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Wrong spin theories.

Singlet, triplet are NOT "spin".

Spin-orbit coupling is wrong (14/8/17)

- Spinning speed is more than 100 times light speed !
- Fermions do NOT return to the original by one rotation.
- What is the spin g-factor 2 ?
- Anomalous Zeeman effect is NOT by electron spin.

*(Fig.1) Spinning speed is superluminal ! → electron spin is illusion.*

Though ordinary textbooks treat the "spin" as if it was a real object, electron spin is completely **unrealistic**.

An electron is very light and tiny.

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**. ( websites this p.5, this. )

Furthermore, fermions such as electrons **cannot** return to their original states by 360^{o} rotation ! See this site.

At this point, you easily find electron spin does **NOT** exist in our real world.

There is No proof of spin in Pauli exclusion and ferromagnetism.

*(Fig.2) Angular momentum is constant. → a tiny particle spins faster. *

The angular momentum is given by **L = mrv**, where "m" is mass, "r" is radius, and "v" is tangential velocity.

When this angular momentum is constant, the rotational **velocity** becomes **higher**, as its **radius** becomes **smaller**

*(Fig.3) In a tiny electron, the spinning speed becomes faster-than-light !*

An eletron is known to be a **point-like** partice, which means its radius is extremely small.

But this electron has **finite** spin angular momentum of **1/2 ħ.**

As a result, its rotational speed becomes almost **infinite**, of course, faster-than-light !

*(Fig.3') Spinning speed is much faster than light.*

So you easily find that the electrons spin is a completely **unrealistic** concept.

This is the main reason why Wolfgang Pauli resolutely **objected** to the existence of "spin" in 1920s, at first.

*(Fig.4) Proton's spin OK. Electron spin is illusion !*

On the other hand, **heavy** proton with some size can produce "real" spin, **NOT** exceeding light speed.

So nuclear spin is **real**, and electron spin is an **illusion**, it's only **circular** orbit like Bohr model.

In fact, the magnetic moment of electron spin is about **1000** times bigger than that of proton !

This wide discrepancy indicates electron spin is rotating in circular orbit with some radius, **NOT** on its own axis.

*(Fig.5) Fermions return by two ( NOT one ) revolutions (= 720 ^{o} ).*

In the quantum mechanics, the spinor (= spinning electron) **doesn't** go back to its original configuration by one rotation ( 360^{o} ).

It must be rotated **twice** (= 720^{o} ) to be back, as shown on this site.

This is called "two-valued" or "orientation entanglement".

Surprisingly, the quantum theory says various *fermions* such as electrons, neutrons, and protons **can't** return after 360^{o} rotation !

*(Fig.6) Fermions must be rotated twice (= 720 ^{o} ) to be back*

Considering spin angular momentum of 1/2ħ, the wavefunction of the spinor becomes like Fig.6.

The rotation of 360^{o} ( θ = 0 → 360^{o} ) **changes** this spinor ( from plus to **minus** ).

So the spinor, which represents fermions, is **NOT** a real object.

Surprisingly, they argue this strange phenomenon had been observed in actual experiments, as shown in this section.

Of course, it is **natural** that real object can return after one rotation.

The trick of these experiments exists in the **wrong** assumption of angular momentum.

*(Fig.7) Spinor = tray or string ?? Impossible !*

Various people tried to express this strange spinor using **realistic** things.

These models use the similar principle in 720

But these models **cannot** describe the strange property of spinor rotation in **all** directions.

( If you try to do so, an arm or threads are **twisted** or entangled. )

So the spinor rotation itself **cannot** be expressed using real things.

*(Fig.8) Bohr's orbit = Bohr magneton, Spin = Bohr magneton !*

The magnetic moment of the magnitude of **Bohr magneton** (= μ_{B} ) was first obtained from Bohr's orbit, as shown on this site.

Surprisingly, the **spin** magnetic moment is the **same** as this Bohr magneton, as shown on this site and this site.

Though orbital motion and "spin" are completely **different** concepts, this fact is very **strange**.

The magnetic moment is given by *g-factor × angular momentum*.

