Tek bir fotonun kuantum mekaniği

The first quantized theory of a single photon has been the subject of interest since the early days of the emergence of quantum mechanics. Despite many successes and achievements, there is no systematic approach for quantum mechanics of a single photon. In fact there are still some major controversies and problems in this field including: a well-defined wave function in coordinate representation, a positive definite probability density, a position operator, and localizability of a single photon. In this dissertation we provide a systematic quantum mechanical treatment of photon fields and address all of these problems. We first use dynamical equations of Maxwell fields to establish the physical space of photon fields. Next we obtain the most general admissible positive-definite inner product on this space. Imposing necessary conditions on this inner product not only makes it relativistically invariant, but also suggests a three-parameter family of Lorentz invariant inner products which can be used to introduce a genuine Hilbert space for the quantum mechanics of photon fields. We can derive the unitary quantum system by determining the generator of time-translation to be Hamiltonian. This represents a quantum mechanics of a single photon which is not the result of the restriction of Hilbert space to the positive-frequency fields. Next we obtain the generalized parity (P), generalized time-reversal (T), and generalized charge or chirality (C) operators for this system and offer a physical interpretation for its PT-, C-, and CPT- symmetries. We introduce a current density corresponding to relativistically invariant positive-definite inner product and observe that it is both conserved and manifestly covariant. We establish the mathematical formalism of observables and wave functions for the photon. We determine the photon wave function and prove that it is a generalization of both the Landau-Peierls well-known wave function for photon and the Riemann-Silberstein complex vector. Subsequently we obtain a positive-definite probability density for the spatial localization of photon fields and examine the conservation of total probability. We construct a photon position operator with commuting components. This position operator commutes with helicity and chirality operators and is equal to the position operator of massless Klein-Gordon fields with an additional term that corresponds to the Berry phase of propagating photon. As far as we know, this is the first systematic study of the first-quantized theory of a single photon.