Youhei Tsubono
Setagaya-ku Tokyo 155-0033, Japan
URL: http://www7b.biglobe.ne.jp/~kcy05t/index.html
♦ Japanese version ♦
New Bohr's worlds (atoms, molecules,bonds.) (11/7/23 new!)
♦ End of Quantum mechanics ! ( Fine structure, Sommerfeld, General relativity ) (12/ 4/9 new!)
♦ Bohr model vs. nonlocality (Bell inequality violation, Leggett inequality,Teleportation) (11/ 11/17 new!)
♦ End of Quantum field theory (QFT). (Dirac equation, Quantum electrodynamics, Higgs ) (11/ 12/25 new!)
In the hydrogen atom, the Bohr-Sommerfeld model completely agrees with the experimental data including the fine structure.
But it was impossible to express the three-body Helium atom by the Bohr model in 1920's due to lack of computers.
Here we show that Bohr's model-based methods can calculate the experimental value (-79.005 eV) of Helium ground state energy correctly.
The standard Helium model has the spin-up and spin-down electrons, so it seems to generate no magnetic fields.
But to be precise, the magnetic fields are produced in almost all space, because the two electrons stay apart from each other by the repulsive Coulomb force.
If they move to cancel the magnetic fields out, the (electro)magnetic field changes, and they radiate the electromagnetic waves.
So the standard Helium model of the quantum mechanics contains the self-contradiction.
Here we suppose the orbital planes of the two electrons are perpendicular to each other.
By a computational method, we calculate the Coulomb force among the particles, and the number of de Broglie's waves contained in the short segment at short time intervals.
Our results demonstrate that two electrons of Helium are actually moving on the orbits of just one de Broglie's wavelength.
The two orbits are symmetrical, crossing perpendicularly, and wrapping the whole Helium atom beutifully, which can explain the strong stability and the closed shell property of Helium due to the de Broglie's wave nature.
If you want to know correct and new Bohr model Helium right now, please proceed to this section.
In 1913, Niels Bohr postulates the Bohr's model which agreed with the observed hydrogen spectrum [1].
Later Sommerfeld developed his theory to explain the fine structure completely [2].
His fine structure values accidentally coincided with the Dirac hydrogen model which uses the new idea of the spin-orbital interaction [3].(See also this page)
In 1923 Louis de Broglie suggested that electrons might have wave aspect and its wavelength λ is equal to h/mv, where h is Plank's constant (= 6.62606896 x 10-34 Js) and m is the electron mass (me= 9.1093826 x 10-31 kg).
λ = h / mv = h / p
where p = mv means the momentum of the particle.
In 1927, Davisson and Germer experimentally confirmed de Broglie's hypothesis in the interference experiment [4].
Recently the results of the two-slit experiment of an electron showed its wavelike properties [5].
About the "photons", see the page Photons particles or waves?
The usual textbooks often say that the Bohr model is wrong because the electron is moving on the circular orbit and radiating energy by the "accelerated" charge.
This explanation about the Bohr model is not correct. (See also this page.)
In the Bohr model, when the orbital length is an integer times the de Broglie's wavelength, it doesn't radiate energy.
The Bohr-Sommerfeld theory includes not only the Maxwell's law (equations) but also the de Broglie's theory.
[ Quantum mechanical model is wrong, and Bohr model is stable. ]
On the other hand, the quantum mechanical standard model contains self-contradiction about the reason why the electrons don't fall into the nucleus.
They say that the quantum mechanical electrons are static as electron clouds obeying the probability density, so they are not accelerated.
If the quantum-mechanical (QM) electrons are static and aren't actually moving, how can we explain about the following phenomena ?
If we consider the atomic nuclear movement around the center of mass, and use the reduced mass of an electron, the calculation result of the energy levels (for example, of the hydrogen atom) becomes better.
This means the atominc nucleus is "actually" moving around the center of mass. And of course this means the electron is moving around the center of mass, too.
How can the relativistic effect be explained ?
