Patent Application
20100028840
This application is based on Japanese patent application serial No. 2008-215948, filed in Japan Patent Office on Jul. 30, 2008, and No. 2008-305468, filed in Japan Patent Office on Nov. 4, 2008. The contents of these two applications are hereby incorporated by reference.
1. Field of the Invention
The present invention relates to an educational tool useful for education of sciences, in particular quantum physics.
2. Description of Related Art
It has been warned for a long time that younger people are away from sciences. However, it cannot be said that an effective measure has yet been taken. Although liberal arts handle familiar and approachable subjects, and therefore can easily be approached, sciences have become more and more difficult to understand and approach. Especially, it seems appropriate to say that textbooks on quantum mechanics, which is the most basic one of all sciences, almost describe an advanced mathematics rather than the sciences. The sciences in general, and physics as a typical example thereof, are academics that comprehend and explain how material, fluid and electricity function as concretely as possible. It is because mathematics makes expressions more concise or easier to understand than words and sentences that the sciences use mathematics. Concrete images should be main players, and mathematical expressions should be backseat ones.
For example, the solution of a differential equation is handled in such a manner that terms that diverge to an infinite value are thrown away, only terms that converge with a finite value are left, and among even and odd functions emerging in the solution, only the even functions are left and the odd ones are thrown away if a concrete system to be analyzed is symmetrical. Thus, a concrete image takes priority over the result of mathematical operation. Further, a concrete image makes the theory easy to approach and interesting.
However, in quantum mechanics, we can only see, as such concrete images, spheres showing the distribution of probability density shown for example in FIG. 3.22 of a non-patent document 1, which is captioned with “Boundary surfaces for p-and d-orbitals” and is shown in
The non-patent document 1 is “Molecular Quantum Mechanics” written by Peter Atkins and Ronald Friedman and published by Oxford University Press. The non-patent document 2 is “Quantum Mechanics II” written by Shin-ichiro Tomonaga and published by Misuzu Shobo (Japan).
However, the three pairs of bisected shallow spheres shown at the top of
It is therefore an object of the present invention to solve the above-mentioned conventional problem, and to provide the distribution of an electromagnetic field of a quantum physical system, such as a hydrogen atom which is the origin of quantum mechanics, to thereby enable an educand to have a concrete image, infer an effect of an applied external magnetic field, and accordingly feel familiar with quantum mechanics, which is the most basic one of the sciences.
One aspect of the present invention is directed to an educational tool. The educational tool according to the aspect of the present invention comprises a drawing or three-dimensional model expressing a magnetic field obtained as a result of a rotational vector operation applied to a vector potential. The rotational vector operation is a well known vector operation that applies a rotational operator “rot” to a vector. The vector potential is here a wave function of quantum mechanics regarded as a vector potential only having a component in a θ-direction in polar coordinates (r, θ, φ).
The magnetic field provided by the educational tool visualizes the figure of an atom, i.e., a constitutional unit of material, generates interest in learning sciences among educands, and enables the educands to easily understand that difference in a wave function results in difference in the distribution of a magnetic field and results in difference in the effects of the application of an external magnetic field.
Thus, the educational tool of the present invention visualizes the figure of an atom constituting material, is useful for a phenomenon analysis, and also can generate interest in learning sciences among educands.
These and other objects, features, aspects and advantages of the present invention will become more apparent from the following detailed description of the present invention when taken in conjunction with the accompanying drawings.
Hereinafter, preferred embodiments of the present invention will be described with reference to the accompanying drawings. For the simplicity of explanation and understanding, a hydrogen atom will be shown as an example. Therefore, an atomic number Z will be replaced with 1 in wave functions.
(1/π)1/2(1/a0)3/2 exp(−r/a0)
If this formula is regarded as a vector potential having a component only in a θ-direction of polar coordinates (r, θ, φ), the is orbital can be expressed as follows, using iθ as a unit vector in the θ-direction.
(1/π)1/2(1/a0)3/2 exp(−r/a0)iθ
Once a rotational operator in a vector space is applied to this formula, a magnetic field expressed as follows can be obtained.
(1/π)1/2(1/a0)3/2 exp(−r/a0)(1/r−1/a0)iφ
Here, iφ is a unit vector in a φ-direction.
