Top page ( correct Bohr model including helium )
Schrodinger equation belongs to Bohr Sommerfeld model.
Computing actual fine structure and its defect. (14/8/15)
In the Bohr-Sommerfeld model, the orbital length is a integer times the de Broglie's wavelength.
Surprisingly, also in the Schrodinger equation, the orbital length is a integer times the de Broglie's wavelength !
Schrodinger's hydrogen introduced very unreasonable concept of "uncertainty principle" by changing the name of de Broglie wavelength, as shown on this page.
And this is why the Schrodinger equation gives the same energy levels as the Bohr-Sommerfeld model.
On this page, we have proved this fact "mathematically".
(Fig.1) Dirac's hydrogen = Bohr-Sommerfeld model.
Many textbooks say that the fine structure of the hydrogen atom means the energy difference between 2p1/2 and 2p3/2, which is showed by the relativistic Dirac equation.
Surprisingly, as shown on this page, this Dirac's solution coincided with the "relativistic" Bohr-Sommerfeld 's solution.
We need the detailed calculation methods of the relativistic Bohr-Sommerfeld model for comparison.
Unfortunately, ordinary textbooks say nothing about the Bohr-Sommerfeld model.
A few textbooks, for example, "The Principles of Quantum Mechanics (4th), p.272" by Dirac, and The Story of Spin by Tomonaga says about Sommerfeld successful model.
Bohr-Sommerfeld theory first gived the fine structure constant (= 1/137.036..).
This is important.
By the way, what is the "physical" meaning of this fine structure constatnt in Bohr-Sommerfeld model ?
In this section, we calculate the Bohr-Sommerfeld model, and explain the "original" meaning and derivation of this fine structure constant.
(Fig.2) Sommerfeld model by "real" de Broglie wave (= "ether" ).
In fact, "relativistic" Sommerfeld model can be explained using "ether" theory naturally, which is explained later. See "real" ether theory.
To be precise, Bohr-Sommerfeld model does NOT use the special relativity itself.
It uses the classical limit.
(Fig.2') ↓ ① Einstein relativity causes fatal paradox.
If we admit "earth ether", we can explain relativistic mass by maximum transmission speed in the ether without fatal paradoxes in de Broglie wavelength.
The most important difference between Bohr-Sommerfeld model and relativsitic quantum field theory such as Klein-Gordon and Dirac equations is "tachyon".
( Fig.A-1 ) Bohr-Sommerfeld model.
In Bohr-Sommerfeld model, de Broglie waves travel at the speed of "v", which is slower than the light ( v < c ).
de Broglie wavelength λ is equal to h/p, where h is Planck constant, and p is momentum including maximum transmission speed.
( Fig.A-2 ) Relativistic quantum field theory.
Relativistic quantum field theory such as Klein-Gordon and Dirac equations originate in Einstein energy momentum relation, as shown on this page.
Of course, also in the relativistic QFT, de Broglie relation is used.
The problem is the frequency of their wavefunction is too big due to the mass energy.
This unrealistically big frequency causes faster-than-light tachyon !
( Fig.A-3 ) Frequency and momentum of wavefunction.
The wavefunction satisfying the equation of Fig.A-2 becomes Fig.A-3.
As you know, the velocity of the wave is gotten from frequency f × wavelength λ ( v = fλ ).
So the velocity of this wavefunction becomes
( Fig.A-4 )
This speed is faster-than-light.
This unreasonable wavefunction is caused by the relativistic invariance of the equations.
Ironically, the relativistic restriction causes imaginary "tachyon".
They insist if we combine negative and positive energy solutions in Dirac equation, this tachyon vanishes.
But each electron or positron is faster-than-light. This is strange.
( Tachyonic virtual photon is caused by the relativistic Maxwell equation, too. )
( Bohr-Sommerfeld model. )
The following explanation is from the Sommerfeld's original paper ( Annalen der Physik  51, 1-167, 1916 ).
About the special relativity, see this page.
In the hydrogen-like atom based on the central force, the angular momentum is constant.
So the angular momentum ( = p ) is,
where m0 is the rest mass of an electron, and m is the relativistic mass.
If the velocity (= v) is equal to the light speed (= c ), the relativistc mass becomes infinite.
As shown on this page, this relativistic mass causes "right-angle lever paradox".