In **Bohr's** orbit, g-factor = **1** and angular momentum = **ħ**

In electron **spin**, g-factor = **2** and angular momentum = **1/2ħ**

So these are the **same**. Too good !

*(Fig.9) Stern-Gerlach experiment = Bohr's orbit.*

It is said that Stern-Gerlach experiment showed the existence of **spin**.

But Bohr's classical orbit **can** explain this phenomenon naturally, too.

As you see in Fig.9 middle, when the orbital plane is **parallel** to the magnetic field B, this plane tends to **rotate**.

So, also in Bohr's orbit, **two** split lines can be observed in Stern-Gerlach experiment.

As shown in Fig.8, the magnitudes of magnetic moment are **Bohr magneton** in both Bohr's orbit and electron spin.

The split intervals are the **same** in them. **Too good** to be true.

*(Fig.10) Magnetic moment.*

As shown on this site, the magnetic moment μ is given by the electric current I × the area (= A ) of this circuit.

This magnetic moment gives the same magnetic field of two magnetic poles, as shown on this site.

So this is called "magnetic **dipole** moment".

*(Fig.11) *

In Fig.11, one electric charge is orbiting around the nucleus at a velocity of "**v**".

This orbital radius is "**r**".

The electric current I means the amount of electric charges passing some point per second.

So I is given by ev/2πr.

The area of this circular loop is A = πr^{2}, so
the magnetic moment μ_{B} = IA = Iπr^{2}.

*(Eq.1) When the orbital length is 1 × de Broglie wavelength.*

According to de Broglie relation, the wavelength of an electron is given by λ = h/mv.

Eq.1 shows the case in which the orbital length (= 2πr ) is just equal to this de Broglie wavelength.

In this case, the orbital angular momentum (= mvr ) becomes **ħ**.

From Fig.11 and Eq.1, the magnetic moment μ_{B} becomes

*(Eq.2) Bohr magneton.*

This value of magnetic moment is called "**Bohr magneton**".

Bohr magneton plays an **important** role as the unit of magnetic moment **also** in the current quantum mechanics.

So the quantization of **de Broglie** wavelength is the main mechanism, which causes the quantized magnetic moment.

*(Fig.12) Spin g factor = 2 ??*

As I said in Fig.8, the spin angular momentum is **1/2ħ**, so sping g-factor has to be "**2**" to give Bohr magneton ( 1 = 1/2 × 2, see Eq.3 ).

By the way, what does this spin g-factor **really** mean

In ordinary classical electromagnetism, g-factor is always "1".

So in g-factor "2", charge and mass must be **separated** like Fig.12.

*(Eq.3) Spin magnetic moment = Bohr magneton *

Then why this spin g-factor **cannot** be the values such as "2.3", "2.4", "2.7" ...

Do you think the value of just "**2**" is **too good** to be true ?

*(Fig.13) Spin magnetic moment is too weak to explain sodium D-lines.*

As shown on this site, **hydrogen** fine structure (= doublet ) between *2p3/2* and *2p1/2* is about **0.000045 eV**.

On the other hand, the fine structure between *3p3/2* and *3p1/2* of **sodium** is as big as **0.0021 eV**, as shown on this site.

If we assume **hydrogen** doublet is caused by spin-orbit interaction, **sodium** doublet **cannot** be explained by spin-orbit interaction. See also this page and this page.

It is known that Sommerfeld model and Dirac equation give exactly the **same** values in fine structure doublet.

Considering the **same** mechanism is working in both hydrogen and sodium D lines, the fine structure is caused by the **Coulomb** energy difference, when one electron can come **closer** to nucleus than another, like **Sommerfeld** model.

*(Fig.14) Core charge of sodium is much bigger than Z = 1 !?*

Approximately, we can consider the outer 3p ( or 3s ) electron of sodium is moving around **Z = +1** central **core** charge.

( "Core" is the total charge of Na nucleus and **all** electrons contained in n = 1 and 2 orbits. )

From the viewpoint of this outer electron, the core charge is moving around in the opposite direction, which causes magnetic field at the point of the electron having "spin".

