52 The Open Optics Journal, 2009, 3, 52-55 Open Access Electron-Photon Interactions: Time and Energy Relationships Tibor Berceli* Budapest University of Technology and Economics, Hungary Abstract: The paper presents a simple, new approach providing results on some properties of electron-photon interactions like their processing time, energy density, optical pulse length. The main goal of the investigation is to establish some basic relationships. Based on the results of the present approach, a general relationship is obtained between the processing time and energy: the processing time is inversely proportional to the utilized energy in the specific process. That is an inherent property of material. Keywords: Electron-photon interaction, energy density, optical pulse length, processing time, time-energy relationship. INTRODUCTION [4, 5]. According to that when an electron is in the conductance band, it is at a stable energy level, consequently Processing time is an important, characteristic property there is no radiation. When the electron transits from the of every process in physics. Therefore, a question can be conductance band to the valence band a photon can be arisen: is the processing time an inherent property of the emitted. Then, when the electron is in the valence band it is material? In this paper that very generic question is again at a stable energy level, there is no radiation. investigated in details. For that purpose the electron-photon Accordingly, radiation can occur during the time interval, interaction is used as a good example. when the electron gets from the conductance band to the The electron-photon interactions have always been very valence band. The time interval of this transition is equal to interesting phenomena. They have already been discussed in the processing time of photon generation. many excellent papers. So far the best explanations come Now the question arises: can the transition time be an from quantum physics. Spontaneous emission can be immediate energy change without taking any time, or does it calculated very accurately using quantum electrodynamics take some time although a very short time? We can get the [1, 2]. This way, statistical results are obtained, because the answer to this question by considering that the generated process is taken statistical. By other words, it is possible to photon has a specific frequency. On that basis we propose an calculate probabilities of large numbers of interactions. approach to estimate the time interval of electron transition The aim of this paper is to find understanding and which is equal to the processing time of photon generation. explanations for electron-photon interactions based on a In a direct band gap semiconductor material, there is an simple new approach. The transition of an electron from an energy difference between the conductance band and the upper energy level to a lower level and the accompanying valence band, called band gap energy. The energy of the photon emission can be assumed as an instantaneous radiated photon is equal to the band gap energy. The process. That is utilized in this paper because currently a frequency, f of the photon is given by the well-known single photon emission is already feasible for performing ph equation [6, 7]: experiments [3]. E The main goal of the investigation is to establish some b fph = (1) basic relationships for the process of photon generation. Its h properties can be described more easily if we consider the where Eb is the band gap energy, and h is the Planck generation of photons in a semiconductor material. In this constant. paper that method is used for the investigations which are based on the present new approach to obtain some more When we try to estimate the processing time of photon results for the properties of photon generation. The derived generation we can use Eq. (1) as a basis. This equation tells relationships are in good conformity with the well-known us that the photon has a specific frequency and this properties of photon generation, however, they provide some frequency is dependent only on the band gap energy [5, 6]. new results as well. We consider the photon as the smallest amount of PHOTON GENERATION radiated energy at a specific frequency. Accepting this concept there are still some questions which can be arisen. First, we consider the photon generation in a direct band One of these questions is: what is the processing time of gap semiconductor material based on its well-known process photon generation? PROCESSING TIME *Address correspondence to this author at the Budapest University of The question is the following: is the radiating transition Technology and Economics, Hungary; E-mail: [email protected] of the electron an immediate process in time or does it need 1874-3285/09 2009 Bentham Open Electron-Photon Interactions: Time and Energy Relationships The Open Optics Journal, 2009, Volume 3 53 some time? In the first case the transition process could be Ec is the energy level of the electron in the conductance described by a Dirac delta function in time [8]. That would band and Ev is the energy level of the electron in the valence mean that the generated photon would not have a specific band. Their difference is the band gap energy Eb. frequency which is not in agreement with the experimental Further on, t is the time elapsed from the start of the results or by other words it is not a realistic approach. process up to the end of the process, or by other words t is Therefore the second assumption is more likely, i.e. the the processing time, i.e. the duration of photon generation: transition of the electron from the conductance band to the valence band takes some time, although this time is t = ph (7) extremely short. That extremely short transition time is the duration of photon generation or by other words the Based on Eqs. (3), (5), (6) and (7) we can write: processing time of photon generation. h h = (8) How can we estimate the processing time? A photon can ph Eb 4Eb be represented by an extremely short electromagnetic pulse which has the duration equal to the generation time of the Consequently, the estimation for the duration of photon photon. That pulse has to be in connection with the photon generation is in conformity with the Heisenberg uncertainty frequency. That means the fundamental frequency of the principle [9-11]. Fourier series representing the electromagnetic pulse has to PROCESSING TIME AND ENERGY RELATIONSHIP be equal to the photon frequency. Based on that requirement a possible assumption is to take a single, one period of a We presented an approach to estimate the duration of raised cosine function representing the electric component of photon generation. Although we consider the photon having the pulse with the duration equal to the period time of the a specific generation time and energy it cannot be separated photon frequency. This way the processing time of photon into smaller parts because the photon has been generated by generation is: a single or unique energy transition of an electron in the atom. ph = Tph (2) As the duration of photon generation is inversely Here ph is the processing time of photon generation and proportional to the photon frequency and its frequency is Tph is the period time of photon frequency: Tph = 1/fph. In that directly proportional to the band gap energy - see Eq. (1) - case the energy is concentrated into the period of the photon we can state: the generation time is inversely proportional to frequency. In the following investigations that assumption is the band gap energy. used. Furthermore, it can be stated for a general case: the The next question is as follows: is the generation time processing time is inversely proportional to the energy constant or is it dependent on some effect? The answer is: utilized for performing a specific process. Here it is assumed the generation time is not constant, it is dependent on the that the other parameters of the process e.g. electric charge, band gap energy. From Eqs. (1) and (2) we get: mass, temperature, material, etc. are unchanged. h The main goal of the investigation on the duration of = (3) ph E photon generation is not to determine its exact value but to b establish some basic relationships. Evaluating the results of If the band gap energy is higher the generation time is the present approach we can formulate some relationships: shorter, or by other words the radiated pulse or burst is - the transition of an electron from one energy level to shorter. another one takes some time, UNCERTAINTY PRINCIPLE - the duration of photon generation is determined by We use now the uncertainty principle for estimating the the transition time of the electron emitting the photon, processing time in a different way. Based on Eq. (1) there is - the duration of photon generation is estimated to be a relation between the period time Tph of the photon equal to the period time of its frequency, frequency and the band gap energy Eb generating the photon: - the duration of photon generation is inversely h proportional to its frequency, Tph = (4) Eb - the processing time or duration of photon generation When we want to determine the processing time we have is inversely proportional to the band gap energy. to consider the Heisenberg uncertainty principle [9, 10]. Based on the previous statements a more general rule can Accordingly, we get: be suggested: the processing time is inversely proportional to the energy involved in the process. Furthermore we can h tE (5) state: time is an inherent property of processes in nature.