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About a photon

Dear all

Please be relaxed.  And have some coffee while reading this blog.

I am reading a book.  It is still the "Schroedinger's Kittens."  According to the book, a single photon can now be made and injected into an experiment.  In order to make a single photon, an isolated single calcium atom is trapped in space by electromagnetic fields and illuminated with a laser pulse.  The frequency of the laser is exactly tuned to excite one electron in the calcium atom by just two energy steps.  The electron then goes down the two steps sequentially.  At the moments when the electron goes down each step, two single photons each having the specific frequency that corresponds to the energy difference between the stairs, are emitted successively.  A human experimenter then uses the secondly emitted single photon for, e.g., the two-slit experiment, says the book.  So far, so simple.

Copenhagen school which was established in the era of pioneering works of quantum mechanics by such a strong personality like Niels Bohr, is a robust party even today.  The Copenhagen school insists on the dual nature of the behavior of quantum entities like an electron, an atomic nucleus or a photon.  The dual nature means the wave-particle nature.  The wave-particle nature is like the relationship of the top and the tail of a coin.  When it is placed on a table flat, we see only one face, that is, the top or the tail, but we cannot see both faces at once.  The Copenhagen school says the same thing about quantum entity.  If you do an experiment which is designed to observe the wave nature like an interference fringe, then you certainly observe the wave.  On the contrary, if you do an experiment which is designed to observe the particle nature, e.g., by using Pockels cell in the delayed choice experiment, then you certainly observe the particles.  But you cannot observe both of the two natures simultaneously, says the Copenhagen school.

The book also introduces us another experiment.  It was proposed by Indian researchers and experimented actually by Japanese researchers.  The book says as if the experiment has succeeded in showing the dual nature of a photon at once.  The experiment employs a bi-prism with very narrow air gap as a beam splitter.  This bi-prism splits the incident photon in two arms, one being straight path, the other being the reflected path by right angle.  The passing of the photon through the former path is only possible if the photon behaves as a wave.  At each end of the two arms, there is a detector which clicks if it has detected a photon.  The experimental result showed a perfect anti-correlation of the clicks.  Though they say that this experimental result shows that the dual nature of a photon could be observed at once, I am not still convinced of their assertion, because the two arms are physically different, so it is not symmetric for the two paths.

Here let us go to the 19th century.  In 1864, James Clerk Maxwell published his famous four equations that can describe the electric phenomena and magnetic phenomena all together.  The most spectacular thing was that the equations predicted the existence of an electromagnetic wave that was thought to propagate through the space by the speed c.  Then soon, the light was found to be a kind of the electromagnetic wave.  That is, the light or the photon should , in principle, be described based upon the four Maxwell's equations.  In fact, the light or the electromagnetic wave can be conceived as a harmonic combination of changing electric field and changing magnetic field.  The two fields can be imagined to be inducing each other, enabling the electromagnetic wave to propagate through the space.  Thus the light is essentially made of electromagnetism.

Then, we ask, how to generate the light?  Then the Maxwell's equations give us the answer.  The answer consists from equations that relate the electric field and magnetic field with the relative position and relative distance betwenn the observing point and the charged particle, and the velocity and the acceleration of the charged particle at a time which further needs to be resolved by using the relative distance.  That is, the following equations are the answer.

E(x,t)=f(x,z(t'),dz(t')/dt,d^2z(t')/dt^2)          (1)

B(x,t)=g(x,z(t'),dz(t')/dt,d^2z(t')/dt^2)          (2)

t'=t-(x-z(t'))/c                                           (3)

where z(t') is the position, dz(t')/dt the velocity and d^2z(t')/dt^2 the acceleration of the charged particle at the time t'.  The time t' must be calculated by solving the equation (3).  The most important point here is that the definite relations between the field variables E(x,t), B(x,t) and the position, velocity and the acceleration of the charged particle have been obtained.  Thus, we can, at least in principle, calculate the E field and B field at any point at any time, only if we could know the exact position and the motion of the electron that goes down from the higher energy level state to lower energy level state during the transiotion in a single calcium atom.

OK, we have understood that the position and the motion of the electron during its transition process dtermines the light or, in other word, the photon that will be emitted when the electron follows the transitional trajectory, if such a concept of transitional trajectory is valid or allowed to imagine.

Unfortunately we come up against a snag.  Let us go back to 21st century.  In this century, everything that is microscopic is treated by using the quantum mchanics.  The quantum mechanics strictly bans the use of the classical concept like a definite path, a definite orbit, a difinite position or a difinite momentum for any quantum mechanical entity.  So we cannot say anything about the exact position and the motion of the electrons in an atom.  We are only allowed to imagine a vague distribution of the electrons which surround the nucleus of the atom under consideration.  This distribution is regarded as the electron cloud.  But actually its comprehension is much more difficult because, firstly the distribution is not a material distribution but a probability distribution, and secondly if the atom under consideration includes more than two electrons then the distribution function is in fact a 6-variable, or a 9-variable,..., many-variable function which we humankind can never image in this 3-dimensional space.

Due to the orders from the quantum mechanics, we cannot identify both of the starting position and the ending position as well as the shape of the transitional trajectory itself of the electron when it undergoes the transition.  Therefore, we are not allowed to use the equations (1) through (3) in order to calculate the E field and the B field which must be generated somehow when the electron changes its course in an atom.  Thus the quantum mechanics bans us from thinking the transitional trajectory and gives us no hint of it.  Then what is left for us may either be a spherical wave front or a plane wave front for the photon which will be emitted from the calcium atom.  This is my current personal inference.  It might be right, or it might be wrong.  But whatever the correct answer will be, it is definitely certain that from somewhere on a spherical surface that encloses the excited calcium atom, an energy quantum in the form of electromagnetic wave must escape to the outer space for the single photon emission to be be possible.

In spite of the various restrictions imposed by the quantum mechanics, the photon is yet considered to be made of electromagnetism.  The quantum mechanics add to this electromagnetic wave a further another wave.  It is the probability wave, in other word, it is the quantum mechanical wave function.  Then we will ask natuarally, do the electromagnetic wave and the wave function go hand in hand?  First of all, the mechanisms for generating the electromagnetic wave and for the probability wave seem to be different.  The origin of the former wave is related by the equation (1) through (3) to the temporally ephemeral transition trajectory which the electron follows.  On the contrary, the quantum mechanics does not say anything about how the probability wave is generated and about which way the probabilty wave is going to propagate.

In an experimental setup, the photon that is made of electromagnetism in principle and is associated by the probability wave will finally arrive at the detector screen.  At the very instance when the photon impinges on the detector screen, the spatially spreading probability wave and at the same time the spatially spreading electromagnetic wave shrink into a single spot.  This is the collapse of the wave function.  It is itself really amazing!  How can this happen at all?

Thank you for your long time


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