So it is natural that we explain this relativistic mass using the maximum transmission speed in the aether.
Change the rectanglar coordinates into the polar coordinates as follows,
When the nucleus is at the origin, the equation of the electron's motion is, (Coulomb force condition)
where Z is the atomic number.
Here we define as follows,
The coordinate r is a function of φ, so we can express the differentiation with respect to t (=time) as follows, (using Eq.1)
Here we define as follows,
Using Eq.2 and Eq.5, each momentum can be expressed by,
Using Eq.3, Eq.5, and Eq.7, the equation of motion becomes,
From Eq.8 we obtain the same result of
This β is changing with time, so we need to replace this β by other things.
In the hydrogen-like atoms, the total energy W (= T+V = the relativistic energy (E) - m0c2) of the Bohr Sommerfeld model is,
where W is the constant energy value, which means the sum of the kinetic and potential energy.
As shown on this page, the relation in which the kinetic energy = relativistic energy - rest mass energy of Eq.11 is NOT special relativity itself.
This relation is what we call "classical" limit.
As shown on this page, a square of E and p causes strange "virtual" particle.
So the constant W of Eq.11 is not equal to the special relativity.
From Eq.11, the following equation is gotten using the replacement of Eq.6.
From Eq.10 and Eq.12,
The solution σ of Eq.13 becomes,
Substituting Eq.14 into Eq.13, γ and C are,
Eq.14 means that the r (= 1 / σ) returns to its original value, when the electron moves the angle of 2π /γ (not 2π).
So the orbit of the "relativistic" Bohr-Sommerfeld hydrogen is "precessing".
For example, after one rotation, the perihelion of the orbit moves the angle of
We suppose the first position of the perihelion is at φ = 0.
So the B of Eq.14 becomes zero, as follows,
(Fig.3) "Elliptical" orbit of hydrogen-like atom.
Here, the nucleus is at the focus (F1), and eccentricity (=ε) is,
From Eq.14 ( B=0 ), Eq.18, and the condition of the perihelion and aphelion of Fig.3,
From Eq.19 and Eq.17, A and B of σ in Eq.14 become,
So the r and σ are expressed as,
From here, we deal with the hydrogen atom ( Z=1 ).
We define as follows, (substituting Z=1 into Eq.15),
If the angular momentum (p) of the electron is p0, γ becomes zero according to Eq.22.
This means that the precession speed of Eq.16 becomes infinite, when p = p0.
So this p0 is the lower limit of the angular momentum of the Bohr-Sommerfeld hydrogen.
(Of course, in this case the "elliptical" orbit is broken.)
The p0 to ħ ratio ( = p0/ħ ) is the "famous" fine structure constant (α).
where ħ ( = h/2π ) is the minimum "quantized" angular momentum of Bohr model.
This means that the fine structure constant α (=1/137) is related to the precession speed of the "relativistic" electron's orbital motion in the Bohr-Sommerfeld model.
We can get this fine structure constatnt α (=1/137) by another simple method ( see also this page. )
By solving the equations of the usual Bohr model hydrogen, the electron's velocity of n=1 orbit becomes
( Electron's velocity in Bohr model hydrogen (n=1). )
The ratio of this velocity (v) to the speed of light (c) is the fine structure constant α, as follows.
( Fine structure constatnt )
From Eq.21, when φ is zero, σ, v, and β become,
at perihelion, the velocity is perpendicular to the axis.
So when φ =0, the following equation is gotten using Eq.1, Eq.24 and Eq.15 (Z=1),
Similarly, from Eq.12, Eq.24 and Eq.15 (Z=1), we can get the equation of
From Eq.22, we obtain the relations of
Using Eq.25, Eq.26 and Eq.27, we delete β as follows,
Eq.28 is equal to
In the Bohr-Sommerfeld quantization condition, the following relations are used,
where the angular momentum p is constant.
So p becomes an integer times ħ
Using Eq.1, the "radial" momentum pr can be expressed as
We integrate pr of Eq.32 using Eq.33 and Eq.21 as follows,
where the original integration interval is form 0 to 2π/γ due to the precession.
