Top page. ← 6/30/2024

Particle physics, standard model is wrong

Quantum field theory is just unphysical math.

- Relativistic Dirac's hydrogen does Not use spin or spin-orbit interaction
- Calculation of unphysical Dirac hydrogen's fine structure.

*(Fig.1) Historical magic or "cheating" ? = "accidental agreement" between Bohr-Sommerfeld's hydrogen fine structure without spin in 1916 and unphysical Dirac's hydrogen's fine structure with fake spin-orbit interaction in 1928. *

Spectral lines of hydrogen atom (= H ) show that its energy levels with the same principal quantum number ( n = 2 ) splits into two closely spaced doublets or energy levels called fine structure.

This very small energy difference between 2p1/2 (= principal quantum number n = 2, orbital angular momentum L = 1, and total angular momentum J = j = 1/2 with down-spin ) and 2p3/2 (= n = 2, L = 1, J = 3/2 with up-spin, this p.2 ) energy levels is said to be caused by relativistic (paradoxical) spin-orbit interaction according to the current mainstream quantum mechanics based on relativistic Dirac equation.

But in fact, this tiny, tiny energy splitting in hydrogen atoms has **nothing** to do with (unphysical) electron's spin or its spin-orbit magnetic interaction.

In 1916, realistic Bohr-Sommerfeld atomic model could successfully obtain the **exact** fine structure energy levels using the realistic moving electron **without** unreal spin ( this p.5-upper ).

Later in 1928, the present mainstream quantum field theory based on the (unphysical) relativistic Dirac equation allegedly using "spin" or spin-orbit interaction could also "accidentally" obtain exactly the **same** fine structure values as the Bohr-Sommerfeld model ( this 5th-paragraph, this p.14, this lower, this p.8-left-lower, this p.1-last~p.2-upper, this p.8 4~5th-paragraphs ).

↑ It's too good to be true, and it's impossible and **unnatural** that the completely **different** atomic models, one of which lacks (unreal) spin and the other has spin, could get completely the **same** atomic energy level formula of fine structure.

This **accidental agreement** shows that the later Dirac hydrogen theory "**artificially copied**" the earlier successful Bohr-Sommerfeld fine structure to get exactly the same solution.

And in fact, even the current mainstream relativistic Dirac hydrogen does **Not** use the (unreal) spin or spin-orbit interaction, which is why Dirac hydrogen could **agree** with the realistic Bohr-Sommerfeld model **without** spin.

Dirac hydrogen uses the total angular momentum J = L (= orbital angular momentum ) + S (= spin 1/2, half-integer ), but the final energy solution uses only the **integral** values (= J + 1/2 = integer, so **No** spin-1/2 is left in Dirac hydrogen solution, this p.4, this last, this p.29 ) which accidentally agreed with the earlier realistic Bohr-Sommerfeld's **integer** orbital quantized values.

↑ It means Dirac hydrogen uses its fake spin-orbit interaction operator denoted by σL (= σ means Pauli spin matrices ) that produces total angular momentum quantum number j as a part of the electron's large (angular) kinetic energy agreeing with realistic Bohr-Sommerfeld's kinetic energy quantum number (= n_{φ} ) instead of the tiny-tiny relativistic spin-orbit magnetic energy. ← So relativistic spin-orbit interaction is illusion, and Bohr-Sommerfeld model **without** spin is right.

Contrary to the mainstream narrative, the realistic Bohr-Sommerfeld atomic model does **Not** use the paradoxical Einstein **relativity**, because the Einstein relativistic theory **contradicts** the orbital motion, the conserved angular momentum, Coulomb potential (= Coulomb inverse square law is **non**-relativistic, this p.3-Remark(2) ), de Broglie theory, kinetic energy and time.

The paradoxical Einstein relativistic energy = mc^{2} copied the original classical Maxwell's **authentic** mc^{2} (= which is used in realistic Bohr-Sommerfeld fine structure ) in the wrong way.

This Dirac equation for hydrogen fine structure (= accidentally agreed with Bohr-Sommerfeld fine structure ) is the **only**
(pseudo-)relativistic equation that can be analytically solved, and we show this only-solvable Dirac hydrogen does **Not** use the tiny relativistic spin-orbit magnetic interaction (= σL ) as fine structure.

So there is **No** evidence of (unreal) relativistic spin-orbit interaction, theoretically or experimentally.

*(Fig.2) Einstein paradoxical relativistic effect = a electron is at rest, and a much heavier nucleus is moving around the lighter electron (= spin-orbit interaction ) ? ← This is impossible due to violating momentum conservation. ↓*

The present unphysical mainstream quantum mechanics heavily relies on the **paradoxical** concept called (relativistic) spin-orbit interaction or spin-orbit coupling like in fine structure tiny energy splitting of a hydrogen atom.

This relativistic spin-orbit interaction is self-**contradictory**, impossible, and it can **disprove** Einstein relativistic theory, too.

When an electron is moving around a heavy nucleus at rest, the nucleus at rest can**not** produce magnetic field (= No charge motion = No electric current I = No magnetic field B ), hence, the electron can only feel electric field and can**not** feel magnetic field from the **stationary** nucleus.

But Einstein relativistic theory does **Not** have the absolute frame, all things must be relative.

So in the crazy Einstein relativistic world, we have to think about the electron's rest frame where the originally-stationary heavy nucleus appears to be moving around the stationary electron that is originally moving ( moving electron ↔ moving nucleus ).

↑ In this unphysical electron's rest frame, the fictitiously-stationary electron with spin can feel (fake) magnetic field from the nucleus that appears to be moving around the electron, though it is **impossible** for a heavier nucleus to be moving around a lighter electron (= so Einstein is **false** ).

This **fake** magnetic field is said to cause the (fictitious) spin-orbit magnetic interaction in the (fake) stationary electron with spin, depending on the electron spin's up (= magnetic energy higher ) or down (= magnetic energy lower ) directions ( this p.3-4 ).

Quantum mechanics or mainstream relativistic quantum field theory called Dirac equation claims this (fake) spin-orbit magnetic interaction is the origin of fine structure tiny energy splitting between 2p1/2 (= down-spin ) and 2p/3/2 (= up-spin ) of the hydrogen atom.

But as you notice, this relativistic spin-orbit (fake) magnetic interaction is physically **impossible** and paradoxical.

Because this fake relativistic spin-orbit magnetic interaction or fine structure energy splitting should be generated only in the (unphysical) electron's rest frame, and in the normal nuclear rest frame (= generating No magnetic field ), this spin-orbit interaction or fine structure energy splitting does **Not** occur, according to their logic ( this p.10 ).

