JP2014021481A - Hydrogen atom model - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an educational tool that prevents educands from going away from science due to lack of a proper educational tool of science and raises their interest in quantum mechanics inclined to be biased only toward mathematical research.SOLUTION: A hydrogen atom model is configured so that supposing that a wave function of a hydrogen atom represented by polar coordinates (r, θ, φ) or parabola coordinates (ξ, η, φ) singularly, or a sum of or a difference between plural ones is regarded as a potential (electrostatic potential), that a result obtained by applying a gradient of a vector calculation and a negative sign is set to an electric field or an electric flux line 2, and that a result obtained by multiplying the same wave function by a unit vector in a θ direction or a ξ or η direction and thereafter applying turning of a vector calculation is set to an electric field or electric flux line 3, one or both of the electric fields are configured into a three-dimensional structure or drawn on a plane, or a potential in a cross section is drawn on a plane. The drawing or the model visualizes the figure of an atom, enables educands to have a close feeling toward sciences, and enables calculation of energy from the electromagnetic field, so that contribution to a science education and research can be expected.

Description

  The present invention relates to a hydrogen atom model as a science teaching material.

In quantum mechanics, which tends to run only for mathematical analysis, there are very few educational materials that can be used to elicit specific images. For example, there are FIGS. 8 and 9. FIG. 8 is an atomic model diagram of the famous de Broglie, and FIG. 9 is a diagram of FIG. 10 is a diagram showing the existence probability distribution of 2pz, 2py, and 2px trajectories from the left. Unlike these so-called classical hydrogen atom images, US Patent Publications US2010 / 0028840 A1 and US2011 / 0086333 A1 are few examples that have stepped into the internal structure. FIG. 10 shows the former FIG. 3 and is a diagram of electric lines of force 2 of the hydrogen atom 1s orbit. The latter describes a contradiction in the existence probability in the s orbit.

US Patent Publication PR 2010/10/0028840 A1 US Patent Publication US2011 / 0086333 A1

By Schiff Takeshi Inoue Quantum Mechanics Yoshioka Shoten Co-authored by Toshikatsu Saegusa and Noriaki Seto Quantum Mechanics Exercise (Schiff's Problem Description) Yoshioka Shoten Hiroshi Serizawa 3D Fractal Tour Morikita Publishing Co., Ltd.

However, the idea that the wave function is a vector potential is merely speculation, lacks persuasive power, and the shape of the electric force line 2 extending outward from a positively charged proton returns to the proton is against physical laws. It is also strange that an electric field having anisotropy in the z-axis direction is derived from a simple spherically symmetric wave function.

The present invention solves the above conventional problems, and what is the wave function of hydrogen atoms, the origin of quantum mechanics? Going back to, providing an electromagnetic field distribution as a result of elucidation, and having a concrete image, such as facilitating physical considerations of quantum mechanics that form the basis of science, etc. It is said.

In order to achieve the above-mentioned object, the present invention is a three-dimensional drawing or three-dimensional configuration using the electric field as the result of applying a vector calculation gradient to the wave function of hydrogen atoms and adding a negative sign.

The hydrogen atom model as a teaching material of the present invention is a three-dimensional drawing or three-dimensional configuration obtained by applying a gradient of a vector operation to the wave function of a hydrogen atom and adding a negative sign as an electric field. The inside of the atom can be witnessed, and it is useful for analyzing phenomena, as well as creating interest in learning science.

Solid model showing electric lines of force 2 and lines of magnetic force 3 of hydrogen atom 1s orbital according to Embodiment 1 of the present invention Solid model showing electric lines of force 2 and lines of magnetic force 3 of hydrogen atom 2s orbit of Embodiment 1 of the present invention Vertical sectional view showing electric lines of force 2 of hydrogen atom 2 pz orbit of Embodiment 2 of the present invention Vertical sectional view showing electric lines of force 2 of a hydrogen atom 2 px orbit of Embodiment 2 of the present invention Equipotential diagram of horizontal cross section of hydrogen atom 2px orbit of embodiment 3 of the present invention Stereoscopic perspective view of potential of horizontal section of hydrogen atom 2px orbit of embodiment 3 of the present invention Horizontal sectional view showing lines of magnetic force 3 of the hydrogen atom 2 px orbit of the fourth embodiment of the present invention Plan view of standing wave orbit where electrons orbit around protons in the conventional example 3D perspective view showing the existence probability distribution of 2p orbit in the conventional example Vertical sectional view showing the electric field lines 2 of the 1s track of the conventional example

