Top page ( correct Bohr model )

QED Lamb shift is doubtful.

"Science" should NOT avoid questioning.

*(Fig.1) Interaction between electron spin and nuclear spin ?*

This site says hyperfine structure is caused by the interaction between electron spin (= 1s ) and nuclear spin.

But the explanation by quantum mechanics is very strange and **unrealistic**.

They say Schrodinger's 1s electron is always **penetrating** proton (= nucleus ) !

Because the orbital angular momentum of "s" states is **zero** ( L = 0 ).

Suprisingly, **only** when this electron exists **inside** a small proton, hyperfine *magnetic* interaction (= 5.9 × 10^{-6} eV ) can occur, they insist.

Thinking commonsensically, it is **impossible** that such a **small** magnetic interaction can precisely happen **inside** proton.

*(Fig.2) QED contradicts experimental results.*

As shown on this site, the recent experiment showed that quantum electrodynamics ( QED ) is **inconsistent** with the actual value.
( Proton radius, QED -- *0.877* fm, vs. Experiment -- *0.841* fm. )

Basically, we **cannot** directly know the electronic and magnetic distribution of **nucleus** (= proton ).

Experimental data are **indispensable**. So, QED is **NOT** first principle at all.

QED just **estimates** proton's radius from the **difference** between the **experimental** Lamb shift ( ? ) and its prediction. So, various webnews sites tend to "**exaggerate**" the meaning of "tricky" QED.

Furthermore, the nuclear magnetisation (= Bohr-Weisskopf correction ) is the main cause of **uncertainty** affecting QED, as shown on this site.

*(Fig.3) Fermi contact accounts for the leading order of hyperfine structure.*

It is said that Fermi contact interaction is the main cause of hyperfine structure.

The important point is that this Fermi contact value has **NO** physical meaning at all.

In Fig.3, μ_{e} and μ_{p} are magnetic moments of electron and proton.

The probability of ψ(0) means 1s electron can interact with nucleus **only** when it exists **inside** proton !

They just **artificially** adjust this primary Fermi contact by QED, as shown on this site (p.26).

So, the **convenient** QED is used only for **small** adjustment, though it gives infinity.

The problem is how this **8/3 π** coefficient is gotten.

Unfortunately, there are **NO** convincing reasons for this value, as I say later.

*(Fig.4) Quantum mechanics (QM) argues magnetic field exists only inside proton !*

According to Fig.3, they use only the electron probability at the origin (= nuclear position, r = 0 ).

This means the magnetic field is produced **Only** inside proton.

Of course, this is just a **far-fetched** interpretation with **NO** physical entity.

But they had to rely on the coefficient of this ψ(0) to get hyperfine value, though it **lacks** reality.

Thinking commonsensically, extremely **weak** hyperfine energy (= 5.9 × 10^{-6} eV ) is easily **disturbed** and gone in the circumstances filled with **huge** Coulomb force and other effects **inside** a proton.

*(Fig.5) Electron spin enters proton → spinning speed is faster than light.*

The quantum mechanics argues the whole electron **must** enter the proton, and interact with proton's magnetic field (= B ).

As shown on this page, electron spinning speed becomes 1644 × 10^{8} m/s (= more than **500** times light speed ! ) inside proton !

So the idea that electron inside proton causes hyperfine structure is unrealistic and **impossible**.

*(Fig.6) Electron's probability at the origin (= nucleus ) is zero.*

Electron's probability is given by wavefunction × radius (= **r** ).

This means the electron's probability at the origin (= nucleus ) becomes **zero** !

So the hyperfine structure itself **cannot** occur according to Schrodinger's hydrogen ?

( Lamb shift by QED is impossible, too. )

*(Fig.7) Magnetic field near the origin (= proton ) becomes infinite.*

According to this site (p.2) and this page, the magnetic field (= B ) generated by the magnetic moment (= μ_{p} ) is **inversely** proportional to the **third** power of r (= distance ).