( The effect of the relativistic mass change by the high speed electron was actually observed especially in the heavy atoms).
Even if the electrons are static (= not moving), the relativistic mass change can occur ??
As a result, the quantum mechanics contains self-contradiction and Bohr model is very "natural" model.
Actually the atoms or molecules in the air (in front of you) is moving and stable.
So we can not think about the electron's stability without the concept of an integer times de Broglie's wavelength.
As one electron, de Brolgie's wave is produced, and also as a whole atom which includes an electron and a proton, de Broglie's wave is produced, too.
It is quite natural that we think as a means of transport, de Broglie's waves are produced.
How can the quantum-mechanical (QM) model explain about various stable atoms (like helium) or molecules actually flying in the air in front of you ?
It is impossible in QM model, because they are actually moving and "accelerated".
As shown in this page, we can express de Broglie's waves by the classical mechanical methods correctly.
Many textbooks or websites often say that Bohr model is wrong, because it can not explain about various phenomena.
But these comments misunderstand the history of the physics.
(Now in several phenomena, the Bohr's model is known to provide good accuracy ([6], [7], [8]).
For example, Bohr model provides reasonable estimates about the Stark effect.
But under high field strengths, Bohr model is said to be different from the experimental results.
These explanations in the textbooks or websites is speaking about the old Bohr model in 1920's.
[ Three-body problems. ]
The important point is that the analytical calculation in Bohr model is possibble only in a atom which has only one electron and nucleus !
Under high electric field, we have to consider the "complex" changes of the electric field in addition to the hydrogen's nucleus.
This means that this condition is a three (or more) - body problem.
So we have to use computers to solve these problems.
(Of course, there are no computers in 1920's !)
This is the reason why we had to accept the very strange quantum mechanical worlds such as many worlds and entanglement instead of "real" Bohr model.
[ Many Worlds = Quantum-mechanics. ]
The same thing can be said about the condition under the complex electric and magnetic fields or multi-electron atoms, metals and molecules.
But as shown in this page, we have succeeded in visualizing various atoms by new Bohr model using simple computer programs.
And using Virial theorem, we can express various Bohr's molecules such as water and ethylene correctly.
It is said that Bohr model doesn't satisfy the anomalous Zeeman effect of multi-electron atoms such as sodium.
Of course, it is a story when we didn't have computers.
And the standard interpretation of the anomalalous Zeeman effect is very unnatural.
The 3s electron of the sodium D line comes very close to the nucleus through the inner electrons.
But the Lande-g-factor of the standard quantum mechanics doesn't contain the influence of the inner electrons at all.
The delicate spin and orbital precession is so stable to withstand the strong influence of the inner electrons ?
This is very strange.
As shown in this page, it is natural that we think the anomalous Zeeman effects is caused by the inner electrons rather than the "strange" spin.
The hydrogen atom shows the normal Zeeman effect, and the Lithium tends to show the Paschen-Back effect.
So the anomalous Zeeman effect was studied mainly using the multi-electron atoms such as the sodium and magnesium.
Of course, also in Bohr-Sommerfeld hydrogen atom, the fine structure between 2s and 2p levels exists. (See Truth of fine structure constant.)
So if we can apply very weak magnetic field ( less than fine structure's level ), the even-numbered splitted lines may be observed also in the hydrogen atom.
But as shown in this page, this fine structure of Bohr-Sommerfeld model is NOT caused by the electron spin.
As the fine structure of the hydrogen atom is very small (about 0.00005 eV), it is impossible that we confirm the Lande g-factor in the hydrogen atom. (See also this page.)
One electron is very light and small.
So by equating the angular momentum of the spinning sphere of the electron to 1/2 ħ, the sphere speed leads to more than one hundred times the speed of light ! ([10])
This is why Pauli strongly objected to the existence of "electron spin" in 1920's.
(About the detailed explanation, see this page)
[ Electron spin is an illusion. ]
And the fermion (such as spinning electron) doesn't go back to its original configuration by one rotation (= 360 degrees = 2π ).