Hereinafter, embodiments of the present invention will be described with reference to the drawings. The four energy functions in the wave function of hydrogen atom expressed by polar coordinates (r, θ, φ) and parabolic coordinates (ξ, η, φ) described later are compared with the wave functions that are equal to each other. Showed. The reason for limiting to four is that, in addition to covering the general features, higher-order trajectories are more complex and the features are difficult to represent. Hereinafter, the vector is represented in bold italics. a ₀ is the Bohr radius, and since the normalization coefficient is not necessary for obtaining the electromagnetic field, it is omitted.

２ｓ軌道

２ｐｚ軌道

２ｐｘ軌道

1s orbit

2s orbit

2 pz orbit

2px orbit

The result obtained by applying a gradient of vector calculation (expressed as grad or ∇ in the mathematical expression) to these and attaching a negative sign is an electric field, and a line continuously connecting the electric field directions of each point is an electric field line 2. . The 1s and 2s trajectories are a three-dimensional model, in which a part of a sphere having a constant r is cut out at a surface where θ and φ are constant, and electric force lines 2 obtained from the above formula are formed on the surface thereof, as will be described later. Adopting a combination of both lines of magnetic force 3 This is referred to as a first embodiment. Since 2pz and 2px trajectories are somewhat complicated, a distribution diagram of electric field lines 2 in a vertical section is referred to as a second embodiment. The potential is also a physical quantity and has the same meaning as the electric field. Therefore, a distribution diagram in a horizontal section is referred to as a third embodiment. A sectional view of only the magnetic force lines 3 is referred to as a fourth embodiment.

となる。ｉ_ｒはｒ方向の単位ベクトルである。
電場がｒ方向成分のみであるから、電気力線２は中心から外側に向かう放射状直線となる。
後述の如く、磁力線３は前記従来例１で開示されたものを採用、式は次式で示す。ただし、磁束と磁場の正確な区別や係数μ_０等は磁力線３に必ずしも必要でないので省略した。

ここでｉ_φはφ方向の単位ベクトルである。電磁場両者の立体的関係を示すべく、半径６センチの白色プラスッチック球４を、φ＝０およびπの垂直面５と、θ＝π／２の水平面６とで四分の一切り取り、外周の球面７を含め、これらの面上に、電気力線２または磁力線３の、面と平行な成分を描いたものである。垂直面５には電気力線２が、水平面６には電気力線２と磁力線３とが平行成分を持つので、数６数７のｒ方向の変化を反映すべく線間距離を、中心付近は密に、周囲に向かって徐々に粗とした。球面７には磁力線３が平行成分を持つので、数式中にθが含まれていない事を反映させ、等しい線間距離で描いた。半径６センチの中に、ｒ＝６ａ_０内部の電気力線２を赤色、磁力線３を緑色で描いた。FIG. 1 is a hydrogen atom model as a teaching material according to Embodiment 1 of the present invention. The electric field E obtained by applying a gradient of vector calculation to the polar coordinate wave function of the 1s orbit shown in Equation 1 and adding a negative sign is given by

It becomes. i _r is a unit vector in the r direction.
Since the electric field has only the r-direction component, the electric field lines 2 are radial straight lines from the center to the outside.
As will be described later, the magnetic field lines 3 are those disclosed in the conventional example 1, and the formula is shown by the following formula. However, the precise distinction between magnetic flux and magnetic field, the coefficient μ _{0 and the} like are omitted because they are not necessarily required for the magnetic force lines 3.