When the electron comes near the proton, the distance between them is closer to zero ( **r → 0** ).

As a result, their magnetic energy ( field ) diverges to ± **infinity** !

*(Fig.8) Unrealistic magnetic field in hyperfine structure.*

As shown on this site (p.2), the magnetic field in the hyperfine structure is expressed using Dirac **delta** function (= δ(r) ).

So the magnetic field diverges to **infinity** at the origin (= nucleus ) !

They argue this **infinite** magnetic field B can **cancel** the zero probability of electron at the origin (= Fig.6 ).

Because **∞** (= B ) × **0** (= probability ) at the origin becomes some "**finite**" value.

As you see, this idea of quantum mechanical hyperfine structure is very **ad-hoc** and too good to be true.

*(Fig.9) Magnetic energy becomes ±∞ only at the origin !?*

Using the magnetic field (= B ) based on delta function, we can know magnetic energy distribution in the hyperfine state.

Surprisingly, this magnetic energy **diverges** to ±∞ at the origin.

In all other places, this magnetic energy always becomes **zero** !

How this electron can absorb or emit **high** energy photons **depending** on its position !? ← **Impossible.**

*(Fig.10) Magnetic dipole energy becomes 3 × 10 ^{8} eV inside proton ! Impossible !*

Using the proton's radius (= 0.8 fm = 0.8 × 10^{-15} m ), the magnetic dipole energy ( see Fig.7 ) becomes as big as **3 × 10 ^{8} eV** inside proton !

It is **impossible** that such a big magnetic energy (= **3 × 10 ^{8} ** eV ) suddenly appears and is given to ( or emitted from ) electron

*(Fig.11) 1s hydrogen wavefunction contains Bohr radius (= a _{0} ).*

As shown on this site (p.3), the hydrogen 1s wavefunction contains the reciprocal of the third power of Bohr radius (= 1/a_{0}^{3} ).

So, in fact, the probability at the origin possesses the **information** of the mean **distance** between electron and nucleus, like classical model of this page.

This is the important **trick** in hyperfine structure by quantum mechanics.

*(Fig.12) Magnetic interaction between nuclear spin and Bohr magneton ?*

First we try to calculate the magnetic interaction between nuclear spin (= μ_{p} ) and electron's magnetic moment (= Bohr magneton. )

Bohr magneton was first obtained using Bohr's classical orbit.

**Accidentally**, the magnetic dipole moment of electron spin is just equal to this Bohr magneton !

The magnitude of proton's magnetic moment is **too small** in comparison with electron spin.

( Proton magnetic moment = about **1/1000** × electron magnetic moment. See this site. )

Considering finite proton radius (= 0.8 fm ) and heavy mass, the spinning speed of proton does **NOT** exceed the light speed, as shown on this page.

*(Fig.13) *

As shown on this page, the magnetic field (= B ) generated by classical magnetic moment (= Iπa^{2} ) becomes like Fig.13. Magnetic moment is given by the area × current I.

Here, electron (= e- ) is moving in the circular orbit, which generates electric current I.

"B" is the magnetic field at the point "p".

*(Fig.14) Magnetic field caused by circular current I*

When z = 0 in Fig.13, the magnetic field B at the nucleus (= e+ ) becomes Fig.14.

As I said, this magnitude is **inversely** proportional to the **third** power of the distance (= a_{0} ).

*(Fig.15) Bohr magneton = electron magnetic moment.*

"μ_{e}" is Bohr magneton, which is just **equal** to electron ( **spin** ) magnetic moment.

*(Fig.16) Bohr radius. *

"a_{0}" is Bohr radius (= 0.529 Å ), which is the mean radius of hydrogen ground state.

*(Fig.17) Proton spin magnetic moment (= μ _{p} ) is much smaller*

Proton's magnetic moment (= μ_{p} ) is much **smaller** (= 1/700 ) than electron due to its heavy mass (= m_{p} ).

