanding for natural ruler; meanwhile as a result of time limit, also has a flaw and the insufficiency respectively, and afterwards makes up for high level optics. Collecting all these understanding accomplishing, that is contemporary photon optics, it bases on the basic concept of quantum mechanics-light is constructed by the photon. This paper mainly elaborate and do proof how photon optics contains the electromagnetism optics, has made up the electromagnetism optics any insufficiency, also explained the optical phenomenon which are not explained before.The Maxwell equation for the optical wave which has established the electromagnetic wave attribute of the light, is the foundation of electromagnetic optics; The Schrodinger equation for the photon which has established the microparticle attribute of the light, is the foundation of photon optics. The innumerable experimental fact and the observation experiment indicated repeatedly these two kind of judgments all is conclusive without doubt, therefore clearing off the relations between those is the key for knowing the nature of light which can not simply interpreted by "the wave-particle duality". This paper starting point based on this kind of judgment:the first-order optical wave equation equivalent in Maxwell equation and the photon Schrodinger equation in mathematics is the identical partial differential equation, therefore their formal solution in mathematics must be accommodating. At the same time, because the authentic condition, the first-order optical wave equation solution must be a real function; On the other hand, because the quantization condition, photon Schrodinger equation solution must be a complex function; According to above basic judgment of compatibility, therefore the former may take is merely the latter real part.The people already have the understanding to the photon Schrodinger equation, its solution namely is the state function describing photon quantum state, as a result of the photon quantum characteristic, the state function must simultaneously satisfy the photon energy eigenvalue equation and the photon momentum eigenvalue equation as the quantization condition, as a result the state function must be a complex function. Therefore, for the state function for any microparticle including the photon, and the generalized Schrodinger equation which any microparticles must follow, its domain of definition is not in real physical space, but is in the abstract Hilbert space, this is the basic concept for understanding the quantum mechanics. This paper in these foundations, for the first time the photon angular momentum eigenvalue equation the third quantization condition which must simultaneously satisfy as the photon state function, as a result the photon state function must take the vector form, we name it the state-vector function (SVFs).Using the one-dimensional SVFs which can describe all quantum behavior of a photon, we will be able to carefully examine and to solve some difficult classical optics question in the view of the photon, such as the orthogonal decomposition of natural light. The well known experience fact is:After a beam of natural light through the ideal plane polarizer, it is transformed the planar polarized light which has the same polarization orientation with polarizer, the intensity is 1/2 of the initial natural light. According to what we know, until now any explanation for this phenomenon has no theoretical basis; using the one-dimensional SVFs, we have first completed the strict proof.Famous physicist Feynman has had the extremely penetrating elaboration to the double-hole interference experiment, he said:"a phenomenon which is impossible, absolutely impossible, to explain in any classical way, and which has in it the heart of quantum mechanics. In reality, it contains the only mystery". He also further pointed out that, "In quantum mechanics any other situations, in the final analysis, always may explain like this that, you also remember the double-hole experiment? They are same". But until now, the related double-hole interference experiment quantitative analysis, still limited to the classical way; But all quantum mechanics explanation, also all pauses in the qualitative stage. Using the one-dimensional SVFs, we can first give the double-hole interference effect the quantum description with the analytic expression.As early as in 1930, Dirac, in his famous textbook on quantum mechanics, made a basic statement on optical interference:"Each photon interferes only with itself. Interference between two different photons never occurs." It has brought a continual argument to researchers for three quarters of a century long. While this assertion was ceaselessly quoted, there always existed some doubts, the matter really are such? Since then an impressive progress in the development of both light sources and detectors has been made. Especially in 1963 the interference effect arising from two independent laser beams had observed, Paul asserted:"Hence, the second part of Dirac's famous statement quoted above has actually been disproved." Using the photon one-dimensional state-vector function to search interference nature of light, our conclusion is the Dirac's basic statement is by no means false, through the analysis and proof, we attempts to make its indication consummation:"Ever y photon interferes only with itself that possesses a set of the selfsame eigenvalues. Interference between two different photons those possess different eigenvalues never occurs"。In 1992, Allen first proposed high-order angular momentum (AM) of light, we expand the photon AM eigenvalue from the usual L2=±h to the unusual Lz=lzh, where lz=0,±1,±2….We mainly discussed the influence on the photon SVFs and its macroscopic performance due to the expansion of the photon angular momentum eigenvalue. Through solving photon Schrodinger equation and the corresponding quantization condition, we expand the one-dimensional (1D) SVFs for the ordinary photon to three-dimensional (3D) SVFs for unusual photon. The patterns of l-order Laguerre-Gaussian modes of a laser beam are well theoretically modeled. And zero-order LG modes of a laser beam represent longitudinal light waves which be constructed from longitudinal photons with zero angular momentum eigenvalue.Finally, we forecast the application prospect of the photon state-vector function which has contained the angular momentum characteristic.