Here, _iφ is a unit vector in the φ direction. In order to show the three-dimensional relationship between the two electromagnetic fields, a white plastic sphere 4 having a radius of 6 cm is cut into quarters by a vertical plane 5 with φ = 0 and π and a horizontal plane 6 with θ = π / 2, and an outer spherical surface. The components parallel to the surface of the lines of electric force 2 or the lines of magnetic force 3 are drawn on these surfaces including 7. Since the electric force line 2 is on the vertical plane 5 and the electric force line 2 and the magnetic force line 3 are on the horizontal plane 6, the distance between the lines is set near the center to reflect the change in the r direction of several 6 Was dense and gradually rough towards the periphery. Since the magnetic force line 3 has a parallel component on the spherical surface 7, the fact that θ is not included in the mathematical formula is reflected, and the spherical surface 7 is drawn with an equal line distance. In a radius of 6 centimeters, the electric force lines 2 inside r = 6a ₀ are drawn in red and the magnetic lines 3 are drawn in green.

磁場は次式で示す。

基本的に図１と同様であるが、２ｓ軌道の電場の式がｒ＝４ａ_０でゼロになる為に、垂直、水平両面ともその付近に赤色の電気力線２が描かれていない事が異なる。FIG. 2 is a hydrogen atom model as a teaching material according to Embodiment 1 of the present invention, and represents the electromagnetic field distribution of the hydrogen atom 2s orbit. When the polar coordinate wave function of the 2s orbit shown in Equation 2 is subjected to a vector calculation gradient and attached with a negative sign, the following equation is obtained.

The magnetic field is given by

Basically the same as in FIG. 1, but since the electric field formula of the 2s orbit is zero at r = 4a ₀ , the red electric field lines 2 are not drawn in the vicinity of both vertical and horizontal surfaces. Different.

ここで、ｉ_ξおよびｉ_ηは各々ξ、η方向の単位ベクトルである。この式に前記特許文献２に開示された方法を適用してｒ＝６ａ_０の範囲の電気力線２を複数本描いた。FIG. 3 is a hydrogen atom model as a teaching material according to Embodiment 2 of the present invention, and is a diagram depicting electric force lines 2 of 2 pz orbit in an arbitrary φ constant vertical plane. When the gradient calculation is performed on the parabolic coordinate wave function u ₁₀₀ of the above-described formula 3 and a negative sign is added, the electric field E is expressed by the following equation.

Here, i _ξ and i _η are unit vectors in the ξ and η directions, respectively. A plurality of lines of electric force 2 in the range of r = 6a ₀ were drawn by applying the method disclosed in Patent Document 2 to this equation.

このままでは複雑、特徴把握困難なので、垂直断面としてφ方向ベクトル成分ゼロになるφ＝０およびπの面を選択した。図１から図３と同様にｒ＝６ａ_０の範囲を、図３と同一方法で描いた。FIG. 4 is a hydrogen atom model (drawing) as a teaching material according to Embodiment 2 of the present invention. If the gradient calculation is performed on the difference between the two wave functions represented by the polar coordinates of the above-mentioned formula 5, 2px orbit and a negative sign is added, the electric field E is expressed by the following equation.

Since this is complicated and it is difficult to grasp the features, planes with φ = 0 and π that have zero φ-direction vector components are selected as vertical sections. Similar to FIGS. 1 to 3, the range of r = 6a ₀ was drawn in the same manner as in FIG.

FIG. 5 is a hydrogen atom model (drawing) as a teaching material according to Embodiment 3 of the present invention. The electric field represented by the above equation 11 is a potential gradient, and therefore should be perpendicular to the equipotential line.
Therefore, a part of the drawing program was modified, and a second point that was separated by a small fixed distance in the direction perpendicular to the direction of the electric field obtained by substituting an arbitrary point was obtained. electric field and the direction perpendicular to the above in the same small distance are those drawn following the procedure of performing again the electric field calculation seeking third point familiar equipotential diagram of 2px orbital range of r = 6a ₀ It is.