As a result, spin-orbit interaction is produced, they insist.

But the discrepancy between these H and Na spin-orbital interactions are **too wide**.

This means the effective central charge in sodium is much **bigger** than Z = 1 (= about Z = **3.54** in Na ), which is very **unreasonable** and unrealistic.

*(Fig.15) Anomalous Zeeman effect of H atom ? ↓ = Bohr-Sommerfeld model. *

Anomalous Zeeman effect is mainly observed in **alkali** metals, which is **inconsistent** with actual energy interval, as I said above.

In fact, the spectrum of one-electron **hydrogen** is so simple that Bohr-**Sommerfeld** model can explain anomalous Zeeman effect in hydrogen.

*(Fig.16) J Mehra's book. *

On page 659 of the Historical Development of Quantum Theory v.1 part2 ( J. Mehra ).

--------------------------------

In 1922, Otto Oldenberg obtained the following results :

For weak fields, each component of Balmer H_{α} **doublet** was separated in a **normal** Zeeman triplet, but for strong fields, the Zeeman patterns of the **doublet** changed into the Zeeman pattern of a singlet line.

---------------------------------

So the anomalous Zeeman effect of hydrogne is so **simple** (= 2 × **triplet** lines ).

These doublets can be completely explained by Bohr-Sommerfeld model, **Not** depending on strange "spin".

*(Fig.17) History of Zeeman effect. *

According to the above book, anomalous Zeeman effects are often seen in **multi**-electron atoms such as Na, Cd, Mg..

In mercury (= Hg ), as much as 11 split lines were observed under magnetic fields in 1898.

The important point is that they **didn't** mention hydrogen atom, when they say about **anomalous** Zeeman effect. Instead they introduced convenient **Paschen-Back** effect in H and Li.

As a result, Heisenberg and Lande thought anomalous Zeeman effect is caused by **core** electrons, Not spin, at first.

And various atoms and metals do **NOT** obey Lande g-factor.

This fact made Russell and Saunders create **new** concepts such as jj coupling and jk coupling (= **artificial** ).

About the calculation of Lande g-factor, see this section.

*(Fig.18) Triplet, Singlet of Magnesium (= Mg ) mean large "Coulomb" energy ( NOT spin ).*

The current quantum theory insists that "triplet" and "singlet" states of the alkaline earths are caused by **spin-spin** and spin-orbit interactions.

But in fact, the energy **intervals** in singlet and triplet are **much bigger** than the **magnetic** energies among **spins**, as shown on this page

( The Story of Spin by S.Tomonaga says they are as big as those of **electric** origin. )

Because spin-spin magnetic dipole energy is too **weak** (= less than 1 × **10 ^{-4}** eV ), which

*(Fig.19) Spin-orbit coupling based on relativistic effect is too weak. ← illusion !*

They insist spin orbit coupling is caused by relativistic effect (= see Wiki ).

But if so, the **magnitude** of theoretical spin orbit coupling becomes much **smaller** than the experimental values
( this site p.6 and this site ).

On this site ( p.2, left ), it is written

-----------------------------------

"For example, for GaAs one finds λ = **5.3** Å^{2}.

Thus, the SO (= **spin-orbit** ) coupling in n-GaAs is by **six** orders of magnitude **stronger** than in vacuum ( λ = **-3.7 × 10 ^{-6}** Å

This enhancement of SO coupling is critical for developing large extrinsic spin currents.

------------------------------------

This wide **discrepancy** between theoretical spin-orbit coupling and actual value shows the concept of spin-orbit coupling is completely **false**, as shown on this page.

*(Fig.20) Spin magnetic moment is too weak for Pauli exclusion.*

According to Pauli exclusion principle, two electrons cannot be the same state with respect to orbital and spin.

Ordinary textbooks say as if this phenomenon was caused by **spin-spin** interaction.

But the magnitude of Pauli exclusion principle is as much as **10 eV**, which **cannot** be explained by very weak spin magnetic energy ( **~10 ^{-5} eV** ).