This p.2-Fig.3 (or this p.3-Figure 3 ) says

"Although in the **rest** frame of the nucleus, there is **No** magnetic field
acting on the electron (= No spin-orbit interaction ), there is a magnetic field in the rest frame of the
electron (= spin-orbit interaction )" ← paradox !

↑ Different frames seen from different observers (= depending on seen from stationary nucleus or stationary electron ) cause spin-orbit interaction or No spin-orbit interaction, which is clearly **paradox**, wrong, and **disproving** Einstein relativistic theory and quantum mechanical fine structure theory.

And it is physically **impossible** for a heavier nucleus to be unrealistically moving around a lighter (stationary) electron due to momentum conservation and action-reaction law (= when two objects attract each other, the lighter object or electron must be moving faster than the heavier one or nucleus to conserve total momentum ).

This **paradoxical** relativistic spin-orbit fine structure
energy splitting is said to be explained by relativistic Dirac equation for hydrogen, but this is wrong.

Because the Dirac equation for hydrogen deals **only** with the nuclear rest frame (= with only static Coulomb electric potential V ) with moving electron (= Dirac hydrogen can**not** deal with the electron's rest frame or fictitious magnetic field potential from the unrealistically-moving nucleus).

Actually, Bohr-Sommerfeld model **without** spin could perfectly explain the hydrogen's fine structure energy splitting, and later, Dirac hydrogen **copied** it and obtained exactly the **same** fine-structure energy values as the older Bohr-Sommerfeld model ( this-lower, this p.14 ). ← Dirac equation with spin = Bohr-Sommerfeld model without spin !

In fact, the relativistic Dirac equation's fine structure energy splitting value is just a part of electron's kinetic energy (= Dirac equation's γ or σ matrix allegedly representing spin comes from electron's kinetic energy, this p.30, this-2.(1)-(5) ), which has **nothing** to do with the relativistic spin-orbit magnetic interaction (= Dirac equation changing spin from down to up increasing j by one corresponds to Bohr-Sommerfeld's angular kinetic energy n_{φ} increasing by one ).

In condensed matter physics like spin-Hall-effect, physicists artificially changed the original Einstein's tiny relativistic spin-orbit interaction constant by using **fake** electron's effective mass, which is unreal, **inconsistent** with the original Einstein relativistic theory ( this p.5, this p.5-6, this-p.24-pseudo-spin ).

Quantum mechanics has to rely on artificially-created **fictional** pseudo-potentials to (wrongly) explain the (fake) spin-orbit interaction.

As a result, the relativistic spin-orbit interaction is paradoxical and **non**-existent.

*(Fig.3) Hydrogen tiny energy splitting called fine structure by real Bohr-Sommerfeld orbits and unphysical Dirac equation with pseudo-spin. *

The current mainstream physics **baselessly** claims the hydrogen atomic tiny energy splitting called fine structure is due to the (imaginary) relativistic spin-orbit interaction.

But in fact, the current mainstream relativistic quantum field theory based on **unphysical** Dirac equation for hydrogen can **Not** give the spin-orbit interaction at all, contrary to the ordinary explanation.

The tiny, tiny relativistic spin-orbit (fictitious) magnetic interaction is expressed as the unphysical Pauli σ matrices × orbital angular momentum L = The (fictitious) spin-orbit coupling interaction operator is supposed to be σL

This alleged relativistic Dirac equation's (pseudo-)spin-orbit interaction expressed by unphysical Pauli matrix σ ( this p.8, this p.7, this p.3-left ) multiplied by the electron's (angular) momentum corresponds to the large electron's kinetic energy or orbital angular **kinetic energy** (= we explain this in detail, later ), which has **nothing** to do with the tiny, tiny imaginary relativistic spin-orbit magnetic interaction.

Dirac equation for hydrogen is the **only** solvable relativistic equation, and this only solvable relativistic Dirac equation's (pseudo-)spin-orbit interaction energy is as **big** as the electron **kinetic** energy (= ~ 3 eV ) treated as the same energy-scale as the electron's radial kinetic energy expressed as the derivative (= d/dr, this p.16-17, this p.6-7 ), which is **far larger** than the tiny, tiny relativistic spin-orbit magnetic energy (= only 0.000045 eV ).

This is why Dirac hydrogen's **pseudo**-spin-orbit interaction operator = σL **switches** into the irrelevant electron's large (angular) **kinetic energy** incorporated into the final principal quantum number (= n = n' + j +1/2 where j + 1/2 means electron's **orbital** angular kinetic energy, Not the tiny relativistic spin-orbit magnetic energy, this last, this p.14 ).

↑ Actually, Dirac hydrogen's total angular momentum J (= orbital L + spin S or σ angular momentums ) allegedly expressing the tiny, tiny spin-orbit interaction energy is treated as the **same magnitude** as the Bohr-Sommerfeld hydrogen's bigger kinetic energy's quantum number (= n_{r}, n_{φ} ) as shown in their same fine structure's energy solution.

↑ The current mainstream quantum field theory or paradoxical relativistic Dirac hydrogen" does **Not** represent the tiny spin-orbit interaction energy as fine structure energy splitting at all, **contrary** to the textbooks' explanation, which means the hydrogen's fine structure is due to realistic Bohr-Sommerfeld model instead of the phony Dirac hydrogen's spin.

As a result, the nonphysical mathematical Pauli σ matrices ( we are repeatedly "brainwashed" into thinking this nonphysical Pauli matrices mean "spin" without the detailed mechanism, this p.5 ) used in the only solvable relativistic Dirac hydrogen have **nothing** to do with (imaginary) spin or spin-orbit interaction.

This is why Dirac equation (= which actually does Not use spin or spin-orbital interaction ) could ( artificially rather than accidentally ) obtain the same fine structure energy solutions as the realistic Bohr-Sommerfeld model without the spin ( this p.9-middle ).

*(Fig.4) Unphysical Dirac hydrogen's 2s1/2 and 2p1/2 energy levels ( n=2, j=1/2 ) "accidentally agreed" with Bohr-Sommerfeld's elliptical orbit's energy level ( n _{r}=1, n_{φ}=1 ). ← Historical magic or "cheating" ?*

Unphysical relativistic Dirac equation for hydrogen atom in 1928 "**accidentally agreed**" with the earlier realistic Bohr-Sommerfeld fine structure energy formula in 1916.

Compare the fine structure formula of Dirac hydrogen ( this p.19 ) and Bohr-Sommerfeld model ( this p.12 ). ← These two completely-different atomic models gave exactly the **same** fine structure results ( this p.14 ) !

