Two-photon wave mechanics and correlation matrices

M. A. M. El-Nadi

The present work is concerned with de Broglie's quantum theory of light. It is assumed that the photon is described by a Hermitian wave function with 16 components. Using this wave function it is shown that the 32 of de Broglie's equations are reduced to one set of 16 equations in the form: [equation] where the [alpha]'[rho] are Dirac matrices and [psi] is a matrix with 16 components. The electromagnetic quantities associated with the photon are described by means of the Dirac matrices operating on [psi] in a specified way. It is shown also that these electromagnetic quantities satisfy Maxwell's equations as a result of the equation the interaction between an electron and a photon is developed and the matrix elements for the radiation transitions are calculated. It is further shown that the above wave equation can be considered as the superposition of two similar wave equations, one for the positive energy photons and the other for the negative energy photons. To each of these states there corresponds electromagnetic quantities defined by the above method. It is the superposition of these fields which gives rise to the reality of the electromagnetic field, found in experience. The wave mechanics of the positive energy photon is discussed and the method of second quantization is applied to its wave function, from which we deduce the commutation relations for the complex fields.

Lajos Jánossy

Ole Keller

This chapter culminates with a discussion of the photon wave mechanical picture of the correlated spontaneous emissions from a three-level atom. The photon spin entanglement of the two-photon cascade J = 0→ J = 1→ J = 0 is briefly analyzed in Sec. 32.5, paying attention to the correlations in the spin state space. Measurements are always carried out in direct space, and for this reason alone it is of interest to study the space-time correlations of the cascade process. The existing theories focus attention on the far-field correlations [214]. In the far field the spatial extension of the rim zone of the atom is unimportant. However, if one imagines that the two detectors needed to measure the second-order correlation matrix at two different space-time points are moved into the rim zone of the atom, the lack of spatial localization of the transverse photons mixes with the quantum mechanical entanglement phenomenon. The standard theory of two-photon cascade entanglement is extended to the domain of near-field electrodynamics by relating the final two-photon state in Hilbert space to photon wave mechanics in direct space.

Alain Aspect

Yongqin Wang, Kang, Lifeng

In this article, we discard the bra-ket notation and its correlative definitions, given by Paul Dirac. The quantum states are only described by the wave functions. The fundamental concepts and definitions in quantum mechanics is simplified. The operator, wave functions and square matrix are represented in the same expression which directly corresponds to the system of equations without additional introduction of the matrix representation of operator. It can make us to convert the operator relations into the matrix relations. According to the relations between the matrices, the matrix elements will be determined. Furthermore, the first order differential equations will be given to find the solution of equations. As a result, we unified the descriptions of the matrix mechanics and the wave mechanics.

Augusto Garuccio