By two rotations (= 720 degrees), it returns. This is called "two-valued" or "orientation entanglement".
Surprisingly, this strange phenomenon was actually observed in the experiment in 1975. ( [11] )
But it is quite natural that things in this real world return to their original states by one rotation.
As shown in this page, if we use Bohr model instead of the quantum mechanical model, the interpretation of this experiment changes to the very natural one in which the fermions return by 2 π rotation !
This is a clear evidence that the Bohr model is correct and the quantum mechanical spin is an illusion.
In the Bohr-Sommerfeld model, when the orbital length is a integer times the de Broglie's wavelength, the electron's motion is stable.
Quantum mechanics neglects this fact, and denies the Bohr-Sommerfeld theory.
Surprisingly, also in the Schrodinger equation, the orbital length is a integer times the de Broglie's wavelength !
And this is why the Schrodinger equation gives the same energy levels as the Bohr-Sommerfeld model.
In this page, we have proved this fact "mathematically".
But the Schrodinger equation has an serious problem.
If we cut the Schrodinger wave function on some borders, the differential coefficient becomes infinite.
So the region of the electron's probability density must be from zero to infinity in the "radial" direction.
But if so, the "radial" kinetic energy (= Tr) becomes minus in some regions ( 1/2 mv2 < 0 ) !
This is the reason why the Schrodinger equation is wrong !
(This strange phenomenon is different from the "quantum tunnelling".)
Also in the quantum mechanics, the particle we can find must have the "positive" kinetic energy ( 1/2 mv2 > 0 ).
This is quite natural. So the Schrodinger equation includes "self-contradiction".
(The Bohr model, which contains the forces of de Broglie's waves, can explain the phenomenon of "quantum tunnelling", too.)
[ Schrodinger's 2P "radial" wave function (angular momentum = 1). ]
As shown in the upper figure, the 2P "radial" wave function ( χ = rR21 ) contains the regions of the minus kinetic energy ( r < a1, a2 < r ).
When the r is bigger than a2, the Coulomb potential energy becomes higher than some maximum value.
In this region, to keep the total energy (E < 0) constant, the "radial" kinetic energy must be minus !
And in the region of r < a1, the "tangential" kinetic energy increases at the inverse square of the radius.
To cancel this "rapid" increase in the tangential kinetic energy, the "radial" kinetic energy becomes minus ! (See this page.)
So this is also a clear evidence that the Schrodinger equation is wrong.
The helium atom has the two electros and the +2e nucleus.
The three-body calculation like the helium was much more difficult than the two-body hydrogen atom.
This is a reason why Bohr model could not give the correct energy value about the helium atom in 1920's.
If the helium structure has not been defined, the development of all the physics and chemistry would have stopped at that point.
On the other hand, in 1928-1930, Hylleraas succeeded in getting the approximate value of the helium ground state energy using the Schroedinger equation.
Now the latest calculation value of the helium ground state energy is about -79.015 eV [9].
But it is a little different from the experimental value -79.005147 eV (Nist Data), because the helium of the Schroedinger equation can't calculate the effect of the nuclear movement correctly.
The advantage of the quantum mechanics is to be able to calculate the values "approximately", even if there are no computers (for example, in 1920's).
(Though in the many-electron atoms and molecules, the quantum mechanical methods are "unrealistically" difficult. )
On the other hand, the Bohr model must find the concrete orbits from the beginning.
So the Bohr model must use the computers except in the hydrogen atom.
This fact caused the various strange illusions such as the many-worlds, collapse, entanglement, uncertainty principle, virtual particles and electron spin.
The quantum mechanics has more serious problems which are explained in detail in the latter part of this paper.
So we try to go back to the Bohr model and solve those illusions of the quantum mechanics.
First we try the helium model using the Bohr's theory based methods.
In (Fig.1), the two electrons of the helium are on the opposite sides of the nucleus and moving on the same circular orbit.