Z is the nuclear charge, in hydrogen atom, Z equals 1.

n is principal quantum number ( n = n_{r} + n_{φ} ), n_{r} is radial kinetic energy quantum number, n_{φ} is angular kinetic energy quantum number, j is total angular momentum, α is fine structure constant, m is rest mass of electron, c is light speed.

For example, one of Bohr-Sommerfeld elliptical orbits (= radial kinetic energy n_{r} = 1, angular kinetic energy n_{φ} = 1 ) gives exactly the **same** energy level as the later ad-hoc Dirac hydrogen 2s1/2 and 2p1/2 energy levels ( principal quantum number = n = 2, total angular momentum j = 1/2 equals the orbital angular mometum L = 1 minus spin 1/2, this p.16 ).

↑ This accidental coincidence between the earlier Bohr-Sommerfeld fine structure **without** spin and the later Dirac hydrogen with (pseudo-)spin is extremely **unnatural**.

*(Fig.5) Unphysical Dirac hydrogen's 2p3/2 ( n=2, j=3/2 ) orbital "accidentally agreed" with realistic Bohr-Sommerfeld's circular orbit's energy level ( n _{r}=0, n_{φ}=2 ) ← Historical magic or just "cheating" ?*

In addition to this energy level, all other hydrogen' energy levels including fine structure splitting of relativistic Dirac hydrogen **accidentally agreed** with the earlier Bohr-Sommerfeld model.

In the upper Fig.5, one of Bohr-Sommerfeld hydrogen's circular orbits (= radial quantum number or kinetic energy n_{r} = 0, angular quantum number or kinetic energy n_{φ} = 2 ) gives exactly the same energy level as Dirac hydrogen's 2p3/2 energy level (= principal quantum number n = 2, total angular momentum j = 3/2 equals orbital angular momentum L = 1 plus spin 1/2, this p.2 ).

↑ This "too good" accidental agreement clearly shows the later mainstream quantum field theory based on unphysical Dirac equation "**artificially copied**" the earlier realistic Bohr-Sommerfeld atomic fine structure energy levels.

*(Fig.6) Total angular momentum J = L + 1/2σ where L is orbital angular momentum and 1/2σ of Pauli matrix is spin ? ← false. *

Quantum mechanics tries to artificially associate nonphysical Pauli σ matrices with the electron's spin ( this p.6 ), though there is **No** evidence or No experimental verification indicating that these nonphysical Pauli matrices mean "spin."

Total angular momentum J is artificially defined as the sum of the orbital angular momentum (= L ) and spin angular momentum (= 1/2σ ).

The square of this total angular momentum J^{2} gives the alleged spin-orbit interaction terms (= σL or LS this p.7,10, this-(5.7.10), this p.13-14 ).

Relativistic Dirac hydrogen uses this alleged spin-orbit interaction term or operator expressed as the product of nonphysical Pauli σ matrices and the orbital angular momentum L ( this p.3-4, this-middle, this p.6-7, this p.28-29(or p.16-17) ).

But this Dirac hydrogen's (pseudo-)spin-orbit interaction term (= σL ) has **nothing** to do with (imaginary) relativistic spin-orbit magnetic interaction or tiny fine structure energy splitting.

Dirac hydrogen uses this (pseudo-)spin-orbit interaction term (= σL = ±k+1, k = j+1/2 or J + 1/2 = integer ) as the electron's large kinetic energy ( this p.3, this p.30, this p.4-5, this p.14-15(or p.7-8 ) ) in the tangential (= angular ) direction = Not as a tiny relativistic spin-orbit (fictitious) magnetic energy.

↑ This is why Dirac hydrogen's **pseudo-spin-orbit** interaction term (= σL = ±K+1, K = J+1/2 ) is used as the major **angular kinetic energy** term of the final total energy principal quantum number n = n_{r} (= radial ) + J+1/2 (= angular kinetic energy quantum number originating from pseudo-spin-orbit term, this p.17,29, this p.7,11 ).

Dirac hydrogen is the only solvable relativistic atomic equation with potential energy (= though it uses **non**relativistic Coulomb energy potential and nonrelativistic Schrödinger hydrogen's angular wavefunction Y ), so this result proves that the nonphysical Pauli matrices are completely **different** concepts from the electron spin, which means there is No evidence that fine structure energy splitting is caused by (imaginary) spin-orbit interaction.

*(Fig.7) Pauli σ spin matrix is a part of kinetic energy irrelevant to spin-orbit interaction.*

As shown in Eq.9, Pauli σ matrix (= allegedly indicating spin ) actually means a part of **large kinetic** energy (= 3~4 eV ) = Pauli σ matrices appear when original Einstein relativistic energy momentum (= p ) quadratic equation (= **without** spin ) is divided into two linear ( Dirac ) equations ( this p.5, this-2.(1)-(5) ).

So the spin-orbit operator σL or σp used in Dirac
hydrogen actually means large orbital angular **kinetic** energy ( this p.30, this p.1-right, this p.14(or p.7) ) **irrelevant** to tiny relativistic spin-orbit magnetic energy.

Quantum mechanical Dirac hydrogen's fine structure is also unrelated to the unphysical spin or spin-orbit interaction like realistic Bohr-Sommerfeld's fine structure without spin ( this p.1-left-3~5th-paragraphs ).

As shown in the above figure, the kinetic energy part of Dirac hydrogen splits into radial (= ∂/∂r) kinetic energy and **orbital angular kinetic** energy (= σL/r ) consisting of **fake** spin-orbit interaction (= σL ) operator ( this p.14, this p.7 ).

In fact, Dirac hydrogen's fake spin-orbit operator (= σp or σL ) has a **large**-energy coefficient (= light speed c or its abbreviation, this p.9-upper, this p.13, this p.16-lower, this-2.(1)-(5), this p.5 ) completely **different** from the tiny relativistic spin-orbit magnetic interaction energy's coefficient = ℏ^{2}/4m^{2}c^{2}r × ∇V ( this p.7, this p.11-(2.3.18) ).

This is why even the current mainstream quantum mechanical Dirac hydrogen's fine structure does **Not** rely on relativistic spin-orbit magnetic interaction or electron spin.

*(Fig.8) Dirac spin-orbit interaction term (= σL ) has nothing to do with relativistic spin-orbit fine structure.*

The current mainstream relativistic quantum field theory based on Dirac hydrogen claims that the tiny atomic energy splitting (= 0.000045 eV ) between 2p3/2 and 2p1/2 states called fine structure is caused by the (imaginary) relativistic spin-orbit magnetic interaction.

But in fact, the current mainstream relativistic Dirac equation for hydrogen allegedly proving the spin-orbit interaction completely **disagrees** with the original relativistic tiny spin-orbit interaction in many ways.