The important point is that proton's g factor (= 5.58 ) is an **experimental** (= NOT QED ) parameter, which is **different** from Dirac equation. Dirac equation gives only "2"

*(Fig.18) Interaction between electron-nuclear spins by simple model.*

Using Fig.14 - Fig.17, we can know the energy difference between "up" and "down" nuclear spins.

This energy interval becomes **2.2 × 10 ^{-6}** eV

*(Fig.19) Hyperfine structure.*

Experimental value of hyperfine structure is **5.9 × 10 ^{-6}** eV.

So, these values have almost

*(Fig.20) Very weak hyperfine structrue is susceptible to other small effects.*

Proton's magnetic moment is extremely small ( about **1/700** × electron magnetic moment ).

So the very weak hyperfine structure is easily **affected** by other small effects such as nuclear cyclotron and vibration.

First, proton itself is always **rotating** and moving by the electron's attractive force (= **reduced** mass ).

It is quite natural that the proton has small energy difference **depending** on its rotating directions due to "Magnus" effect or something.

*(Fig.21) Only when electron is inside proton, it can interact with proton ?*

On the other hand, the hyperfine model by quantum mechanics is **crazy**.

They argue that the electron can interact magnetically with proton, **only** when it exists **inside** proton !

*(Fig.22) Fermi contact is the leading term.*

According to quantum mechanics and this site, Fermi contact interaction (= part of delta function ) is the **main** factor of 1s hyeperfine, they insist.

*(Fig.23) Magnetic field (= B ) created by circular current I.*

The important point is that there are **NO** clear reasons for this Fermi contact value (= 8/3π ).

For example, relativistic Dirac equation can only give g-factor = 2 (← NOT proton ) in all fermions.

Surprisingly, "*Classical Electrodynamics*" ( J.D. **Jackson** ) says this Fermi contact is based purely on **classical** electromagnetism ! See this site. So quantum mechanical "spin" is **illusion** ?

As shown on this site, the magnetic field B produced by circular electric current I is given by Fig.23.

*(Fig.24) The total magnetic field included inside proton's sphere ?*

Next, they integrate the magnetic field of Fig.23 **over** the volume of the sphere (= 4/3πa^{3} ) to get total magnetic field.

This value of "**4/3**" related to sphere is the key point.

As a result, this **total** magnetic field inside the proton's sphere gives Fermi contact
term of Fig.22, they insist.

As you see, this calculation is based purely on **classical** ( NOT quantum ) mechanics.

*(Fig.25) Other magnetic field is cancelled out.*

Basically the magnetic field is "closed" system.

So the total magnitudes of "up" and "down" magnetic fields are the **same** in all space.

If you integrate the magnetic field **outside** the proton's sphere, you will find the total magnetic field becomes **zero**.

This means "spin-spin" magnetic interaction ( ex. triplet, singlet ) **cannot** happen according to this logic.

*(Fig.26) Dirac delta function (= δ ).*

More suprisingly, they **concentrate** that total magnetic field into the one single point ( r = 0 ) !

So, the magnetic density (= B ) at the origin becomes **infinite** !

There are No magnetic fields in other places, they insist.

Of course, this situation is completely **out** of touch with reality.

*(Fig.27) Magnetic field B is given by the curl of vector potential A.*

As shown on this site, the magnetic field B is defined by the curl of vector potential A.

This vector potential is defined to be consistent with **classical** Ampere's ( Biot-Savart ) Law.

They integrate the magnetic field B **over** the volume of proton's sphere. (← **Trick** ! )

"R" is the **radius** of the proton's sphere.

*(Fig.28) *

"Ω" is solid angle (= 4π ) in the spherical coordinate.

*(Fig.29) Biot-Savart law = "classical" electromagnetism !?*

They use the definition of vector potential "A" based on classical electromagnetism.

The relation of Fig.29 gives classical **Bio-Savart** law. See this site (p.11)

*(Fig.30) Total magnetic field.*

After the long complicated calculation using Fig.27 - Fig.29, they get Fig.30.