(Fig. 1)
Two electrons are moving on the opposite sides of the nucleus on the same one orbit.
Equating the centrifugal force to the Coulomb force, we have
(Eq. 1)
where r is the circular orbital radius (meter), e is the electron charge ( e = 1.60217653 × 10-19 C, e > 0 ), and ε is the permittivity of vacuum (= 8.854187817 × 10-12 C2/Nm2).
The circular orbital length is supposed to be an integer (n) times de Broglie's wavelength of the electron, we have
(Eq. 2)
where h/mv means the de Broglie's wavelength.
The total energy E of the helium is the sum of the kinetic and the Coulomb potential energy of the two electrons, so
(Eq.3)
Solving the above three equations (Eq.1-3), the ground state energy (n=1) becomes - 83.33 eV.
This value is lower than the experimental value -79.005 eV.
In this model, the two electrons are on the same one orbit of one de Broglie's wavelength.
But if the two electrons can be in one small orbit, this means that the ground state electron of the Bohr hydrogen model can come closer to the nucleus than the original orbit.
And in (Fig.1) orbit, the two electrons are just at the opposite positions, so the wave phases of them interfere with each other and vanish.
Because in the Fig.1 model, the orbit is circular (not elliptical), which is confirmed by the motion's equation of Eq.1.
(As shown in this page, we can express de Broglie's waves by the classical mechanical methods correctly.)
To avoid the problems of vanishing de Broglie's wave in the upper section, we suppose another model as shown in (Fig.2) and (Fig.3).
In this model, the electron 1 moves on the X-Y plane, the electron 2 moves on the X-Z plane.
So only the x-direction is common.
As the two orbits are symmetrical and same-shaped, any points on the electron 1 orbit are at the same distance from the points on the both-side (± z) electron 2 orbit.
The both-side (± z) electron 2 orbit have the opposite ± x-directions.
On the electron 1 orbit, the x-direction de Bloglie's waves of the electron 2 interfere with themselves and vanish.
So the wave of electron 1 would not be affected by the wave of electron 2.
The same thing can be said on the electron 2 orbit.
(Fig. 2) Two same-shaped orbital planes are perpendicular to each other.
Fig.2 shows one quarter of the orbitals.
Electron 1 starts at (r1, 0, 0), while electron 2 starts at (-r1, 0, 0).
The two orbits are avoiding each other by the "stable" de Brolgie's waves.
(Fig. 3) The two electrons have moved one quarter of their orbitals.
In Fig.3, the electron 1 is crossing y axis perpendicularly, while electron 2 is crossing z axis.
When the two orbits are crossing perpendicularly, the motion pattern as shown in Fig. 2 and Fig. 3 is the most stable one.
( If the electrons are moving like Fig. 2 and 3, the potential energy becomes the lowest ).
The state in which the two electrons are the closest to each other is Fig. 3. ( = when the two electrons are passing each other. )
If this closest state is not Fig. 3, the two electrons come closer to each other than Fig. 3., which makes the potential energy higher by the repulsive Coulomb force.
Here we investigate how the electrons of the helium are moving by calculating the Coulomb force among the two electrons and the nucleus at short time intervals.
The computer program (class filename: MathMethod) written in the JAVA language (version 1.5.0) to compute the electron orbit of the helium is shown in the link below.
new sample JAVA program 1 SS = 1 × 10-25 sec (See Eq.4.)
old sample JAVA program 1 SS = 1 × 10-22 sec--fast but the result and Eq.no are a little different
In the new sample JAVA program, we divide the orbit into more than one hundred million short segments for the calculation.
So the calculation takes a little time. (About 1 minute per calculation.)
But of course, the calculation result is the most accurate.
(For example, input the values 3074 (=r1) and 79.0035 (=|E|) of Table 1 after running the program, which gives 1/4 WN = 0.25000000 as de Broglie's waves)
In the old sample JAVA program, the calculation speed is very fast, in which we divide the orbit into about one hundred thousand short segments.