If the tiny energy splitting between 2p3/2 (= electron's spin-up with respect to its orbit ) and 2p1/2 (= spin-down with respect to its orbit ) is really caused by relativistic spin-orbit magnetic interaction, their energy splitting intervals must be equal like the spin-up-magnetic energy = +μB and the spin-down magnetic energy = -μB.

Because the spin-magnetic energy is expressed as the ±spin magnetic moment (= ±μ ) × B (= magnetic field ) = ±μB whose energy interval must be 2 × μB ( this p.16 ).

But the unphysical quantum mechanical spin theory gives the **wrong** unequal spin-orbit energy splitting interval like |+μB| = 1/2 × |-μB| whose energy interval is Not 2 × μB ( this p.2-4 ).

As shown in the upper figure, the spin-orbit interaction of 2P1/2 (= down-spin ) state becomes -2ℏ, and the spin-orbit interaction of 2P3/2 (= up-spin ) state becomes +ℏ. ← the energy splitting intervals are different between spin-down and spin-up, which **disagrees** with the original relativistic spin-orbit magnetic energy definition.

Furthermore, relativistic Dirac hydrogen's (pseudo-)spin-orbit (= σL ) has **nothing** to do with the tiny, tiny relativistic spin-orbit magnetic energy (= 0.000045eV ).

Because Dirac hydrogen's (pseudo-)spin-orbit interaction operator (= σL = ±k+1 ) is used as the completely-**irrelevant** electron's kinetic energy in the angular direction, which is as large as 3 eV in 2p state = **far larger** than the tiny relativistic spin-orbit magnetic energy (= only 0.000045 eV ).

And the Dirac's hydrogen gives the energy solution in the **rest** frame of the **nucleus**, which can**not** cause the relativistic spin-orbit magnetic energy in the electron's rest frame where a (imaginary) heavier nucleus orbiting around the stationary electron allegedly causes the pseudo-magnetic energy.

Dirac hydrogen uses only **Coulomb electric** potential energy (= V = Coulomb electric energy, this p.6, this p.2-2nd-paragraph ) **without** the magnetic energy term as the potential energy. All other parts (including fictitious spin-orbit term ) of Dirac hydrogen are electron's **kinetic** energy.

↑ **No** tiny magnetic potential energy between the fictitious magnetic field and electron's spin is included in Dirac hydrogen (= its electron's spin experiences **only electric** field caused by the static nucleus ), hence, distinguishing spin-up and spin-down (magnetic moment) for giving (fictitious) spin-orbit magnetic energy is **impossible** (= electron's spin magnetic moment can interact only with **magnetic** energy or magnetic potential, the spin's direction can **Not** be distinguished through Coulomb electric energy. ← Dirac pseudo-spin-orbit term corresponds to the irrelevant large orbital angular kinetic energy, Not spin-orbit magnetic energy ).

So it is theoretically **impossible** that Dirac hydrogen using **only** Coulomb **electric** potential energy gives the (fictitious) spin-orbit magnetic energy. ← Dirac hydrogen whose total energy equals the normal Coulomb energy plus kinetic energy uses the **same** mechanism as Bohr-Sommerfeld hydrogen **without** spin, and this is why their fine structure solutions (= Both of the atomic models **lack** the relativistic spin-orbit magnetic interaction ) agreed.

As a result, there is **No** evidence of the spin or relativistic spin-orbit interaction, theoretically and experimentally.

*(Fig.9) Dirac hydrogen needs too many unnatural coincidence !*

Quantum mechanical atoms are said to contain electron's spin, so its hydrogen atomic energy levels should be split into more different energy levels (= 2s1/2, 2p1/2, 2p3/2 should give completely-different energy levels ) than the actual hydrogen's energy spectral lines or realistic Bohr-Sommerfeld hydrogen energy.

Naturally, the 2s1/2 state with No orbital angular momentum and No spin-orbit interaction must give completely **different** energy levels from 2p1/2 state with orbital angular momentum and spin-orbit interaction, because the 2s1/2 and 2p1/2 have the completely-**different** atomic structures.

But Dirac hydrogen "accidentally agreed" with Bohr-Sommerfeld fine structure energy levels.

It means ad-hoc Dirac hydrogen needs many **unnatural accidental agreements** between the originally different atomic energy levels like 2s1/2 = 2p1/2, 3s1/2 = 3p1/2, 3p3/2 = 3d3/2.. ( this p.6, this p.45, this p.27 ).

↑ These many unnatural lucky agreements shows Dirac hydrogen is just the **artificially**-created product **copying** the original successful Bohr-Sommerfeld fine structure.

Negligibly tiny doubtful Lamb shift also does **not** need electron spin.

*(Fig.10) Unphysical Dirac hydrogen needs unreal 1P1/2 energy level in the hydrogen ground state.*

In fact, the unphysical relativistic Dirac equation for hydrogen atom always needs the **unrealistic** energy levels such as 1P1/2, 2D3/2 .. which paradoxical states must Not exist in hydrogen energy levels, so wrong.

Because Dirac hydrogen solution must always contain **two different** orbital angular momentums expressed as two different nonrelativistic Schrödinger's hydrogen's angular wavefunction = spherical Harmonics = one has orbital angular momentum L = J - 1/2, and the other's orbital angular momentum is L = J + 1/2 ( this p.15, this p.15(or p.8 ), this p.4, this p.17-lower, this p.2, this p.20 ).

For example, this unphysical Dirac hydrogen's ground state (= J = 1/2 ) must consist of the original 1S1/2 (= L = J - 1/2 ) state and the unrealistic 1P1/2 (= L = J + 1/2 = 1 ) with orbital angular momentum L = 1 despite the lowest-energy ground state n = 1, which should Not exist in quantum mechanical hydrogen atomic wavefunction ( this Table-8.2.3 ).

Dirac hydrogen's imaginary 1P1/2 wavefunction is the product of the unreal radial wavefunction and nonrelativistic Schrödinger's hydrogen's angular wavefunction or spherical Harmonics ( this p.3, this p.4 ) with orbital angular momentum L = 1.

So the Dirac hydrogen's ground state (= n=1, j=1/2 ) solution must include the unrealistic angular momentum state or spherical Harmonics like conθ sinθe^{iφ} ( this p.2, this p.70(or p.55), this p.3, this p.41 ). ← Dirac hydrogen ground state includes the orbital angular momentum, but it does not cause spin-orbit energy splitting in n = 1 energy level, which is **contradiction**.

↑ And the use of non-relativistic Schrödinger hyrogen's solutions means the relativistic Dirac hydrogen is **Not** the relativistic theory at all.