And the last term of Eq.34 uses the next replacement,
Here as shown in Eq.39 - Eq.48 of this page, we do the partial integration and use the following complex integration formura,
Here we use the following known formula (complex integral),
Using Eq.39', Eq.40' and γ of Eq.22, the result of Eq.34 is,
Using Eq.32 and Eq.22,
Eq.36 changes to the following equation using Eq.37
Substituing Eq.38 into the term ( red line ) of Eq.29, and using the fine structure constant α of Eq.23,
From Eq.39, the relativistic energy ( E = W + m0c2 ) becomes, (adding the charge Z)
Bohr-Sommerfeld solution of Eq.40 is the same as the Eq.41 solution of Dirac equation.
About the detailed calculation method of Dirac's hydrogen, see this page.
This means that the enegy levels of Eq.41 are just equal to those of Eq.40, as follows,
2p1/2 (n=2, j=1/2) -------- 2s (nr=1, nφ=1)
2p3/2 (n=2, j=3/2) -------- 2p (nr=0, nφ=2)
This is a very surprising coincidence !
As shown in Fig.2, if we suppose some medium (= "ether"), which can transmit de Broglie wave, it is natural that we think that medium has its maximum transmission speed.
( In case of light or electric field, which accelerates the charge, this maximum speed is c. )
In this medium, as a charged particle is closer to the light speed c, it is more difficult to accelerate the charge.
And of course, like the light wavelength, the accelerated particle's kinetic energy ( or momentum ) is stored as its de Broglie wavelength, which maximum transmission speed is c.
( As its momentum is bigger, the de Broglie wavelength is shorter, which is used in the quantization of Eq.32. )
Based on these facts, we can use the equation of Fig.2 as the particle's momentum.
In this case, the energy (= dT ) required to accelerate the particle during infinitesimal time dt is
where ux is the velocity in the x direction, and Fx is the force.
where we use
So the total kinetic energy required to accelerate the particle to the velocity v is
Here we use the replacement of
This kinetic energy is completely equal to the first term of Eq.11.
Bohr Sommerfeld model uses the first-order energy term, so it is a little different from the original relativistic theory (= second-order term or Dirac gamma matrices ).
And of course, this model doesn't use the strange concept such as "time delay"
In this theory, we use the next momentum (= p ) of an electron, which becomes more difficult to accelerate as its speed v is closer to c.
( Because it is natural that the medium (= ether) has its maximum transmission speed. )
where λ is de Broglie wavelength.
As shown in Eq.42, and Eq.43, when the electric field accelerates an electron in the moving direction (= ux ), the following force Fx is needed.
When the electric field accelerates this electron ( uy = 0 ) in the y direction, it needs the force Fy of
As shown in Eq.48 and Eq.49, the electric field needs more power to accelerate the electron in the moving direction (= x ) than in the y direction.
Because the field in the x direction is "condensed" due to the velocity ux, which causes more resistance.
( In this case, the velocity uy is zero, so the field in the y direction is not condensed. )
So in the case of Eq.49, it is easier to imagine the situation, because the field is not condensed in the acceleration direction.
(Fig.4) Electron is moving in x direction. We accelerate it in y direction.
In Fig.4, the electric field is trying to accelerate the electron in the y direction.
To do this, the electric field is moving in the x direction at the same speed (= ux ) as the electron, and pushing the electron in the y direction.
The electric field speed is c, so the "pushing" efficiency becomes as shown in Fig.4.
( When the electron's velocity ux= 0, this pushing efficiency becomes c/c = "1". )
As a result, we can explain the heavier electron mass in Eq.49.
Generalizing this effect, we get the momentum of Eq.47.
The important point is that Bohr Sommerfeld model uses the first order energy term, and consider W = kinetic energy + potential energy of Eq.11 as a "constant". ( See also Eq.25. )
In fact, the second order energy equation of the special relativity can NOT do the addition or subtraction of each particle's energy and momentum normally.
( See "Relativistic QED contradicts special relativity ??" of this page. )
So they needed to introduce a very strange concept of "virtual" particles in the relativistic theory.
Here we use the following replacement,
Substituing Eq.50 into Eq.39, the Taylor series for Eq.39 becomes
So the total energy W becomes
The fine structure constant α is small.
So if we suppose A of Eq.50 is nr+nφ, the first term of Eq.52 is just equal to the nonrelativistic Bohr-Sommerfeld model (or Schrodinger equation).