But the calculation results are a little different from Table.1. ( Though they are almost same.)
(For example, input the values 3074 (=r1) and 79.005 (=|E|) after running the program, which gives 1/4 WN = 0.25000 as de Broglie's waves)
As shown in (Fig.2) and (Fig.3), the helium nucleus is at the origin.
The electron 1 initially at (r1, 0, 0) (Fig.2) moves one quarter of its orbital to (0, r2, 0) (Fig.3), while the electron 2 initially at (-r1, 0, 0) moves to (0, 0, r2).
As meter and second are rather large units for measurement of atomic behavior, here we use new convenient units MM
(1 MM = 1 × 10-14 meter), SS (1 SS = 1 × 10-25 second) and MM/SS (1 MM/SS = 1 × 10-14 meter/ 1 × 10-25 second = 1 × 1011 meter/second) (Eq.4).
(Eq. 4)
In this program, we first input the initial x-coordinate r1 (in MM) of the electron 1, and the absolute value of the total energy E (in eV) of the helium.
From the inputted value, we calculate the initial velocity of the electron.
And at intervals of 1 SS, we compute the Coulomb force among the two electrons and the nucleus.
When the electron 1 is at (xx, yy, 0), the electron 2 is at (-xx, 0, yy) (in MM). (See Fig.2, 3.)
Change MM to meter as follows; x (m) = xx × 10-14. y (m) = yy × 10-14.
So the x component of the acceleration (m/sec2) of the electron 1 is,
(Eq. 5) x component of the acceleration.
where the first term is by the Coulomb force between the nucleus and the electron 1, and the second term is by the force between the two electrons.
Considering the helium nuclear mass (= alpha particle), we use here the reduced mass ( rm =1/2 × (2me × nucleus)/(2me + nucleus) = 9.10688561 × 10-31 kg. See Eq.6.) except when the center of mass is at the origin.
(Eq. 6)
Here the two electrons has completely the same mass and charge.
So we can assume one virtual particle of 2 × me at the center of the two electrons.
The center of the two electrons' charges agrees with this virtual particle's position, too. (Though its effective charge which influences the nucleus is changing with time.)
This means that only the force F(t) between this virtual particle and the nucleus affect their motions.
When we fix the motion of the nucleus, we can use the above reduced mass equation.
(See also reduced mass of three-body helium.)
In the same way, the y component of the acceleration (m/sec2) is,
(Eq. 7) y component of the acceleration.
Change m/sec2 to MM/SS2 using the next relation,
(Eq. 8)
Based on that calculation value, we change the velocity vector and the position of the electrons.
We suppose electron 1 moves only on the XY-plane, so the z component of the acceleration of the electron 1 is not considered.
If we consider all components of the Coulomb force against the electrons, the electron's motion becomes as shown in (Fig.1).
But in (Fig.1), the two electrons are packed in one orbit of one de Broglie's wavelength.
We suppose de Broglie's waves are related to some limited spaces.
Actually the two slit behavior of the electron is caused by this de Broglie's wave.
When the de Broglie waves of the two electrons of the helium cross perpendicularly to each other (Fig.2 and 3), their wave phases are independent from each other, and do NOT cancel each other.
( See also this page. )
So if a part of wave phases is cencelled out, the electron space is vanished, and which blocks the electron.
(As shown in this page, we can express de Broglie's waves by the classical mechanical methods correctly.)
We also calculate de Broglie's wavelength of the electron from the velocity (λ = h/mv) at intervals of 1 SS.
The number of that wave (λ in length) contained in that short movement section (the sum of them is WN) is,
(Eq. 9) number of de Broglie's wave contained in the short movement segment for 1SS.
where (VX, VY) are the velocity of the electron 1 (in MM/SS), the numerator is the movement distance (in meter) for 1 SS. the denominator is de Broglie's wavelength (in meter).
Here, the estimated electron's orbit is divided into more than one hundred million short segments for the calculation.