*(Fig.11) Dirac hydrogen is wrong, containing unrealistic energy levels.*

As I said, relativistic Dirac hydrogen's each energy level always needs two orbital angular momentums, one is L = J +1/2, and the other is L = J - 1/2.

It means Dirac hydrogen's ground state energy must contain the normal 1S1/2 ( n = 1, J = 1/2 ) with orbital angular momentum L = 0 = J - L = 1/2-1/2, and the other is **unreal** 1P1/2 state with orbital angular momentum L = 1 = J + L = 1/2+1/2.

In the same way, Dirac hydrogen n = 2 energy level must include the unrealistic 2D3/2 state with the orbital angular momentum L = 3/2+1/2 = 2.

This fact shows Dirac hydrogen's fine structure based on non-existent wavefunctions is wrong, and only the Bohr-Sommerfeld fine structure is left as the legitimate theory with **No** self-contradiction.

*(Fig.12) Dirac hydrogen J = 1/2 solution must contain the paradoxical angular momentum without spin-orbital interaction, so false. *

Relativistic Dirac hydrogen 2P1/2 state whose electron spin is anti-parallel to its orbital angular momentum must cause the relativistic spin-orbit magnetic energy.

But Dirac hydrogen's solution must include the **paradoxical** angular momentum which does **Not** cause the relativistic spin-orbit energy despite 2P1/2 state.

Dirac hydrogen's 2P1/2 (= 2 component spinor ) consists of two different Schrödinger's hydrogen's spherical Harmonics Y^{0}_{1} (= orbital angular momentul L = 1, z component orbital angular momentum m = 0 ) and Y^{1}_{1} (= orbital angular momentul L = 1, z component orbital angular momentum m = 1 ) with the same total angular momentum J = 1/2 and the same z component of total (= spin + orbit ) angular momentum J_{z} = m = 1/2 ( this p.10,13, this p.4, this p.18, this p.44, this p.21, this p.1-right-lower ).

It means Dirac hydogen 2P1/2 state's Y^{0}_{1}'s electron orbital angular momentum ( this p.4 ) is **perpendicular** (= neither parallel nor antiparallel ) to the spin, and causes No spin-orbit interaction, which contradicts their claim that 2P1/2 causes spin-orbit interaction.

This paradoxical solution also **disproves** Dirac hydrogen's fine structure.

*(Eq.1) Unphysical relativistic quantum field theory based on Dirac equation*

The present mainstream (unphysical) relativistic quantum field theory is based on abstract Dirac equation which is said to explain all spin-1/2 particles such as electrons, (fictional) quarks, antiparticles..

Unfortunately, this relativistic Dirac equation completely **lacks** physical picture of particles, so it's impossible to know the detailed physical mechanism of this nonphysical quantum field theory based on Dirac equation.

The tiny-tiny dubious relativistic spin-orbit effect or fine structure energy splitting is said to be obtained by solving this nonphysical Dirac equation with Coulomb potential energy, whose calculation results "accidentally agreed" with the earlier Bohr-Sommerfeld fine structure formula without (unreal) spin.

But in fact, this (pseudo-)relativistic Dirac equation for hydrogen does Not prove the existence of the (fictional) relativistic spin-orbit interaction at all.

So the relativistic spin-orbit (fictitious) magnetic interaction is just **illusion** with No theoretical or experimental basis.

As a result, the (realistic) Bohr-Sommerfeld hydrogen model without spin remains as the only legitimate theory explaining fine structure energy splitting correctly.

*(Eq.2) Einstein relativistic energy (= E ), momentum (= p), mass (= m, rest mass energy is mc ^{2}, c is light speed ) was divided into Dirac equation with unphysical γ or Pauli σ spin matrices.*

Einstein paradoxical relativistic energy (= E ), momentum (= p ) and mass (= m is rest mass ) equation is E^{2} = p^{2}c^{2} + m^{2}c^{4} where c is light speed.

From this original relativistic energy equation, by replacing energy and momentum by derivative operators ( energy E = iℏd/dt, momentum p = -iℏd/dx ), quadratic relativistic Klein-Gordon equation was obtained.

To get the first-degree energy equation, Dirac artificially **divided** the original quadratic relativistic equation (= Einstein energy-momentum or Klein-Gordon equation ) into two first-degree equations with unphysical γ matrices composed of Pauli σ spin matrices( this p.22-26, this p.2-3 ).

By adding Coulomb electric potential V (= in rest frame of a nucleus or proton ) to the energy and using this, we can obtain Dirac equation for hydrogen atomic energy above ( this p.17, this p.2-6 ).

Dirac equation used unphysical 4 × 4 γ matrices of

*(Eq.3) 4 × 4 γ matrices γ ^{0}*

where I means 2 × 2

Other three γ matrices are

*(Eq.4) 4 × 4 γ matrices, γ ^{1}, γ^{2}, γ^{3} σ is 2 × 2 Pauli spin matrix *

where j = 1, 2, 3

σ_{j} mean 2 × 2 **Pauli** spin matrices of

*(Eq.5) σ is supposed to mean Pauli spin matrix, but actually, σ is irrelevant to spin or spin-orbit interaction. ← this is trick of unphysical Dirac fine structure.*

As shown above, the square of the same Pauli matrices is 1 or identy matrix ( σ_{j}σ_{j} = 1 ).

Changing the order of two different Pauli matrices flips the sign like σ_{i}σ_{j} = - σ_{j}σ_{i} ( this p.2 )

Calculations of γ matrices become ( this p.7, this p.6 )

*(Eq.6) The square of the same γ ^{0} matrix is 1. *

*(Eq.7) Unphysical γ matrices used in (pseudo-)relativistic Dirac equation*

*(Eq.8)*

Eq.2-lower contains two equations of

*(Eq.10) Coulomb electric potential V*

V is the ordinary Coulomb electric potential.

First we omit 4πε, and add this later.

The momentum *p* is expressed as space derivative operator based on de Broglie wave theory as follows,

*(Eq.11)*

Using Pauli σ matrices and vector cross product formula, we can obtain the following formula,

*(Eq.12) σ is Pauli spin 2 × 2 matrix, a and b are ordinary non-matrix numbers.*

See this p.7-2.2.9

Using Eq.12, Pauli σ matrices, and momentum p derivative operator, the electron's total kinetic energy term (= σp where σ is Pauli matrix, p is momentum ) of Dirac hydrogen equation becomes

*(Eq.13) Dirac hydrogen's total kinetic energy (= σp ) splits into radial (= ∂/∂r ) and orbital angular kinetic (= σL ) energies.*

As shown above, Dirac hydrogen's kinetic energy (= σp ) splits into radial kinetic energy (= ∂/∂r ) and orbital angular kinetic energy (= σL/r, this p.14-upper, this p.5 ).