*(Fig.21) Repulsive force of Pauli exclusion principle is very strong (= about 11 eV in Li ).*

According to this book ( Quantum chemistry 6th edition, by Ira N. Levine, p 292 ), the ground state energy of lithium becomes lower than **-214.3 eV**, if all three electrons enter **1s** orbitals.

On the other hand, the experimental value of ground state energy is **-203.5 eV** in which one of three electrons enters **2s** ( NOT 1s ) orbital.

This means the third electron is **expelled** from 1s orbital by some force, though Coulomb attraction of force is strong enough.

This mysterious **repulsive** force of about **11 eV** is due to Pauli exclusion principle, as shown in Fig.21.

*(Fig.22) Spin-Spin interaction is too weak.*

In Pauli exclusion principle, "spin" direction is crucial factor for deciding whether each electron enters the same orbital or not.

But magnetic dipole interaction of spin is basically **too weak** (= about fine structure level, **1 × 10 ^{-5}** eV. ), which can

This means we need **other** concepts to describe this strong repulsive force of Pauli exclusion.

If "spin" ( magnetic moment ) is too weak, there is **only one** thing left.

As shown on this page, **de Broglie** wave ( destructive ) interference is the main generator of this Pauli repulsive force.

*(Fig.23) Origin of ferromagnetism = Spin ??*

In ferromagnetism, they insist "spin" of each atom such as Fe tends to be **parallel** to each other.

On the other hand, in antiferromagnetism ( or diamagnetism ) such as MnO, each spin tends to be **antiparallel** to each other.

So the **origin** of (anti) ferromagnetism is "spin", as most textbooks say ??

In fact, spin magnetic ( dipole ) moment interaction is **too weak** to cause (anti) ferromagnetism.

*(Fig.24) Origin of ferromagnetism = Exchange interaction "J" ??*

In the current quantum theory, they describe the force of (anti) ferromagnetism as coupling constant (= J ) in Heisenberg model.

The magnitude of coupling constant (= exchange interaction ) is as big as **1 eV**, which is much **stronger** than spin magnetic dipole interaction (= **10 ^{-5} eV** ).

Spin magnetic moment interaction is much **too weak** to account for most magnetic materials such as ferromagnet, as is referred to in various sites ( See
this (p.7),
this (p.5),
this (p.20),
this (p.7) ).

**Spin** dipole interaction is as small as **0.3 K** as temperature expression, which is easily **broken** at room temperature.

( For example, iron can become ferromagnetism at **1043 K**, which is much bigger than 0.3 K. )

*(Fig.25) De Broglie wave interaction is the origin of ferromagnetism.*

Spin - spin magnetic interaction is unrealistically **too weak** to describe magnetic materials.

Again, there is only one thing left.

Electron movement causing magnetic field is accompanied by **de Broglie** wave. ( See also this page. )

Like strong repulsive Pauli exclusion, it is quite natural that we think these synchronous de Broglie waves makes ferromagnetism **stable** at room temperature.

As shown in Davisson-Germer experiment, de Broglie wave interference has the power to change the direction of Coulomb force.

*(Fig.26) Spin is too weak to explain molecular bonds.*

It is said that each molecular bond contains a pair of **spin** up and down electrons.

But the actual molecular bond strength is as much as **4.7 eV** in H2 molecule, as shown on this site.

Again, spin-spin magnetic interaction (= **10 ^{-5} eV** ) is

As long as we believe these

*(Fig.27) "Spin" is fantasy ?*

It is said **spin** high and low states are involved in Fe orbitals, as shown in this experiment.

But unfortunately, 3d electrons of iron (= Fe ) have **nothing** to do with unrealistic spin.

As shown on this site, six electrons in the iron **3d** orbitals are split into several subsets.

"t2g" orbitals are basically nonbonding and at lower energy than the antibonding eg orbitals.

This experiment investigates how these states change using X rays, which can distinguish these **energy** levels, as shown in this Fig..

*(Fig.28) "Spin" magnetic dipole energy is "too weak" ! *

As shown on this site (Fig.15.1), the energy difference between high and low spin states in iron is as large as **1 eV - 3 eV**, which clearly originates in **Coulomb** energy ( **NOT** weak spin magnetic energy ).