When the electron 1 has moved one quarter of its orbit and its x-coordinate is zero (Fig.3), this program checked the y-component of the electron 1 velocity ( last VY ).
When the last VY is zero, two electrons are periodically moving around the nucleus on the same orbitals as shown in (Fig.2) and (Fig.3).
So, only when -0.000001 < last VY < 0.000001 (MM/SS) is satisfied, the program displays the following values on the screen, r1, VY, preVY (VY 1 SS ago), and (mid)WN (the total number of de Broglie's waves contained in one quarter of the orbit).
Table.1 shows the results in which the last VY is the closest to zero. Fig.4 graphs the results in Table.1.
| E (eV) | r1 (MM) | WN | WN x 4 |
|---|---|---|---|
| -77.5000 | 3134.0 | 0.25241336 | 1.00965344 |
| -78.0000 | 3114.0 | 0.25160304 | 1.00641216 |
| -78.5000 | 3094.0 | 0.25080048 | 1.00320192 |
| -79.0000 | 3074.5 | 0.25000555 | 1.00002220 |
| -79.0030 | 3074.1 | 0.25000079 | 1.00000316 |
| -79.0035 | 3074.0 | 0.25000000 | 1.00000000 |
| -79.0040 | 3074.0 | 0.24999921 | 0.99999684 |
| -79.0100 | 3073.8 | 0.24998972 | 0.99995888 |
| -79.5000 | 3055.0 | 0.24921812 | 0.99687248 |
| -80.0000 | 3036.0 | 0.24843810 | 0.99375240 |
| -80.5000 | 3017.0 | 0.24766535 | 0.99066140 |
As shown in Table.1 and Fig.4, when the total energy of the helium (E) is -79.0035 eV, WN × 4 is just 1.00000000.
The experimental value is -79.005147 eV (Nist). So the relativistic correction to the energy caused by the electron's velocity is -0.001647 eV. This value is proper, because it is just between the helium ion (-0.0025 eV) and the hydrogen atom (-0.000...).
This results demonstrate that two electrons of the helium are actually moving around the nucleus on the orbits of just one de Broglie's wavelength as shown in Fig.2 and Fig.3.
The latest result of the variational methods using the Schroedinger equation is about -79.015 eV ([9]).
This Bohr model-based new method is much simpler and shows more accurate result than the Shroedinger equation-based complicated methods !
The fault of the Shroedinger equation is that it doesn't have the clear electron's orbit.
So it is impossible to know when we should use the reduced mass in the Schroedinger equation.
If we "by mistake" use the reduced mass instead of the electron mass in the condition such as Fig.2 (the center of mass is at the origin), the result becomes inaccurate.
(For example, here we use the usual electron mass (not the reduced mass) in calculating the initial electron velocity from the inputted values in the JAVA program, because this initial state is Fig.2.)
This judgement is possible only in the Bohr model helium which has the clear electron orbits.
Here we use the new unit (1 SS = 1 × 10-25 second) and compute each value at the intervals of 1 SS.
If we change this definition of 1 SS, the calculation results of the total energy (E) in which the orbital length is just one de Broglie's wavelength change as follows,
| 1 SS = ? sec | Result of E(eV) |
|---|---|
| 1 × 10-22 | -79.00470 |
| 1 × 10-23 | -79.00370 |
| 1 × 10-24 | -79.00355 |
| 1 × 10-25 | -79.00350 |
This means that as the orbit becomes more smooth, the calculation values converge to -79.00350 eV.
The standard helium model of the quantum mechanics(QM) has the spin-up and spin-down electrons.
So it seems to generate no magnetic field.
But to be precise, in all areas except in the part at just the same distance from the two electrons, magnetic fields are theoretically produced by the electrons even in the standard helium model.
So as the electrons move to cancel the magnetic field out, they lose energy by emitting electromagnetic waves.
Actually, one-electron atom hydrogen has the magnetic moment, two-electron atom helium has no magnetic moment.
So the standard quantum-mechanical helium model has self-contradiction.