So the spin-orbit oeprator (= σL ) of Dirac hydrogen means just the orbital angular **kinetic** enregy **irrelavnt** to tiny relativistic spin-orbit magnetic energy.

As a result, the fine structure is **Not** caused by relativistic spin-orbit energy but by kinetic energy change like Bohr-Sommerfeld fine structure without spin.

The total angular momentum operator (J) is the sum of the orbital angular momentum (L) and spin (S) expressed by Pauli σ spin matrix as follows ( this p.18 ),

*(Eq.14)*

Here we use the relation of Pauli matrices ( σσ = I = 1, this p.10 ).

And replacing the squared angular momentum operators like J^{2} → J(J+1) and L^{2} → L(L+1)

*(Eq.15) Total angular momentum J = L + 1/2σ → σL = spin-orbit interaction ?*

So when the orbital angular momentum L = J + 1/2 (= like 2p1/2 state with total angular momentum J = 1/2 and orbital angular momentum L = 1 = J + 1/2, this p.2 ),

*(Eq.16) orbital angular momentum L = J + 1/2 case where J is total angular momentum*

where we define k = J + 1/2.

And when L = J - 1/2, (= like 2p3/2 energy level with total angular momentum J = 3/2 and orbital angular momentum L = 1 = J - 1/2 ), See also this p.17.

*(Eq.17) orbital angular momentum L = J - 1/2 case where J is total angular momentum *

See this p.7-(35.30), this p.20-(92), this p.8-last,p.9-lower

*(Eq.18) Dirac hydrogen's spin-orbit interaction (= σL ) means orbital angular kinetic energy without spin*

Using Eq.9-upper equation, Eq.13, Eq.17 (= orbital angular momentum L = j-1/2 case), we can get the above equation ( this p.3-(2.6) ).

As shown above, Dirac hydrogen tried to pretend that the ordinary large orbital kinetic energy (= σL/r ) as (fictional) spin-orbit (= σL ) interaction, but σL clearly means orbital angular kinetic energy irrelevant to spin.

Dirac hydrogen wavefunction is expressed as 4×1 matrix consisting of the upper 2×1 matrix of orbital angular momentum L = J + 1/2 and the lower 2×1 matrix of orbital angular momentum L = J - 1/2 where the total angular momentum J is the common value.

Each Φ consists of two different spherical Harmonics.

So when n = 1 (= total energy principal quantum number ), J = 1/2, Dirac hydrogen wavefunction includes the unrealistic orbital angular momentum L = J + 1/2 = 1 in n=1 state (= 1P1/2 ).

↑ In n=1 of hydrogen, there is only one 1s1/2 state, so this 1p1/2 state is unreal.

When n = 2, j = 3/2, Dirac hydrogen also includes another unrealistic state of 2D3/2 (= n=2, L = J + 1/2 = 2 ).

*(Eq.20) The other equation of Dirac hydrogen*

In the same way, the lower equation of Eq.9 becomes like the upper Eq.20, by using Eq.13 and Eq.16

Using the artificial relation switching angular momentum wavefunctions L = J - 1/2 → L = J + 1/2 ( this p.20, this p.7, this p.4, this p.14-lower ) in spherical Harmonics,

*(Eq.21)*

The spherical harmonics parts (Φ) of Eq.18 becomes the common same wavefunction (= common orbital angular momentum L ), and it can be eliminated as follows ( this p.18-19 ),

*(Eq.22)*

In the same way, by using Eq.21, the angular spherical Harmonics Φ of Eq.20-lower can be eliminated as follows,

*(Eq.23)*

*(Eq.24) 2P1/2 which should have spin-orbit interaction includes the paradoxical solution with No spin-orbit interaction in Dirac hydrogen. orbital angular momentul L = j + 1/2 case*

Dirac hydrogen's solution must contain two different orbital angular momentum L = j + 1/2 and L = j - 1/2 in the same common total angular momentum j.

In the case of L = j+1/2 like 2p1/2 state ( n = 2, j = 1/2, L = j + 1/2 = 1 ), Dirac hydrogen's solution (= upper-half 2×1 spinor matrix ) must consist of two different spherical Harmonics with the same orbital angular momentum (= L = j + 1/2 ) of different z direction like Y_{1}^{0} and Y_{1}^{1} ( this p.13, this p.17 ).

But this Y_{1}^{0} is the state where the orbital angular momentum L = 1 is perpendicular to z direction or spin direction, which means No spin-orbit interaction which needs parallel or antiparallel spin-orbit directions.

*(Eq.25) Dirac hydrogen's solution, one of two component spinors, L = J - 1/2 case*

We replace the radial functions of f(r) and g(r) by

*(Eq.26)*

Substituting Eq.26 into Eq.22, we have ( this p.2, this p.19, this p.22~, this p.21~ )

*(Eq.27)*

Coulomb potential V of Eq.10 is used here.

In the same way, substituting Eq.26 into Eq.23, we have

*(Eq.28)*

We expand u(r) and v(r) as follows,

*(Eq.29)*

Substituting Eq.29 into Eq.27 and Eq.28, and seeing the power of γ-1, we have

*(Eq.30)*

To avoid the solution of a_{0}= b_{0} = 0 in Eq.30, the following relation must be satisfied.

*(Eq.31)*

To avoid the divergence at the origin, the γ must be plus as follows,

*(Eq.32)*

To converge at r → ∞, we suppose u(r) and v(r) satisfiy

*(Eq.33)*

And at r → ∞, Eq.27 and Eq.28 become

*(Eq.34)*

Substituting Eq.33 into Eq.34, we have

*(Eq.35)*

From Eq.35, λ (= plus) becomes

*(Eq.36)*

Considering Eq.29 and Eq.33, we can express u(r) and v(r) as follows,

*(Eq.37)*

As you may notice, the replacement of Eq.37 is very similar to Schrodinger's hydrogen.

So also in Dirac's hydrogen, **unrealistically**, radial wavefunctions are always from **zero** to **infinity**.

If we add 4πε (see Eq.10) to γ of Eq.32, it can be expressed by the fine structure constant (=α)

*(Eq.38)*

where this fine structure constant α is approximately

*(Eq.39)*

Substituting Eq.37 into Eq.27, we can get the following relations ( using Eq.36 and the green part of Eq.35 ).