**Spin** magnetic **dipole**-dipole interaction is only **1 × 10 ^{-4} eV**, when their distance is about Bohr radius. See this site (p.6) and this site (p.7).

In Fe, this "spin" energy becomes very weak (=

These kinds of experiments **always** depends on **vague** wavefunctions ( 3d ) and unreal Spin, which is **inconsistent** with the magnitude of actual energy.

Unless we use more **concrete** and realistic picture of each electron, **true** atomic bonding remains **unknown** and useless, forever.

*(Fig.29) Current in coil changes the motions of atomic electrons.*

It is said that Einstein-de Haas effects showed "Spin" is "**actual** rotation".

But if spin is rotation, its spinning speed **must** be much **faster** than light speed, as I said above.

This is strange.

In Fig.29 left, the magnetic field B in the iron cylinder is zero, and the magnetic dipole moment vectors (= μ ) of its atoms are **randomly** oriented.

In Fig.29 right, when a magnetic field is set up along the cylinder's axis, the μ line up parallel to and the cylinder begins to **rotate**. See this site p.4 or
this.

They insist spin g-factor "2" can be observed in this experiment by measuring the action and reaction between electrons.

But as shown on this page, this experiment just showed the electric current inside **coil** influences atomic electrons through electromagnetic force (= action at a distance ).

So unrealistic spin has **Nothing** to do this experiment.

*(Fig.30) Wavefunction and unrealistic spin are main obstructions.*

The wavefunction just gives vague probability density of a partcile.

So, it is **impossible** to describe **concrete** particle's motion using this **vague** wavefunction.

Under this strict situation, only **unreal** spin is left as a **thing**.

But as I said above, this spin ( magnetic moment ) **diagrees** with various important phenomena at all.

As a result, the current study based on quantum mechanics is **chaotic** and confusing.

Unless we admit concrete particle's motion, the microworld and biological mechanism remains **unknown**, forever.

*(Fig.31) Interpretation of spectum lines is strange.*

As shown on this site (p.5) and this site (p.3), the energy difference between "**singlet**" and "**triplet**" is caused by *Coulomb* force ( **NOT** by weak spin magnetic moment ).

Very **conveniently**, the selection rule **prohibits** transition between this singlet and triplet, as shown on this site (p.2).

Because if singlet and triplet are real, **much more** spectrum lines must be found.

But if this transition is *prohibited*, how an electron can **go up** to triplet states from singlet ground state ? This is **inconsistent**.

Though they argue very **slow** and weak transition may happen ( see this site ), the **slow supply** ( ground state → triplet ) **cannot** explain strong (= fast ) triplet-triplet transifion.

In the Nist data ( He, Mg ), only singlet-singlet, tiplet-triplet transitions can be seen ( "term" section ).

And they insist photon spin 1 allows specific transition ( S-P, P-D states ).

It is obvious that this selection rule was **artificially** introduced to **conceal** various **inconsistent** transition.

Actually, if you compare Mg and singly-ionized-Al+, the energy level of 3p2 ( triplet ) are completely **different**, though these two atoms have the **same** structure.

*(Ap.1) Spinning speed of the sphere and spherical shell.*

The angular momentum L is equal to the moment of inertia × angular frequency ω

The left of Ap.1 is solid sphere, and the right of Ap.1 is thin spherical shell, as shown on this site.

First, we calculate the spinning speed on the surface of the solid sphere (= rω ).

( Thin spherical shell case is 3/5 times solid sphere's speed. )

From Ap.1 left, the spinning speed (= rω ) on the spherical surface becomes **much faster** than light,

*(Ap.2)*

where m is the electron mass, and ħ is h/2π ( h = Plank's constant ).

This spinning speed is *more than 100 times the speed of light* (= 100 × c = 100 × 3.0 × 10^{8} (m/s)) !

Here the electron radius (r) is supposed to be equal to proton's radius (= 0.88 × 10^{-15} m ).

But the actual electron size is **much smaller** than the proton.