In polar coordinates, since a set of points having a constant value of radius r constitutes a spherical surface having a radius r, a magnetic line of force formed by a magnetic field having a component only in the φ-direction on the spherical surface (i.e., a curve whose tangential line coincides with the magnetic field in direction at any point of the curve) corresponds to a latitude line of the earth. Although a magnetic field can be calculated for any value of r in principle,
( 1/32π)1/2(1/a0)3/2 exp(−r/2a0)(2·r/2a0)cos θ iφ
Large difference from the 1s orbital shown in
(1/π)1/2(1/a0)3/2 exp(−r/a0){(1/r2−1/a0r)cot θ ir+(2/a0r−1/a02)iθ}
Although this formula divided by jωe results in a normal electric field defined by Maxwell's electromagnetic equation, the result is not shown here because of unnecessity for a drawing or model.
Since the ir is a unit vector in an r-direction, the electric field shown by the formula has components both in the θ and r-directions.
( 1/32π)1/2(1/a0)3/2 cos φ exp(−r/2a0){(4/r−1/a0)cos θ ir+(1/r)/sin θ iθ+(2/a0−2/r−r/4a02)sin θ iθ}
( 1/32π)1/2(1/a0)3/2 exp(−r/2a0){(2−r/2a0)sin θ cos φ iφ+sin φ ir}
Hereinafter, the operation and function of the educational tools configured as stated above will be described. When solving an electromagnetic equation, in general, use of a vector potential can reduce the number of unknown variables, and enables us to solve the equation and calculate the electric and magnetic fields, even if it is hard to directly obtain the solution of the electric and magnetic fields because of the presence of six unknown variables. This method of solving the equation by use of the vector potential is well known among electrical engineers. In the case of a microwave transmission line, such as a waveguide, and a microwave resonator having a transmission line with its inlet and outlet closed, the vector potential is handled as a vector having a component only in a traveling direction.
If a hydrogen atom is compared to a resonator, the traveling wave can be assumed to revolve along a great circle by analogy with an image of an electron revolving around a nucleus. Since only θ is in the direction of the great circle among the polar coordinates (r, θ, φ), the magnetic and thereafter the electric fields were obtained by regarding the wave function as a vector potential having a component only in the θ-direction. However, regarding the wave function as the vector potential might be blamed for blaspheming against quantum mechanics, and therefore, the explanation will hereinafter be given.
A textbook on quantum mechanics says that a wave function represents the existing probability of an electron.
In contrast,
This is still an inconsistency or a problem to be solved. The present invention provides means for solving the problem. Even if it is not an inconsistency, an electron revolves around a nucleus as a wave motion. Therefore, clarifying a profile, an amplitude, and a direction of polarization of the wave motion will be helpful in research and education.
Regarding a hydrogen atom as an electromagnetic resonator will resolve a further large question. It is said that a microwave oven, which is the best known and popular electromagnetic resonator, was invented on the basis of the fact that chocolate was melted in front of a radar transmitter. On the other hand, it is well known among those skilled in the art that a fly keeps flying inside the operating heat chamber of the microwave oven. Its technological expression is that an output load variation coefficient is reduced drastically at a light load. For example, a microwave oven that raises the temperature of 2 liters of water by 10 degrees centigrade by 2 minutes heating is supposed to raise the temperature of 1 liter of water by 20 degrees and 500 milliliters by 40 degrees, because heating for the same length of time is supposed to produce the same amount of heat. However, the temperature increase is reduced with a decrease in a load, and is almost zero at a load of 1 milliliter of water. A fly is a lighter load than the 1 milliliter of water. The fly is, as a matter of fact, exposed to heat from the wall surface of the heat chamber and the like for 2 minutes, and therefore, the temperature is raised to some extent.
In accordance with Japanese Unexamined Patent Publication No. 2004-184031, which discloses the fundamentals of heating in a microwave oven, the heating obeys an equation for a pointing vector. The following formula is the integral expression thereof
If the domain of integration is defined as the whole food to be heated, the left-hand side of the formula, which is the negative whole surface integration of an inner product of a pointing vector (E×H) and a normal vector n on the surface S of the food, expresses an inflow energy, and the right-hand side expresses the behavior of the inflow energy in the volume V inside the surface S. The first term in the right-hand side shows an energy loss, i.e. Joule heat, inside the food, and the second one shows a stored energy. Even though the mechanism of an energy loss exists, no energy flows in and no energy loss is caused without the pointing vector pointing inward.
Radar is a traveling wave, and the pointing vector (E×H) thereof points in the traveling direction. Chocolate placed on the traveling path thereof would be heated by the pointing vector pointing inward. In contrast, inside the resonator, the pointing vector rotates in synchronization with a frequency because of the superposition of the traveling and reflected waves thereof, and therefore, the integration value within one period is zero. In another expression, the traveling and reflected waves of the pointing vector are identical in amplitude and reverse in direction, and therefore, the superposition thereof is zero. Any way, if the left-hand side of the above-described formula is zero, the right-hand side will also be zero, and therefore, there occurs no heating.