(See also this page.)
Using Eq.50 and the fine structure constant α, 1/A2 can be expressed as
As the α is small, 1/A4 is approximately,
Using Eq.53, Eq.54 and Eq.55, W of Eq.52 can be expressed as, (approximation up to α4)
For example, substituting Eq.57 into Eq.56, the energy difference of 2s and 2p in Sommerfeld model becomes Eq.58 (using Eq.23 and Eq.53).
According to the Sommerfeld's theory, the energy difference (energy level = n) between k and (k-1) sublevels becomes Eq.59
(See also " The Story of Spin " by Tomonaga.)
When we want to know the difference between 2s and 2p, we substitute n=2 and k=2 into Eq.59.
(2s --- n=2, (k-1)=1, 2p --- n=2, k=2)
And aH is the Bohr radius and μ0 is the permeability of vacuum.
So they satisfy the following equations.
Eq.59 is known to coincide with the experimental values of various atoms such as H, He+, Li++, Be+++.
We calculate Eq.59 ( Z=1, n=2, k=2 ), as follows,
Eq.61 is just equal to Eq.58.
This means that the meaning of Eq.59 is the same as that of Eq.56.
As shown on this page, 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.
This is why Pauli strongly objected to the existence of "electron spin" in 1920's.
At that time, it is known that Eq.59 of Sommerfeld's model coincides with the experimental values in various atoms,
To prove the existence of electron spin, they needed to get Eq.59 by the spin-orbital interaction instead of relativistic mass change.
We suppose an electron is rotating around the +Ze nucleus.
From the viewpoint of the moving electron, the nucleus is rotating around the electron instead.
The speed of the nucleus is -v, and the coordinate of the nucleus is -r.
So according to Biot-Savart law, the magnetic field (B) at the point of the electron becomes,
Here we introduce the angular momentum quantum number K (K = 1, 2, 3 ....).
This K satisfies the relation K ħ = m0r × v
So Eq.62 can be expressed as
The magnetic moment of the spinning electron is g-factor (ge) × s (=1/2) × Bohr magneton.
So the interaction energy between the spinning electron and the magnetic field (B) becomes
where μB is the Bohr magneton, s is the spin angular momentum (=1/2).
Substituting B of Eq.63 into Eq.64,
The energy difference between spin up (+1/2) and spin down (-1/2) states of Eq.65 becomes,
(+1/2 - (-1/2) = 1)
According to the Bohr's theory, the average value of 1/r3 satisfies
where aH is the Bohr radius, n means the energy level.
Substituting Eq.67 into Eq.66,
It was known that if K2 of Eq.68 was changed into k(k-1), the results agreed with the experimental values well.
So substituting Eq.69 into Eq.68,
When the spin g-factor (ge) is 1, Eq.70 is just equal to Eq.59 (=Sommerfeld model).
But if the ge is 1, the electron magnetic moment becomes 1/2 × 1 = 1/2 × Bohr magneton.
So Eq.70 is wrong, because the electron magnetic moment must be 1 × Bohr magneton.
To solve this problem, L.H. Thomas appeared.
He had succeeded in justifying the electron spin using the Thomas factor 1/2.
(Though his method is a little complicated and "unnatural".)
Next we explain about this factor.
As shown on this page, the precession angular frequency of the spinning electron under the magnetic field (B) is
According to Bohr's correspondence principle, ħω = | W1 - W2 |.
So mutiplying Eq.71 by ħ,
As shown in Eq.64, the interaction energy under the magnetic field (B) is
So the energy difference between s = ± 1/2 states under the magnetic field becomes
Eq.73 is just equal to Eq.72 !
Thomas found that when the electron is "accelerated", the electron's "specific" coordinate axis is "rotating" from the viewpoint of the "laboratory system".
According to his calculation results, the rotation angular frequency of this coordinate axis is
where a is the acceleration of the electron.
So the "true" precession angular frequency of the electron in the laboratory system is Eq.71 + Eq.74, as follows,
The electron's acceleration is caused by the Coulomb force of +Ze nucleus, as follows,
Here we use Eq.62 and μ0 of Eq.60.
From Eq.76, Eq.62, and Eq.60, the total angular frequency of Eq.75 can be expressed as
Using Eq.77, the interaction energy of Eq.64 under the magnetic field changes into
And Eq.70 changes into
When ge is 2, Eq.79 is just equal to Eq.59 of Bohr-Sommerfeld model !