As you see Ap.2, as the radius becomes smaller, its rotational velocity becomes faster.

So, the electron's spinning speed becomes **much faster** than light ! It's **impossible**.

*(Ap.3) ↓ This can be confirmed by actual experiments ? *

It is said that the strange spinor rotation was confirmed in actual experiments ( [4], [5] ).

But is it really so ?

These experiments used **neutrons** as fermions.

And even when they rotated a neutron by 360^{o}, it couldn't return to its original state, they insist.

*(Ap.4) One neutron is rotated, another remains as it is → interference. *

In these experiments, they **split** neutrons with the same phase into two paths.

One of them **rotates** (= precesses ) by an aribitrary angle under magnetic field, through Larmor precession.

Another neutron remains as it is.

Lastly, they mix and make these neutrons from two paths **interfere** with each other.

This experiment showed neutron **cannot** return by **one** rotation ( two rotations made them return. ), they insist.

*(Ap.5) Torque on the magnetic moment (= neutron ). *

Neutron has neutral charge, but has weak magnetic moment.

According to Larmor precession, the external magnetic field (= B ) exerts a torque on the magnetic moment (= μ ).

"S" is angular momentum of neutron ( in this case, S = 1/2ħ ).

γ is the gyromagnetic ratio which gives the proportional constant between μ and S.

*(Ap.6) Electron's magnetic moment = Bohr magneton.*

We simply think about electron's magnetic moment (= Bohr magneton ).

As shown in Ap.6, when spin angular momentum **S** is **1/2ħ**, the gyromagnetic ratio **γ** becomes "**e/m**".

Here the magnetic field B is in the z direction.

Using cross pruduct, Ap.5 can be expressed as

*(Ap.7) Larmor precession of neutron under magnetic field B.*

From Ap.7, we obtain angular frequency ω of precession.

*(Ap.8) *

Based on the assumption of ω in Ap.8, they **estimated** the rotation angle of neutron.

So, if this assumption is **different**, the rotation angle becomes different, too. ( ← Important ! )

When we use the ordinary angular momentum (= **ħ** ) of Bohr's orbit, **instead of** spin 1/2ħ, the gyromagnetic ratio becomes **half**, as follows,

*(Ap.9) *

If **γ** becomes **half**, the angular frequency **ω** becomes **half**, too.

*(Ap.10) If we assume Bohr's orbit instead of spin ..*

From Ap.10, if we assume Bohr's orbit (= ħ ), the precession angle becomes half of spin's case (= 1/2ħ ).

So, this experiment just showed a neutron **returns** by **one** rotation. It's **natural**.

*(Ap.11) Bohr's orbit vs. spin. *

In gyroscope, as the angular momentum increases, the precession becomes slower.

Bearing in mind that the real objects return by 360^{o} rotation, you find Bohr's angular momentum is **true**, and spin is illusion.

This strange nature of the "spinor" comes from the **spin anglar momentum 1/2 ħ** .

If the angular momentum is *ħ* like Bohr model, the wavefunction becomes,

*(Ap.12)*

As shown here, this wavefunction ( e^{iφ}) returns by **2π** rotation.

But if the angular momentum is 1/2 ħ like "spinor" , the wavefunction becomes,

*(Ap.13)*

This wavefunction ( e^{iφ / 2} ) does **NOT** return by **2π** rotation.

So, if we consider "real" atomic model, we must **forget** about the angular momentum **1/2 ħ**.

*(Ap.14) Angular momentum = ħ → 1 × de Broglie wavelength.*

In Bohr model, the *minimum* angular momentum is **ħ**.

As shown in Ap.14, when the angular momentum is **ħ**, its orbital length becomes just **1 × de Broglie** wavelength.

de Broglie relation is satisfied in **all** particles.

In proton, and neutron, spin + ( small ) rotation causes complicated g factor.

*(Ap.15) 360 ^{o} rotation makes "spinor" a different thing. *

Here we explain the strange property of Ap.15.

We define the following **unit vector** using the polar coordinates,

*(Ap.16)*

and Pauli matrices (= σ ) are

*(Ap.17)*

When the spin points to the direction of this unit vector, the spin operator (= S ) can be expressed as

*(Ap.18)*

This is an inner product of vectors n and S.