Just for reference, when a largish object to be heated, such as food, is placed in a microwave oven, an electric field E causes a displacement current flowing in a dielectric and a conduction current flowing in a conductor in accordance with Ampere's law “rotH=σE+jωεE.” These currents form a magnetic field H. This magnetic field H, together with the electric field E, generates a pointing vector (E×H) which points to the inside of the object, and heats the same.
The foregoing lengthy explanation can be summarized as follows. It can be stated that a traveling wave is in a state where it loses energy, and an electromagnetic wave within a resonator, which is a standing wave, is in a state where it loses no energy, although the two are the same electromagnetic wave.
Quantum mechanics has a basic principle that a particle, such as an electron, has a nature of wave as a photon is related to an electromagnetic wave. It is therefore undoubtedly natural to have a view that a wave motion of a particle has the nature of energy conservation similar to that of the above-stated electromagnetic wave. Therefore, it cannot be thought blasphemous to attribute no photon emission and no energy loss of an electron revolving around a nucleus of a hydrogen atom even with acceleration to the nature of a standing wave in the above-stated resonator.
Without this sort of explanation, an idea that the electron moving with acceleration loses its energy by emitting light and falls to the nucleus moving from one discrete orbital to another is rather close to the common sense of an ordinary person. It is further a question why only an electron being at the lowest energy level on the 1s orbital can remain stable without falling toward the nucleus although an electron on any other orbital than the 1s jumps to another orbital lower in energy level when losing its energy, even if the electron does not emit light due to its accelerated motion. An answer that the wave equation does not allow the idea of an electron falling toward the nucleus would not be physics but mathematics.
Furthermore, the proposed “hydrogen atom resonator theory” can be expected to provide another effect or function. In general, one of magnetic and electric fields varies sinusoidally but the other varies cosinusoidally inside the resonator, and therefore, the total energy of the two fields always maintains a constant value. As a result, if the distribution of one of the two fields is found, the total energy can be calculated. The whole space integration of the square value of the magnetic field of each orbital is partly shown below.
The energy value of each s orbital is (1/a02)/n2, where n is a principal quantum number. Multiplying this value by a coefficient (−h2/2me), which frequently appears in quantum mechanics, results in (−h2/2me)/a02/n2=−mee4/(8ε2h2)/n2 according to a0=εh2/(πmee2). This expression coincides with an energy level formulated by Bohr.
As the last part of the explanation on the functions or actions, the velocities of both the electromagnetic wave and the electron present inside the resonator will be described. Since an electron cannot move at the velocity of light according to common sense, it can be thought that the electromagnetic wave revolves an integer number of times when the electron makes one revolution, and the two synchronize with each other.
However, since an energy exchange between the electron and the electromagnetic wave remains as a problem, an idea that an electron revolves as a wave motion at the velocity of light inside the hydrogen atom resonator is rather favorable. The central idea of the theory of special relativity is that mass reaches an infinite large value at the velocity of light. Since the definition of inertial mass is the amount of acceleration when an external force is applied, it can be said that the mass does not need to be or cannot be considered under a state where the energy exchange with the exterior is not or cannot be performed. This idea is that an electron comes to have the velocity of light upon transiting into the state of resonance.
Here, description will return to the main subject. The preferred embodiments of the present invention advantageously raise educands' interest in sciences and quantum mechanics with the atomic models or the atomic structure drawings and help to prevent them from going away from sciences, and furthermore, enable them to easily understand the relation of a hydrogen atom with its external magnetic field. Since the distributions of the magnetic fields shown in
While the invention has been shown and described in detail, the foregoing description is in all aspects illustrative and not restrictive. It is therefore understood that numerous modifications and variations can be devised without departing from the scope of the invention.
As described above, the educational tool in accordance with the present invention visualizes the figure of a quantum physical system, such as a hydrogen atom, enables educands to have a close feeling toward hardly understood or approached sciences and quantum mechanics, prevents them from going away from sciences, enables them to have an image of phenomenon occurring under the application of an external static magnetic field, and is therefore useful for education and research.
| Number | Date | Country | Kind |
|---|---|---|---|
| 2008-215948 | Jul 2008 | JP | national |
| 2008-305468 | Nov 2008 | JP | national |