In this new helium, the two symmetrical orbits crossing perpendicularly are wrapping the whole helium atom completely.
See also Why "solitary" helium doesn't form compounds ?.
(The Bohr model hydrogen which has only one orbit, can not wrap the direction of the magnetic moment completely.)
It is just consistent with the fact of the strong stability and the closed shell property of helium.
(See also Bohr noble gases.)
These orbits are all just one de Broglie's wavelength.
If we can describe the ground state of helium atom by the Bohr-model based methods correctly (which means "more correctly" than the quantum-mechanical variational methods), the excited states and the atoms with more electrons can be explained by the Bohr-model based methods, too.
For example, in the lithium atom, the third electron is known to be in the 2S state, and we can get the approximate calculation value close to the experimental value using the 2S state function of the hydogen atom which energy levels are the same in both the Bohr model and the quantum-mechanical model.
See also Bohr model lithium (Li) or Bohr's hydrogen molecule ion (H2+).
Wave propagation by Huygens' pronciple. (See Fig.2).
According to the Huygens's principle, the wavefront of the electron 1 's de Broglie's wave is parallel to itself
The propagating direction of this "secondary" wavefront of electron 1's de Broglie's wave is just the same as that of electron 2, as shown in the upper figure.
(This means that the movement of electron 2 as shown in Fig.2 and 3 is naturally occuring.)
The directions of their de Broglie's waves themselves are perpendicular to each other.
(So the de Broglie's wave of each electron doesn't interfere with each other.)
The same thing can be said also in the wavefront of electron 2's de Broglie's secondary wave.
As a result, the two orbital planes of them are just perpendicular to each other like Fig.2 and 3.
In the standard model of the quantum mechanics (QM), it is said that the electrons are stable as electron clouds, which are not actually moving.
They say this is the reason why the electrons don't fall into the nucleus radiating energy in QM.
But if so, how do we explain about the relativistic corrections to the energy (caused by the high electron's velocity) and the use of the reduced mass?
If we use the reduced mass of an electron, the calculation results of the hydrogen energy levels becomes more accurate. Does this mean that the electron and the nucleus are actually moving around the center of mass?
So the quantum-mechanical model contains self-contradiction also in this subject.
The fermions like electrons don't go back to their original configurations when they are rotated by an angle of 2 π. (By the 4 π rotation, they return.)
This is called the "two-valued".
It is very surprising that this two-valued property of the fermions was experimentally observed [11].
But in this real world, does such a strange phenomenon actually happen?
In this experiment, they rotated the neutrons (= fermions like electrons) around the spin axis by using precession (See this page !).
The angular frequency of the precession is,
(Eq. 10)
This means that they "imagine" the rotation angle based on the spin g-factor (g), because we can't directly look at this precession.
The fermion's spin angular momentum is 1/2 ħ.
If we change this angular momentum to ħ, (spin) g-factor becomes half of the original value, which keeps the original (spin) magnetic moment(=g-factor × angular momentum) unchanged.
We can experimentally measure only the (spin) magnetic moment, can't measure the (spin) angular momentum and g-factor. (See also this page)
If we use the precondition of the half (spin) g-factor, the interpretation of the above experimental result changes to the very natural one that the spinning neutrons returned to their original states by the 2 π rotation.
If the (spin) angular momentum becomes ħ (= the (spin) g-factor becomes half), this means that the atomic models change to the Bohr model in which the serious problem of the spinning speed faster than the speed of the light doesn't occur.
Only the Stern Gerlach experiment can not determine the existence of the spin. Because also in the Bohr model, when the plane of the electron orbit contain the direction of the magnetic field, the electron's motion becomes unstable.
There are other problems in the quantum mechanics.
For example, in the hydrogen solution of the Schrodinger equation, the probability density of the ground state electron near the point at infinity is not zero.
It is very strange.
The hydrogen solution of the Bohr-Sommerfeld model completely coincides with that of the Dirac equation.
Why does the Bohr-Sommerfeld model which has no electron spin coincide with the solution of the Dirac equation which includes the spin-orbital interaction?