*(Eq.40)*

and Substituting Eq.37 into Eq.28, we have

*(Eq.41)*

Here we define (using Eq.32 and Eq.36)

*(Eq.42)*

and replace r by x as follows,

*(Eq.43)*

Using Eq.42 and Eq.43, Eq.41 become

*(Eq.44)*

And Eq.40 becomes

*(Eq.45)*

Here we expand ω_{1} and ω_{2} as follows,

*(Eq.46)*

Substituting Eq.46 into Eq.44, coefficient of x to the power of **n - 1** ( = x^{n-1} ) becomes,

*(Eq.47)*

In the same way, substituting Eq.46 into Eq.45, the coefficient of x^{n-1} is

*(Eq.48)*

Summing Eq.47 and Eq.48, we obtain

*(Eq.49)*

Substituting c_{n} and c_{n-1} of Eq.49 into a and b of Eq.47, and using Eq.42, we get the relation of

*(Eq.50)*

When we use the replacements of

*(Eq.51)*

Eq.50 beomes

*(Eq.52)*

From the relation of Eq.52, we can define the function F(x) as follows,

*(Eq.53)*

And using Eq.49, ω_{1} and ω_{2} (= Eq.46) can be expressed using the common c_{n},

*(Eq.54)*

Eq.54 means if we decide the power of c_{n}, the upper and lower spinors become the same as c_{n}.

The power of c_{n} is related to the energy level (= radial part ) of the hydrogen, as I explain later.

( This method is similar to Schrodinger's hydrogen. )

As a result, unrealistic 1p1/2 state and 1s1/2 have the same energy level with the common energy E and common total angular momentum J, but different orbital angular momentum ( L = 0 in 1s1/2, L = 1 in 1p1/2 ).

So Dirac's hydrogen cannot avoid many wrong states such as 1p1/2, 2d3/2 ....

When F(x) is an infinite series, F(x) *diverges* exponentially at r → ∞ as follows,

*(Eq.55)*

To make F(x) a **finite** series, μ of Eq.52 must satisfy ( this p.36-40 )

*(Eq.56)*

where F(x) becomes the *n' th degrees* with resprct to x.

From Eq.51,

*(Eq.57)*

Substituting Eq.42 into Eq.57,

*(Eq.58)*

Here we return 4πε.

Using the fine structure constant α of Eq.39, Eq.58 becomes

*(Eq.59)*

They difine the new integer **n** (= energy level or principal quantum number ) as

*(Eq.60) Dirac hydrogen's energy quantum number*

This n means the major energy level or the principal quantum number, n' is the radial (kinetic energy) quantum number, and |k| = J+1/2 is the angular momentum or **kinetic energy** quantum number which originates from the Dirac pseudo-spin-orbit interaction (= σL ) = relativistic spin-orbit magnetic interaction is illusion.

The energy E of Eq.59 becomes ( using Eq.38 and |k| = j + 1/2 )

*(Eq.61) Dirac hydrogen's energy solution in 1928*

where n means the principal quantum number.

And as shown on this page. this Dirac hydrogen energy values of Eq.61 ( this p.21, this p.9, this p.7 ) are *completely consistent* with those of Bohr Sommerfeld model of Eq.62 ( this p.12, this p.3-lower ).

*(Eq.62) Bohr-Sommerfeld fine structure energy without spin in 1916*

This means that the enegy levels of them are *just equal* to each other, as follows,

Dirac 2p1/2 (n=2, j=1/2) -------- elliptic, (n_{r}=1, n_{φ}=1) Bohr-Sommerfeld

Dirac 2p3/2 (n=2, j=3/2) -------- circular, (n_{r}=0, n_{φ}=2) Bohr-Sommerfeld

↑ Dirac hydrogen's spin change from down to up (= j = 1/2 → j = 3/2 ) has as **large** energy impact as the change of **orbital kinetic** energy of Bohr-Sommerfeld's hydrogen (= n_{φ} = 1 → 2 )

This **unnatural** coincidence shows the later Dirac hydrogen **copied** the earlier successful Bohr-Sommerfeld fine structure result, even by making the electron's large angular kinetic energy look like a **fake** spin-orbit interaction (= σL ).

*(Fig.72) Dirac hydrogen's ground state ( n=1, L=1 ) of hydrogen atom ?*

As I said, Dirac hydrogen must always contain two different orbital angular momentum states in each energy level ( this p.32-33, this p.4 ).

So Dirac hydrogen's lowest-energy ground state 1S1/2 with the principal quantum number n = 1 and orbital quantum number L = 0, total angular quantum number J = L + 1/2 = 1/2 must contain the unrealistic 1P1/2 state with the orbital angular momentum L = 1 !

This is why unphysical Dirac hydrogen's ground state includes angular momentum wavefunction or paradoxical spherical Harmonics of cosθ, sinθ, e^{iφ} ( this p.3, this p.70(or p.55), this p.2 ), which should Not exist in the original 2s1/2 state with zero orbital angular momentum.

↑ Dirac hydrogen ground state includes (unrealistic) orbital angular momentum, but it does not cause spin-orbit doublet. ← paradox !

*(Eq.73)*

Here we try to get the Dirac hydrogen's 1S1/2 eigenfunction or ground-state wavefunction ( Z = 1, hydrogen ).

When the principal quantum number n is 1 (and j=1/2 ), *n' becomes zero* according to Eq.69.

*(Eq.74)*

From Eq.47 and Eq.70, the total energy E and γ are

*(Eq.75)*

where |k| = j + 1/2 = 1.

Substituting Eq.75 into Eq.45,

*(Eq.76)*

where r_{0} is Bohr radius.

Eq.76 means the exponential part of Dirac's hydrogen 1S is equal to R_{10} of Schrodinger's hydrogen.

Substituting Eq.75 and Eq.76 into Eq.51, A becomes

*(Eq.77)*

where 4πε is returned.

From Eq.63, Eq.75 and Eq.77, a_{0} becomes

*(Eq.78)*

The fine structure constant α of Eq.48 is used in the last line of Eq.78.

Eq.78 means, in Eq.73 case (= upper part is 1P1/2 ), eigenfunction of Dirac's hydrogen is zero.

( When k = 1, lower spinor is also zero, substituting k = 1 into Eq.81. )

*(Eq.79)*

When the sign of k becomes opposite ( k = 1 → -1 ), orbital angular momentums are exchanged, as shown in Eq.79.

So when **k = -1**, the upper part is 1S1/2 and lower part is 1P1/2.

In this case, the eigenfunction of 1S1/2 is not zero,

*(Eq.80)*

where k is **minus**.

In the same way, when k = -1, the lower part of 1P1/2 is **NOT** zero,

*(Eq.81)*

*(Eq.82) Ground state of Dirac's hydrogen ?*

According to advanced quantum mechanics by J.J. Sakurai, the upper part of Dirac's hydrogen is related to Schrodinger's hydrogen.