この式を前記非特許文献３の４０−４１頁記載のプログラムに取り入れ、マイクロソフト社のＶｉｓｕａｌＣ＋＋６．０のプログラムに修正、−６ａ_０＜ｘ＜６ａ_０および−ａ_０＜ｚ＜６ａ_０の範囲を描いた２ｐｘ軌道の電位の立体透視図である。FIG. 6 is a hydrogen atom model (drawing) as a teaching material according to Embodiment 3 of the present invention. As in the case of FIG. 5 after substituting φ = 0 in the above equation 5, when expressed in rectangular coordinates (x, y, z), the following equation 12 is obtained.

This formula is incorporated into the program described in pages 40-41 of Non-Patent Document 3 and modified to Microsoft's Visual C ++ 6.0 program. The range of −6a ₀ <x <6a ₀ and −a ₀ <z <6a ₀ is set. It is a three-dimensional perspective view of the potential of the drawn 2 px orbit.

これが２ｐｘ軌道の磁場であり、図１から図３と同様にｒ＝６ａ_０の範囲を、図３と同一方法で複数本の磁力線３を描いた。FIG. 7 shows a hydrogen atom model (drawing) as a teaching material according to the fourth embodiment. Multiplied by the unit vector i _theta of theta direction to the number 5 above, by substituting θ = π / 2 after performing turning operations, get the number 13.

This is a magnetic field of 2 px orbit, and a plurality of magnetic lines 3 are drawn in the same method as FIG. 3 in the range of r = 6a ₀ as in FIGS.

What is the solution to the Schrödinger equation and the wave function before the operation and action of the teaching material configured as described above? Will be explained. The wave function is a potential (electrostatic potential). The reasons are as follows: 1) Since the electric potential is a scalar amount having the work amount as a unit, it is suitable for the solution of the Schrödinger equation which is a scalar equation that handles the work amount (energy). 2) Explain that positive and negative values appear in the wave function. 3) It is consistent with the basic atomic image that the equivalent charge of electrons is distributed around the proton. 4) Since the essence part is the same as the present existence probability theory, there is little contradiction with various existing experimental results. Etc.

電位は重畳可能であるから、ｎ個の電荷による合成電位φ_ｎが、

と定義される。この考えを、正電荷の陽子の周囲を電子の負電荷が等価的に分散分布していると仮定する水素原子に適用すれば、観測点の関数として、これら全ての正負電荷を重畳した電位が定義可能である。それが波動関数であるとの考えである。４）は、任意の点を電子が通過する頻度に着目したのが存在確率であるのに対し、電荷量に着目したのが等価分散電荷分布であり、両者は比例すると考えられる。The above 3) is simply supplemented. As a basis of electromagnetism, a potential φ is defined by the following equation at an observation point located at a distance r from a single charge amount q.

Since the potentials can be superimposed, the combined potential φ _n by n charges is

It is defined as If this idea is applied to a hydrogen atom that assumes that the negative charge of electrons is equivalently distributed around the positively charged proton, the potential that superimposes all these positive and negative charges as a function of the observation point becomes It can be defined. The idea is that it is a wave function. In 4), the existence probability focuses on the frequency of electrons passing through an arbitrary point, while the equivalent distributed charge distribution focuses on the charge amount, and both are considered to be proportional.

If the wave function of a hydrogen atom is an electric potential, an electric field can be obtained by applying a gradient of vector calculation and adding a negative sign. As shown in Equations 6 and 8, the result is very simple, and the direction of the electric field of the 1s and 2s orbits is the radial direction, which is the same spherical symmetry as the wave function itself. All the problems to be solved are solved.

2pz orbital wave functions are derived directly from the Schroedinger equation in parabolic coordinates, u ₁₀₀ (ξ, η, φ) and u ₀₁₀ (ξ, η, φ), and polar coordinate representations of equal and different sums, respectively. Is the wave function of Reasons 1) It cannot be considered that parabola coordinates and polar coordinates are superior or inferior as coordinates expressing the Schrodinger equation. 2) It is considered that the wave function directly derived from the Schrödinger equation has some stronger meaning than the wave function represented by the sum or difference of a plurality of wave functions. 3) When the wave function is re-examined in the potential theory, the directly derived wave functions u ₁₀₀ (ξ, η, φ) and u ₀₁₀ (ξ, η, φ) are the origins of ξ = 0 and η = 0. Indicates the maximum value, and can be used as it is.