For example, the fine structure means the realtivistic energy difference between the 2S and 2P states in the Bohr-Sommerfeld model. (Because the electron's velocities of these states are different).
But in the standard hydrogen model with electron spin, the "interpretation" of this fine structure has been changed to the spin-orbital interaction ( = the energy difference between 2P1/2 and 2P3/2 ).
"Accidentally" this value coincided with that of the Bohr-Sommerfeld model.
Furthermore, the Dirac hydorogen model includes "many unnatural accidental coincidences" of the relativity and spin-orbital interactions.
(For example, 2S1/2=2P1/2, 3S1/2=3P1/2, 3P3/2=3D3/2.........).
It is much more unnatural than the Bohr Sommerfeld model.
How about the singlet and triplet states ?
In the triplet states (S =S1+S2 =1/2+1/2 =1), we can't actually imagine the state in which the total spin angular momentum (S) is perpendicular to the angular momentum L.
In the singlet state (S =1/2-1/2 =0), the spin effect is said to vanish.
But the two electrons of the different orbits are apart from each other.
So around the electron 1, the magnetic moment by the electron 1 exists, and the spin-orbital interaction (by the electron 1 itself) can occur. (if we imagine this state concretely).
The spin-orbital interaction means that the spin effect doesn't vanish.
This is inconsistent with the fact S=0.
How about the anomalous Zeeman effect ?
For multi-electron atoms, for example, the 3s electron of the sodium D line comes very close to the nucleus through the inner electrons.
But the Lande-g-factor does not contain the influence of the inner electrons at all.
The delicate spin and orbital precession is so stable to withstand the influence of the inner electrons ?
The Lande g factor contain the unnatural precession around the total angular momentum J (See this page !).
This J is different from the total magnetic moment μ and the external magnetic field direction.
So J has no relation to the direction of the force.
Why does this unnatural precession occur?
The actual experimental spectrum results under the magnetic fields are much more complicated than the Lande's theory.
And one-electron hydrogen atom is known to show the normal Zeeman effect.
(Some textbooks say that even the hydrogen atom shows the anomalous Zeeman effect. But it is only "theoretical" thing. The experimental results clearly show the normal Zeeman effect in the hydrogen atom.)
It is quite natural to think that the anomalous Zeeman effect is caused by the inner electrons rather than the strange electron's spin.
These electron spin and Lange g-factor are only "mathematical things".
They are not what actually happen.
The same thing can be said about the "virtual particles" in QFT which are used in calculating the Lamb shift and the electron g-factor.
Actually, the QFT is only a "mathematical" product, not about the "real" world.(See this page!)
By the result of the simple Dirac hydrogen model which doesn't use the renormalization theory, the energy levels of the 2S1/2 and 2P1/2 are completely the same.
And the very small energy difference between them is called the Lamb shift (the vacuum polarization) which can be calculated only by the relativistic quantum field theory (QFT) using the renormalization theory.
But the calculation of simple Dirac hydrogen model uses the "nonrelativistc" Coulomb potential (See this page).
So to be precise, it is not a relativistic solution.
So before considering the QFT (vacuum polarization), the energy difference between 2S1/2 and 2P1/2 would occur in the different inertial reference frames !
This means that it is inconsistent with fact that the Lamb shift is caused by the vacuum polarization.
So the QFT contains the self-contradiction in the Lamb shift.
It needs "other" interpretations. (See also this page ( Lamb shift is an illusion ? ) )
If the Bohr model is correct, the problems of the collapse interpletations using the many world interpretaions and so on, would not occur.
And of course, in the Bohr model, the electron doesn't hit the nucleus, because the orbital angular momentum is not zero.
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PDF version(2010/4)
PDF old version(2009/2) (The reduced mass isn't used in this old paper. And here, 1SS = 1 × 10-21 sec. So the calculation values are a little rough and different from the above correct values.)
2011/3/9 updated. Feel free to link to this site.