( About the **lower** part, I could **NOT** find what it really means. )

But of course, **unreal** 1P1/2 is also **indispensable**, because Dirac equation mixes them.

If 1P1/2 state does not exist in Eq.82, the simultaneous equations are **broken**.

As I explained above, the angular momentum operator **actually acts** on 1P1/2 eigenfunction, and get its eigenvalue.

As a result, Dirac equation for hydrogen atom contains *many fictional states* like this.

(1s1/2 and **1p1/2 ??**, 2p3/2 and **2d3/2 ??**, 3d5/2 and **3f5/2 ??** ... )

Then, relativistic spin-orbit magnetic interaction is illusion.

OK. so next we consider about the *n=2 and j=1/2* states.

Fortunately, in this case, both states 2S1/2 ( L= j-1/2 = 0 ) and 2P1/2 ( L = j+1/2 = 1 ) really exist.

Then can we get the *eigenfunctions* of these states ?

*(Eq.83)*

In the case of Eq.83, the eigenfunction of 2P1/2 becomes

*(Eq.84)*

The result of Eq.84 *resembles* that of Schrodinger 2P1/2.

(And the part of the exponential function also resembles Schrodinger's n=2 solution.)

The relativistic effect of the hydrogen atom is **very small**.

So the similar results of the eigenfunctions in Dirac and Schrodinger equations are *reasonable*.

How about 2S1/2 ?

The eigenfunction of 2S1/2 becomes

*(Eq.85)*

The result of Eq.85 is **completely different** from Schrodinger's 2S1/2 solution. !

Schrodinger's 2S1/2 radial part is

*(Eq.86)*

where r_{0} means the Bohr radius.

As I said, the relativistic effect of the hydrogen atom is very small.

So we *can not* accept this strange result of 2S1/2 which is far from the original Schrodinger hydrogen's 1s wavefunction.

*(Eq.87)*

How about the case of Eq.87 ?

(2S1/2 is the upper part of the spinor. Of course, the angular momentum of 2S1/2 is zero.)

In the case of Eq.87, the eigenfunction of 2S1/2 becomes

*(Eq.88)*

Surprisingly, Eq.88 *resembles* Schrodinger's 2S1/2 of Eq.86 !

But the 2P1/2 in the case of Eq.87 is

*(Eq.89)*

this is *completely different* from Schrodinger's 2P1/2, though the relativistic effect of hydrogen is very small.

As you notice, Dirac equation **can not** give the correct eigenstates of 2P1/2 and 2S1/2 **at the same time**.

And it contains the *wrong two other states*, too.

Unfortunately, Dirac equation *can not* distinguish these wrong states from the correct ones.

This is the result of physicists trying to match Dirac's hydrogen with Sommerfeld's fine structure using various "mathematical" tricks somehow.

*(Eq.90) Total angular momentum J and Hamiltonian.*

In this section, we demonstrate total angular momentum J commutes with Dirac's Hamiltonian, which means J and Hamiltonian can have the **same** eigenfunctions.

Because each operator acting on the common eigenfunction can give each **eigenvalue**, which can commute, as shown Eq.90.

*(Eq.91) Orbital angular momentum L and Hamiltonian.*

On the other hand, orbital angular momentum L does **NOT** commute with Hamiltonian.

It means L and Hamiltonian can **NOT** have their common eigenfunction.

As a result, for example, in the states of the energy level n = 2 and j = 3/2, **different** angular momentums l = j ± 1/2 must be included.

Of course, **2d3/2** state (= pair of 2p3/2 ) having l = 2, does **NOT** exist in quantum mechanical hydrogen.

These facts clearly proves Dirac's hydrogen is **wrong**.

From Eq.9 and Eq.11, Hamiltonian is

*(Eq.92)*

**Orbital** angular momentum (L) can be expressed as

*(Eq.93)*

First, we think about **3** (= z ) component of this L,

*(Eq.94)*

Using Eq.94, we show this angular momentum operator (L) *commutes* with the Coulomb potential operator (V) as follows,

*(Eq.95)*

where we use the relations such as

*(Eq.96)*

where the momentum derivative acts on before and after V.

This angular momentum L **doesn't** commute with the momentum "p" in Dirac's Hamiltonian.

*(Eq.97)*

This is de Broglie relation.

So the commutation between orbital angular momentum L and Hamiltonian becomes

*(Eq.98)*

As a result, L and Hamiltonian doesn't commute.

Next we think about **spin** operator S.

*(Eq.99)*

where σ is 4 × 4 Pauli matrix.

This Pauli matrices of spin do NOT commute with α matrices (= Eq.8 ) of Hamiltonian, as follows,

*(Eq.100)*

Here we think about 3 component of spin, like orbital angular momentum of Eq.94.

This σ_{3} does not commute with α_{1} and α_{2} of Hamiltonian, as follows,

*(Eq.101)*

where Pauli matrix's relations of Eq.6 are used.

The total angular momentum operator (J) is the sum of the angular momentum (L) and spin (S) as follows,

*(Eq.102)*

From Eq.98 and Eq.101, this J **commutes** with Dirac Hamiltonian.

*(Eq.103)*

This means both Dirac Hamiltonian and J can have the **common eigenfunction**.

But Dirac Hamiltonian and L can not.

As a result, Dirac's wavefunction **must** always include **pair** state which has the same " **j** " and **energy**, but different L.

So for example, the pair of 2p3/2 becomes the fictional **2d3/2**.

Space parity operator causes the inversion of the space coordinates.

And the polar coordinates change under the **space inversion** as follows,

*(Eq.p1)*

The spherical harmonics function (Y) is known to change under the space inversion as follows,

*(Eq.p2)*

So when the angular momentum (= l ) of Y changes by ± 1, it shows different property under the space inversion.

Some examples of Eq.p2 are,

*(Eq.p3)*

A unit vector can be expressed using the polar coordinates as follows,

*(Eq.p4)*

And the inner product of Pauli matrices (Eq.6) and the unit vector becomes

*(Eq.p5)*

The operator Eq.p5 satisfies

*(Eq.p6)*

And under the space inversion of Eq.p2, the Eq.p5 changes as follows,

*(Eq.p7)*

Eq.p7 means that when the operator Eq.p5 is added, the property under the space inversion changes as +1 → -1 (or -1 → +1)

So when Eq.p5 is added to the spherical harmonics functions (Y), the *angular momentum (= l ) in it changes* by ± 1

(The radial part (r) doesn't change under the unit vector.)

*(Eq.104)*

*(Eq.105)*

*(Eq.106)*

*(Eq.107)*

*(Eq.108)*

2022/9/19 updated. Feel free to link to this site.