Quantized Vector Potential and the Photon Wave-function

Abstract : The vector potential function ${\overrightarrow{\alpha }}_{k\lambda }(\overrightarrow{r},t)$ for a k-mode and λ-polarization photon, with the quantized amplitude α 0k (ω k ) = ξω k , satisfies the classical wave propagation equation as well as the Schrodinger's equation with the relativistic massless Hamiltonian $\mathop{H}\limits^{\sim }=-i\hslash c\overrightarrow{\nabla }$ and finally an equivalent quantum equation for the vector potential amplitude operator ${\mathop{\alpha }\limits^{\sim }}_{0}=-i\xi c\overrightarrow{\nabla }$. Thus, ${\overrightarrow{\alpha }}_{k\lambda }(\overrightarrow{r},t)$ behaves as a wave function for the photon within a non-local representation that can be suitably normalized. It is deduced that the probability for detecting a k-mode photon around a point on the propagation axis depends on the square of the angular frequency. Taking into account the left and right circularly polarized states and weighting ${\overrightarrow{\alpha }}_{k\lambda }(\overrightarrow{r},t)$ by ${\omega }_{k}\sqrt{2{\varepsilon }_{0}}$ we define a six components function as a general function for a k-mode photon. The square of the modulus of the defined general function gives the energy density at a given coordinate which depends on the fourth power of the angular frequency. The amplitudes of the electric and magnetic fields of a single k-mode photon free of cavity are also calculated and it is shown that they are proportional to the square of the angular frequency. In this way, the influence of the photon electric and/or magnetic fields on the energy levels of atoms and molecules might be used for a non-destructive photon detection.