In fact, this question is equivalent to what the space dimensions of a photon? In order to simplify the discussion, we take the system only with two states as an example (see Figure 1) and under the dipole moment approximation, the perturbation terms takes the form,

Then, the Equation (2) can be simplified as

where is the transition dipole moment defined by

The first term in Equation (4) describes the emission of photon from the system; the second one corresponds to the absorption of photon by the system.

Now we only consider the absorption of photon by the system. Then we find, when, the second term becomes 0/0. Based on L’Hopital rule,

So, the Equation (4) becomes

In fact, can be selected as ħω₂₁, but doesn’t lose generality, therefore,

We know corresponds to the probability of the transition from state 1 to state 2. After the system absorbs a photon and be excited to the state 2, should be 1, so,

We obtain,

This result tells that the time for the system to absorb a photon is the period of the light wave and therefore, the space dimension of a photon in its propagation direction is just one complete period or wavelength, therefore, we can call each single period of light wave as a photon. For the different frequency of light/electromagnetic wave, the spacetime dimension for a photon is different, the higher the frequency, the smaller the spacetime of the photon. In fact we can design an experiment to verify the time for a system to absorb a photon is the period of the light. We can put a switch between the light source and system, if the switch turn-on and turn-off time is less than the period of the light wave, then, the system cannot absorb the light. At moment, we just don’t know whether this kind of switch exists or not. If yes, then to verify the conclusion here is much easier. Our result here also reveals that the time for the system to absorb a photon depends on the frequency of light and determines the line width of the absorption spectrum. That means for the absorption spectrum, the nature line width is 2T, where T is the period of corresponding light/electromagnetic wave.

To my knowledge, the light/electromagnetic wave is believed to be continuous wave like water wave, but sometime behaviors like a particle. Nobody tells us why and how. Therefore, we call this property of light/electromagnetic wave as duality.

We still take two levels system as an example, First, the system absorbs a photon from its surroundings and be excited from L level to K level, then, gives up a photon to its surroundings and systettm decay from K level to L level. So, if we take this two levels system as a source of light, then, the light cannot be continuous but like dashed line instead (See Figure 2). The minimum gap between two photons is.

From the discussion above, we know, but normally, , therefore,. From the literature [4], we know that the Einstein coefficient A

represents the decay rate per unit time per atom (here we don’t consider the stimulated emission). For two levels system, the 1/A represents the lifetime of the excited state and correspondingly, the Einstein coefficient B represents the probability rate of the system excited from L level to K level (see Figure 1) per unit time per atom. Therefore, the 1/B represents the lifetime of the system at the ground state under the radiation, so, the minimum gap between two photons for this two levels system is

From the discussion in paragraph 1 and literature [4]we know and, we obtain,

Equation (12) tells us that t_g is proportional to ω, that is, the higher the frequency, the larger the t_g.

Now, we understand that the discontinuity of the light/electromagnetic wave makes the light/electromagnetic wave like a series of pulse or energy corpuscle. If the time resolution is close or beyond the T, we will find the interaction between photon and matter will last a certain time (period of light/electromagnetic wave), not like a real particle in ideal elastic collision. If the space resolution is beyond the wave length of light/electromagnetic wave, then we will see the light/electromagnetic wave more like a wave instead of particle. This is the real reason why sometime the light/electromagnetic wave behaviors like a particle, sometime, the light/electromagnetic wave becomes more like a wave. Therefore, we call the property of the light/electromagnetic wave as the particle-wave duality.

Based on the result above, we speculate how to modulate the photon individually. If we can modulate the photon individually, then it will find great application in many fields, such as communication, electronics, etc.

Generally, the Maxwell’s equation [8] is a continuous equation, but now, we demonstrate the light/electromagnetic wave is not continuous at all. That is the reason why the Maxwell’s equation cannot describe the particle behavior of light/electromagnetic wave. Then, what is the correct expression of the Maxwell’s equation or how to make the Maxwell’ equation still valid at the micro range? If the light/electromagnetic wave is not continuous, then how about the electric field and magnetic fields (even including gravity fields)? My understanding is that if the light/electromagnetic wave is discontinuous, then there is no reason for the electric fields and magnetic fields (even including gravity field) to be continuous. The electric fields and magnetic fields (even including gravity fields) must be discontinuous. The meaning for the discontinuity of the electric fields and magnetic fields (even including the gravity fields), at moment, is still unclear. Maybe it is the origin of the quantized electric fields and magnetic fields (even gravity fields). May be it is the discontinuity of the light/electromagnetic wave that is the basis of the quantized electric fields and magnetic fields (even gravity fields). We will keep working on these topics and present the result later.