Eq.79 = 1/2 × Eq.70.
So this 1/2 is called "Thomas precession factor".
Finally, Pauli accepted Thomas's spin model "reluctantly".
( Finally, Thomas got Pauli's sanction about the electron spin.)
By the way, which fine structure do you think is more natural, Thomas's spin-orbital or Bohr-Sommerfeld model ?
If the fine structure is caused by the spin-orbital interaction, this model includes "many unnatural accidental coincidences" of the relativity and spin-orbital interactions. ( 2S1/2=2P1/2, 3S1/2=3P1/2, 3P3/2=3D3/2.........).
And the replacement ( K2 → k(k-1) ) of Eq.69 can be gotten "naturally" in Bohr-Sommerfeld as shown in Eq.59.
But in the spin-orbital interaction model, this replacemant of Eq.69 must be gotten from quantum mechanical "mathematical" trick like l(l+1).
Eq.72 using Bohr's correspondence principle is strange, too.
The two-valued experiment in 1975 showed that the fermions don't return by one rotation. (By two rotations, they return.)
So we should consider the angle of 4π as one cycle in the electron spin
This means that the angular frequency of Eq.72 becomes half.
But if Eq.72 becomes half, Thomas factor 1/2 becomes meaningless.
This includes self-contradiction.
As a result, it can be said that the spin-orbital interation model is a more "unnatural" thing than the Bohr-Sommerfeld model.
Recently, using the Penning trap  which can trap single electron, the anomalous magnetic moment was measured.
(Fig.5) Measurement of anomalous magnetic moment by Penning trap.
As shown in Fig.5, the Penning trap has the minus-charged (-) caps in the upper and lower parts.
And it has the plus-charged (+) ring around it.
So when the center is the origin, the electric potential of the Penning trap becomes
This means that the electron is attracted to the origin in the z direction by the force of
where ωz is the angular frequency in the z direction.
The electron is attracted outward by the positive-charged ring.
They apply a magnetic field (B0) in the z direction.
So the equation of the electron's motion in the x-y plane becomes
Eq.82 includes the centrifugal force and the Lorentz force.
To measure the spin magnetic moment, they need to add a small magnetic field of
Due to the magnetic field gradient (2βz) of Eq.83, the magnetic moment of the electron receives the force of
where μB is the Bohr magneton, S is the spin angular momentum (± 1/2), and gs is the spin g-factor.
(Eq.85) Bohr magneton.
In the Penning trap, various motions are mixed as shown in Fig.6
So the electron's motion is very complex.
Of course, as the electron is very light, the influence of the thermal noise is very big.
(Fig.6) Axial, magnetic and cyclotron motions in the Penning trap.
The magnetic force acting on the cyclotron motions of various quantum numbers is
As a result, the total angular frequency in the z direction is
Furthermore, they apply weak oscillating potential ( V =K sin ω t ).
When this ω is equal to Eq.87, it causes "resonance".
This ω can be measured.
The electron spin is changing (flipping) randomly.
(Of course, the various cyclotron motions are changing randomly, too)
If they pick and measure the anglar frequency difference between ± S of Eq.87, they can get the exact spin g-factor .
But as shown in Eq.88, the magnetic moment caused by the minimum (ħ) orbital motion is the same as the spin magnetic moment.
So if we use the minimum orbital motion instead of the "strange" spin, the g-factor becomes half.
(Only the interpretation changes.)
( References )
 W.H.Louisell, R.W.Pidd, An Experiment of the Gyomagnetic Ratio of the Free Electron, Phys.Rev.94 (1954) 7-16.
 D.T.Wilkinson, H.R.Crane, Precision Measurement of the g Factor of the Free Electron, Phys.Rev.130 (1963) 852-863.
 Dehmelt, Experiment with an Individual Atomic Particle at Rest in Free Space, Am.J.Phys.58 (1990) 17-27.
 H.Dehmelt, New Continuous Stern-Gerlach Effect and a Hint of the Elementary Particle, Z.Phys.D10 (1988) 127-134.
 V.Gerginov, K.Calkins, et.al. Phys.Rev. A 73 (2006) 032504.
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