So the **eigenfunction** expressing "spinor", which points to the **direction of n** is

*(Ap.19)*

because

*(Ap.20)*

In Ap.20, we use

*(Ap.21)*

The spinor of Ap.15 and Ap.19 does **NOT** return by one revolution.

So the spinor itself is a completely **unreal** object.

It is said that the anomalous Zeeman effect is the evidence that "electron spin" really exists.

But one-electron atom **hydrogen** is known to show the **normal** Zeeman effect.

And **lithium** atom showed Paschen-Back effect, which spectral lines are much **fewer** than the sodium D line.

It is natural that we think the anomalous Zeeman effect is caused by the **inner electrons** rather than the "strange" spin. See anomalous Zeeman effect.

In the weak magnetic field (= z axis direction ), some of the lines of the **sodium** obey the Lande g-factor as follows,

*(Ap.22)*

In Ap.22, μ_{z} is the z component of the total magnetic moment (μ) of the sodium.

μ_{B} is the Bohr magneton.

j (= ± 1/2, ± 3/2 ...) is the z component of the total angular momentum.

"g" is the Lande g-factor.

And s and l are the z components of spin and orbital angular momentums, respectively.

How can this Lande g-factor (g) be gotten?

*(Ap.23) ↓ Anomalous Zeeman effect ?*

The total (j), angular (l) and spin (s) angular momentums satisfy the above "triangle" relation.

So the cosine formula become,

*(Ap.24)*

In the same way

*(Ap.25)*

Each **magnetic moment** of the orbital (l) and the spin (s) becomes as follows, (using the spin g-factor 2)

*(Ap.26)*

Here, μ_{j} points to the direction of the **total** angular momentum of,

*(Ap.27)*

The projection of the above orbital and spin **magnetic moments** onto the axis of the μ_{j} is,

*(Ap.28)*

As a result,

*(Ap.29)*

And the projection of this μ_{j} onto the z axis becomes

*(Ap.30)*

where the μ_{B} is the Bohr magneton, and then, the μ_{z} is the z component of the magnetic moment.

Substituting Ap.29 into μ_{j} of Ap.30,

*(Ap.31)*

Finally, we can arrive at the Lande g-factor (g) of Ap.22.

This means that in the weak magnetic field, **two** *strange precessions* are occurring ( see Ap.32 ).

Precession (1) ---- total magnetic moment μ precesses about total angular momentum ( direction of the μ_{j})

Precession (2) ---- this μ_{j} precesses about z axis.

*(Ap.32) Two "unnatural" precessions in anomalous Zeeman effect.*

So the anomalous Zeeman effect is **very complicated** !

Here, if you look at these Precessions (1 and 2) carefully, you will notice something "strange".

Generally, things precess about *the direction of some forces* such as the magnetic field and gravity.

But the direction of total "angular" momentum (= the direction of μ_{j}) has *no relation to the direction of forces*.

( The direction of μ_{j} is neither the z axis direction (external magnetic field) nor the total "magnetic" moment direction (μ) as shown in the above figure.)

And moreover, this Precession (1) must be **much faster** than the Precession (2) by this calculational method.

So *the Precession (1) is a very unnatural thing*, and probably it is not occurring.

This Precession (1) was **made intentionally** to explain some of the sodium spectrum lines (like D line).

As you know, when the z components of orbital (l) and spin (s) angular momentums are 1 and 1/2, respectively, the z component (j) of the total angular momentum **must be** 1+1/2 = 3/2 even in this weak magnetic field.

But, due to this **Precession** (1), the z component of the spin (or orbital) angular momentum is *changing with time*. ( The s can be the value besides 1/2 ! ) .

For example, the case [ s (spin) = 1/2 + 0.05, l (orbital) = 1 - 0.05 ] is possible in the Lande's theory.

But the sum of them, j must be always 3/2 (= 1/2 + 0.05 + 1 - 0.05 = 3/2 ).

Do you think it is *inconsistent*?