The direct solutions corresponding to 2s and 2pz orbits in parabolic coordinates are u ₁₀₀ (ξ, η, φ) and u ₀₁₀ (ξ, η, φ), and the wave functions of the 2pz orbit claimed by the conventional probability amplitude theory are both The difference must be taken. For the reason 2), it can be interpreted that the Schrodinger equation suggests "wave function = potential".
If the wave function of the conventional 2 px orbit is reconsidered based on this idea, the maximum value is not reached at the origin. What is necessary is just to obtain | require the difference with the wave function of 2s orbit to make it the maximum at an origin. The result is Equation 5.

で定義される座標であり、波動関数は３つの量子数ｎ_１、ｎ_２、ｍを順に添字としてｕ_１００の様に表現される。This will be explained for reference. The fact that the wave function of hydrogen atoms is derived even in parabolic coordinates (ξ, η, φ) is introduced in a few textbooks on the market. Non-Patent Document 1 pages 110-113 and Non-Patent Document 2 pages 66-68 are described in detail. Based on these, the above formulas 1 to 5 were calculated.
According to them, parabolic coordinates (ξ, η, φ) are between rectangular coordinates (x, y, z) and polar coordinates (r, θ, φ), respectively.

The wave function is expressed as u _{100 with} the three quantum numbers n ₁ , n ₂ , and m in order as subscripts.

Next, in order to explain the magnetic field, we return to the hydrogen atom image “negatively charged electrons exist as standing waves around the positively charged proton located at the center”. There are two ways to form a standing wave, one is formed by two traveling waves that are opposite to each other, and the other is, as seen in FIG. It is formed by catching up with. In any case, a standing wave appears on top of a moving wave. Even as a wave, if the charge is moving, it is a current. The idea that the instantaneous state of the current is a standing wave and the potential (electrostatic potential) formed by the standing wave is a wave function is the basis of the present invention.

となる。速度ｖの方向は電子の軌道方向であろうが、電子の軌道そのものは不明である。
しかし少なくともｓ軌道では、統計上の平均的軌道を陽子を中心とする円と考えて差し支えなかろう。さすれば電流の方向は大円方向であるθ方向が最適である。ベクトル速度ｖをｖｉ_θと変更すれば上式は次の様になる。

このベクトルポテンシャルを転回演算すれば磁束が得られ、μ_０で割れば磁場となる。前記特許文献１の「波動関数＝ベクトルポテンシャル」説に根拠が与えられた事になる。This is expressed mathematically. Wherein the aforementioned potential, and each of the charge q _i of Equation 13 is moving in the same vector velocity v, becomes the vector potential A _n Changing the coefficients. Furthermore, when the relationship with the potential φ _n is seen by taking v out of the sigma,

It becomes. The direction of velocity v will be the direction of the electron trajectory, but the electron trajectory itself is unknown.
However, at least in the s orbit, the statistical average orbit can be considered as a circle centered on protons. In this case, the direction of the current is optimal in the θ direction, which is a great circle direction. If the vector velocity v is changed to vi _θ , the above equation becomes as follows.

A magnetic flux is obtained by rotating this vector potential, and a magnetic field is obtained by dividing by μ ₀ . The basis is given to the “wave function = vector potential” theory of Patent Document 1.

However, the θ direction is the south direction compared to the earth, and the east-west direction and the vertical direction are all excluded. Even in the s orbit, the east-west direction cannot be denied. Furthermore, in other orbits, as estimated from FIG. 9, it is considered that the ellipse orbits the sphere symmetrically arranged, so the up and down direction should be considered. In any case, the r direction component essential for directions other than θ, particularly for an ellipse, should be considered. However, unfortunately, if the vector potential in the r direction is rotated, it becomes zero.