The important point is that this method completely **depends** on the assumption that the proton is a pure **sphere**.

*(Fig.31) Magnetic moment μ _{p} of proton.*

The magnetic moment μ is given by the area × current.

The integral of Fig.31 expresses the **total** magnetic moment inside proton.

*(Fig.32) Total magnetic field → Fermi contact ?*

From Fig.30 and Fig.31, we obtain the total magnetic field of Fig.32.

And they **concentrate** this total B only into the **single** point ( r = 0 ) using Dirac delta function.

This is also one of artificial **tricks**.

As a result, they can get Fermi contact term of Fig.22.

*(Fig.33) Tiny electron spinning speed is superluminal !?*

They derived the Fermi contact term using purely **classical** mechanics (= circular **current** + Biot-Savart ).

If **classical** mechanics dominates even **inside** proton, the electron spin magnetic moment **must** be classical, too.

But as shown on this page, the electron spinning speed is **much faster** than light ( > c ), when its rotating radius is equal to that of a proton.

( Of course, an electron is much smaller, so its speed becomes much faster ! )

*(Fig.34) Magnetic field outside proton is cancelled out.*

As I said, the integral of the magnetic field over the space (= **sphere** ) outside the magnetic moment becomes zero.

Because the magnitudes of "up" and "down" magnetic field are just the **same**.

So the total magnetic field in all space becomes Fig.32.

Of course, this value **changes** depending on the electric current's **shape**. ex. circular → elliptic.

This means this obtained value is "model-dependent".

*(Fig.35) We can arbitrarily choose the magnetic field area.*

In Fig.34, I said the **total** magnetic field outside proton is cancelled out.

But of course, the magnetic density in each area is **NOT** zero.

So if we choose other areas as contributing to the total magnetic field of Fig.32 instead of the origin, the probability density of electron's wavefunction (= ψ ) in those areas **changes**. (= NOT ψ(0) ! )

As a result, the resultant hyperfine structure changes **depending** on this chosen area.

This means that this method **cannot** give the definite value as a hyperfine structure, and we can **arbitrarily** change that value by **designating** magnetic areas.

*(Fig.36) Calculated value is still different from experimental value.*

The hyperfine structure expresses the energy difference between "up" and "down" nuclear spins.

As shown in Fig.36, this value based on **classical** Biot-Savart law **cannot** give correct hyperfine structure.

So, they had to use other artificial **tricks** after classical calculation.

*(Fig.37) Electron (= S ) and nuclear (= I ) spins.*

Here "S" is electron spin angular momentum (= 1/2 ), and "I" is nuclear spin (= 1/2 ).

The ratio of magnetic moment to angular momentum (= 1/2 ) is called "quantum" g-factor. See this site.

*(Fig.38) *

So the product of S mutiplied by I is 1/2 × 1/2 = **1/4**.

The different between ±1/4 becomes **1/2**, in this case.

*(Fig.39) *

Here they define total spin (= F ).

*(Fig.40) ↓ Quantum effect ?*

According to the quantum mechanics, the square of F becomes "**2**", even when F = 1.

*(Fig.41) *

As a result, the difference in ±"SI" becomes **twice** (= 1 ) the classical value.

( Compare "1" in Fig.41 with "1/2" in Fig.38. This is the last **trick**. )

*(Fig.42) Artificial trick.*

As a result, they can get the value close to experimental hyperfine structrue.

But this method is very ad-hoc, and **inconsistent**.

As shown in Fig.29, they get the total magnetic field value, depending on **purely** classical **Biot-Savart law**.

In spite of this, they completely **ignore** this definition, and intentionally **double** the magnetic field using quantum's **trick**.

Thinking commonsensically, if the last manipulation based on quamtum mechanics is right, calculation using classical electromagnetism itself becomes **invalid** and **cannot** be used in the first step.

2014/10/13 updated. Feel free to link to this site.