To begin with, the value j=3/2 **originates from the precondition** [s (spin) =1/2, and l (orbital) = 1].

But the Lande's theory *neglects this precondition*, and say that j must be 3/2, *but* s and l must be changing with time.

This means that Lande g-factor includes the self-contradiction and it's one of the "mathematical" tricks.

It is said that the *Larmor spin precession* is related to the EPR (electron paramagnetic resonance), ESR (electron spin resonance), and NMR (nuclear magnetic resonance).

But as I said above, the spin itself can **not** be expressed by the **real** things.

This means that the Larmor *spin* precession is an illusion ?

As shown in Ap.5 and Ap.10, the angular frequency of the *electron* spin precession under the magnetic field (B) becomes

*( Ap.33 )*

where g_{e} is the spin g-factor (almost **2**), and B is the magnetic field.

The Bohr magneton of the electron is

*( Ap.34 )*

In the EPR (or NMR) resonance condition, the absorbing or emitting electromagnetic wave energy is equal to the energy difference between the spin up (S = +1/2) and spin down (S = -1/2) states.

*(Ap.35) The energy of the absorbing (or emitting) electromagnetic waves. *

The energy difference between these two states can be expressed as

*(Ap.36)*

As I said above, the Bohr magneton is the **common** value in both the Bohr model and the electron spin.

So if we use the Bohr model instead of the "strange" electron spin, the energy difference becomes, (using angular momentum ± 1)

*(Ap.37)*

where g_{L} is the orbital g factor (almost **1**), which satisfies the following relation,

*(Ap.38)*

As a result, Aq.36 is equal to Aq.37.

So even if we use the *Bohr model*, we **can explain** the phenomena of EPR and NMR correctly.

This is due to the equation of 2 × 1/2 = 1 × 1 (= Bohr magneton ).

The quantum physicists try to explain these EPR and NMR resonance by the Larmor *spin* precession.

But is it really right ?

The energy difference of Ap.36 (or Ap.37 ) can be expressed using the *spin precession angular frequency* of Ap.33, as follows,

*(Ap.39)*

This is the reason why the physicists claim that the *Larmor spin precession is related to the resonance*.

But this interpretation includes a "strange" idea.

As I said above, the fermions such as electrons and neutrons can **not** return to their original states by one rotation.

So we should consider the angle of **4π** ( not 2π ) as **one** cycle !

This means that the *frequency* (not angular frequency) becomes *half* of the original value.

This is **inconsistent** with the above interpretation.

In the Bohr model, the electron is *rotating* instead of "spin".

Under the magnetic field (B), the **Lorentz force** causes the circular movement of the angular frequency of

*(Ap.40)*

As a result, the resonance energy becomes

*(Ap.41)*

Ap.41 is *equal* to Ap.39.

So it is **quite natural** that the resonance is caused by the usual rotation instead of the "strange" spin.

The Bohr magneton is a "*ideal*" value.

If one electron can be split into the circular orbit equally, the Bohr magneton is correct. (Because the current density becomes uniform.)

But one electron **can not** be split, so the orbital g-factor is a little different from 1.

**Various quantum number** ( n = 1, 2, 3, ...) orbits contain the magnetic moment of the **Bohr magneton**.

For example, in the *n=3* Bohr-Sommerfeld's orbit, the angular momentum (L) of the electron can be *L = 1, 2, or 3*.

The case of L=1 corresponds to the Bohr magneton ( magnetic moment = g-factor (1) × L (1) × Bohr magneton ).

In the n=3 energy level, the L=1 orbit becomes "ellipticall".

[1] G.E. Uhlenbeck and S.A. Goudsmit, Nature 117,264 (1926).

[2] Charles E. Burkhardt, Foundations of Quantum Physics (2008)

[3] J.C. Wesley and A. Rich, Phys. Rev. A4 1341-1363 (1971).

[4] H. Rauch et al., Phys. Lett. 54A, 425 (1975).

[5] S.A. Werner et al., Phys.Rev.Lett., 35 1053 (1975)

2014/5/10 updated. Feel free to link to this site.