となる。数７には無いθ方向の変化はあるものの、ｒ方向の変化は類似であり、何よりもφ方向成分のみの関数である点で数７と同一である。ｒ方向成分を多分に有するξ方向の電流からもφ方向成分のみの磁場が得られたので、１ｓ軌道の水素原子の磁場はφ方向成分のみを有すると判断できる。従って単純な極座標表示の波動関数を採用したものである。Therefore, parabolic coordinates are handled again. When the wave function u ₀₀₀ (ξ, η, φ) of the 1 s orbit is multiplied by the unit vector i _ξ in the ξ direction, the rotation calculation is performed, and the result is the magnetic field H.

It becomes. Although there is a change in the θ direction that is not in Equation 7, the change in the r direction is similar and is the same as Equation 7 in that it is a function of only the φ direction component. Since the magnetic field of only the φ direction component is obtained from the current in the ξ direction having a large r direction component, it can be determined that the magnetic field of hydrogen atoms in the 1s orbit has only the φ direction component. Therefore, a simple polar coordinate wave function is employed.

Move on to action. According to the present invention, so-called visualization of the inside of a hydrogen atom is realized, and it can be understood how the difference in the wave function is reflected in the difference in the internal electromagnetic field. For example 1s orbital is concentrated in the vicinity of the center in the field both, 2s orbital is expanded outward from the 1s orbital, the electric field at r = 2a ₀ is such that is zero. Further, an equivalent dispersion distribution of negative charges, which is not mentioned in Patent Document 1, can be estimated. According to the basic knowledge of electromagnetism, the electric force line 2 is a line segment starting from a positive charge and connecting the direction of the electric field to the negative charge. For example, from FIG. 3 and Equation 10, ξ = 6a in a 2 pz orbit. If the points of ₀ and η = 0 and polar coordinates are displayed, it can be estimated that negative charges are concentrated around r = 3a ₀ and θ = π. From FIG. 2 and Equation 8, it can be inferred that negative charges are distributed around the spherical surface of r = 4a _{0 in} the 2s orbit. From FIG. 1 and Equation 6, it can be inferred that negative charges are concentrated in the vicinity of the proton in the 1s orbit. From FIG. 4 and Equation 11, it can be inferred that negative charges are concentrated around φ = 0, θ = π / 2, and r = 3a ₀ in the 2px orbit.

Furthermore, the electric field energy not calculated in Patent Document 1 can be calculated from the electric field formula obtained by the present invention. If we square the ns orbital electric field equation (including the normalization factor) and integrate it over the whole space, the energy is 1 / (a ₀ ² n ² ), which is proportional to Bohr's energy level. The magnetic field energy calculated in Patent Document 1 also coincides. This result will support the lossless resonator theory claimed by the same document, and will support the wave function = potential.

As described above, the effect of the present invention is not only the science and models that show the internal structure of the atoms that could not be known in the past, but especially the science of quantum mechanics to stop the separation from science, not only to mathematics. It helps to prevent errors that may occur due to biased reasoning. In addition, the energy level of hydrogen atoms can be calculated from the electromagnetic field distribution, which is expected to contribute to quantum mechanics education.

The hydrogen atom model as a teaching material according to the present invention visualizes the inside of the hydrogen atom, makes the quantum mechanics difficult and difficult to attach, prevents the separation of science, and is useful for education and research.

2 electric field lines 3 magnetic field lines 4 sphere 5 vertical surface 6 horizontal surface 7 spherical surface (sphere surface)

Claims (3)

  A hydrogen atom model in which the wave function of a hydrogen atom alone or the sum of a plurality of wave functions, or the difference between sums and differences, and the result of adding a negative sign to the result of adding a negative sign to the plane as an electric field.   Hydrogen atom model in which the wave function of hydrogen atom alone or the sum or difference of multiple wave functions or the sum and difference are drawn as potential (electrostatic potential).   When the wave function is expressed in polar coordinates, the wave function of hydrogen atom or the sum or difference of multiple wave functions or the sum and difference of the wave functions, and the result of applying a gradient of vector operation and adding a negative sign to the electric field. If the unit vector is expressed in parabolic coordinates, the unit vector in the ξ direction or η direction is multiplied by the wave function, and the result of rotation is applied to the magnetic field. .

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