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3-4: Quantum Mechanics of Light |
The Schroedinger equation
learned so far
has been introduced
in order to describe
the wave nature
of the substance particles,
i.e.
the de Broglie wave
accompanying
the motion of the particles.
It has been clarified
that this
Schroedinger equation
can beautifully describe
the duality of the particle nature
and the wave nature.
On the other hand, it has also been emphasized that light possesses the wave-particle duality. In the classical theory, light was originally an electromagnetic wave which obeys Maxwellian equations. It consists of an oscillating electric field E and a magnetic field B as shown in Fig. (A). How can this wave motion get the particle nature? |
Fig. (A): The wave nature of light
In the classical theory, light consists of an oscillating electric field E and a magnetic field B . |
[The Wave Nature of Light]
In the classical theory, light consists of an oscillating electric field E and magnetic field B These two are perpendicular to each other, and have the same frequency _{} and the same wave length _{} Therefore the speed of light is _{} The light speed in the vacuum is |
[The Energy of Cavity Radiation]
According to Maxwell's electromagnetism, the energy U of the electromagnetic waves (or light or radiation) in a cavity (vacuum) is written where the integration should be over the whole cavity. The electric field E and the magnetic field B can be written in the form of superpositions of plane waves. Here, let the wave vector of the plane wave be k and the wavelength be _{} Then there is a relation _{} For example, the electric field E can be expanded as where the polarization vectors _{} and _{} are unit vectors in the vertical and horizontal directions, respectively. The magnetic field is similar too. Here it should be noted that the electric and magnetic fields are superpositions of many types of waves with various magnitudes of wave numbers (wavelengths). Using this results, we can write the energy of the cavity radiation as where The equations (4) are equivalent to the equations of spring motion (or harmonic oscillator) with the mass 1. Let _{} and _{} be the coordinate and momentum of the k th harmonic oscillator, respectively. The first equation of Eqs. (4) is the relation between the velocity of the mass and the momentum, and the second equation of Eqs. (4) is Newtonian equation of motion for the harmonic oscillator. Since it is somewhat tedious to explain how to derive these results from Maxwellian electromagnetism, we omit the details here. You may be asked to accept the following conclusions: Looking at the two equations, Eqs. (3) and (4), we can say that a cavity radiation is equivalent to a set of infinite number of harmonic oscillators. |
[Quantization of
Electromagnetic Wave (Light)]
As stated above, we know that light is a set of infinite number of harmonic oscillators. As explained on the pages, 1-6: Tunnel Effect or 3-1: Energy Eigenvalues, Eigenstates, if we treat a harmonic oscillator with quantum mechanics, or if we quantize it, then the eigenvalues would be discrete and integral multiples of a unit energy quantum, which is given by This is nothing else than Planck's energy quanta. Namely, the energy of the radiation in a cavity exists in the corpuscular (or particle-like) form. Only if we deal with the classical electromagnetic waves in the quantum mechanical way, then the energy quanta appear at once. How splendid this is! |
[Interaction between Light
and Charged Particles]
Because light (electromagnetic wave) consists of electric and magnetic fields, when meeting such a charged particle as an electron, it exerts a force on the particle and there occurs an interaction between them. In order to correctly treat this interaction, we have to combine the Schroedinger equation for the charged particle and the quantum mechanics of the electromagnetic waves. The formulation for this purpose is the quantum electrodynamics. Since its contents are slightly advanced, they are omitted here. Using the theory of the quantum electrodynamics, we can completely solve all the mysteries concerning the interactions between lights and substance particles, e.g. Einstein's theory on the photoelectric effect and Compton effect discussed on the pages of the previous Seminar, 3-6: The Hypothesis of Light Quanta and the Photoelectric Effect and 3-7: Compton Effet. Thus the duality of the particle nature and the wave nature has completely been unified within quantum mechanics and all the mysteries in "the microscopic world" have been solved perfectly. |
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