Top page ( quantum mechanics is wrong )
Particle physics, standard model is wrong
Quantum field theory is just unphysical math.
(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 ) shows 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 (= total angular momentum J = j = 1/2 ) and 2p3/2 (= J = 3/2, this p.2 ) energy levels is said to be caused by relativistic (paradoxical) spin-orbit interaction ( this p.4 ) 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 or quantum electrodynamics (= QED ) 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 5th-paragraph ).
↑ 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 (= nφ ) instead of the tiny-tiny relativistic spin-orbit magnetic energy. ← So relativistic spin-orbit interaction is illusion, and Bohr-Sommerfeld model 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) ) and de Broglie theory.
The paradoxical Einstein relativistic energy = mc2 copied the original classical Maxwell's authentic mc2 (= which is used in realistic Bohr-Sommerfeld fine structure ) in the wrong way.
Relativistic Dirac hydrogen shows the energy levels of 2s1/2 and 2p1/2 have exactly the same energy values sharing the common total angular momentum j = 1/2 ( this p.16 ).
Much smaller energy splitting (= only 0.000004 eV ) is said to exist between these 2s1/2 and 2p1/2 states called Lamb shift which is negligible (= far smaller than the thermal fluctuation ), useless, too small to observe in the ordinary spectral line.
Quantum electrodynamics (= QED ) is said to give this negligibly small Lamb shift energy splitting by using unreal (unobservable) virtual photons ( this p.1-right, this abstract ). ← This QED unreal virtual particle should be replaced by realistic tiny thermal fluctuation energy splitting without spin.
But in fact, the current relativistic quantum field theory or QED can Not give analytical values of this tiny Lamb shift, instead, they just artificially pick up freely-adjustable numerical values for Lamb shift (= fudge factor, this p.3-left ), which is Not the prediction of (unrealistic) QED at all.
QED itself uses the wrong math trick where meaningless divergent infinities allegedly caused by unreal virtual particles (= QED calculation results ) must be artificially removed by other infinities (= infinite bare charge or mass, this p.4-5 ) using the ad-hoc illegitimate method called renormalization ( this p.2-third-paragraph, this 2-3rd paragraphs ).
↑ Of course, this ad-hoc QED renormalization's idea that infinities minus other infinities may give some tiny finite physical values is the illegitimate and wrong math, because infinities are indefinitely uncertain values that cannot be analytically predicted or determined.
So even the founders of the mainstream relativistic quantum field theory such as Dirac and Feynman harshly criticized and dismissed this fraudulent QED renormalization manipulating infinite virtual particles as "hocus-pocus" and mathematically inconsistent ( this 1st-paragraph ).
As a result, 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 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 cannot produce magnetic field (= No charge motion = No electric current I = No magnetic field B ), hence, the electron can only feel electric field and cannot 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 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 cannot 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 nφ 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 incoporatied 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 manetic 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 maginitude as the Bohr-Sommerfeld hydrogen's bigger kinetic energy's quantum number (= nr, nφ ) 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 ( nr=1, nφ=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 = nr + nφ ), nr is radial kinetic energy quantum number, nφ 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 nr = 1, angular kinetic energy nφ = 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 ( nr=0, nφ=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 nr = 0, angular quantum number or kinetic energy nφ = 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 J2 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 = nr (= 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 using nonrelativistic 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) Dirac equation's unphysical Pauli σ matrices are used as the major part of large kinetic energy, Not as the very weak relativistic (imaginary) spin-orbit magnetic interaction. ← After all, Dirac equation does Not use "spin", either.
It is often said that the mainstream relativistic quantum field theory based on (unphysical) Dirac equation proved the tiny, tiny relativistic spin-orbit (fictitious) magnetic interaction (= only 0.000045 eV ) as fine structure, though it's impossible.
In fact, relativistic Dirac equation uses the alleged ( pseudo-)spin-orbit interaction formula (= LS = σL σ is Pauli matrix ) Not as the tiny relativistic spin-orbit magnetic interaction, but as the irrelevant large electron's kinetic energy !
This is why Dirac hydrogen's spin angular momentum number J or j is treated as the same magnitude as the electron's big (angular) kinetic energy's quantum number (= nφ = kinetic energy > 3 eV ) in the Bohr-Sommerfeld model without spin, instead of the tiny,tiny spin-orbit interaction (= only 0.000045 eV).
Unphysical Dirac equation originally includes the Pauli matrices (= σ ) as a part of the total kinetic energy term = σp where p is electron's total momentum ( this p.14(or p.7 ), this p.30 ), Not as a tiny,tiny spin magnetic moment.
Dirac equation divides the total electron's kinetic energy term including Pauli σ matrix into the radial kinetic energy (= expressed as the derivative d/dr ) and angular kinetic energy terms, like the nonrelativistic Schrödinger hydrogen's kinetic energy term splitting into the radial and angular kinetic energies ( this p.2 ).
↑ This electron's angular kinetic energy term consists of the pseudo-spin-orbit interaction (= σL, this p.3, this p.14 ), which has nothing to do with the tiny-tiny relativistic spin-orbit magnetic interaction.
So the relativistic Dirac equation, which is falsely mentioned as relativistic spin-orbit fine structure, does Not use the spin or spin-orbit interaction, and its Pauli σ matrices are used as a part of electron's (angular) kinetic energy expressed as (J+2/3)/r or (K+1)/r ( K = J + 1/2, this p.7, this p.30, this p.14-15 ), Not as a tiny spin magnetic moment.
It's like the ordinary Schrödinger equation's hydrogen's total kinetic energy splitting into the radial and angular kinetic energies (= Dirac hydrogen gives almost the same energy levels as nonrelativistic Schrödinger' hydrogen ) where this electron's large angular kinetic energy part ( this p.3 ) corresponds to Dirac hydrogen's (pseudo-)spin-orbit interaction operator which has nothing to do with the tiny, tiny original relativistic spin-orbit magnetic energy.
And Dirac hydrogen's solution needs the unreal imaginary (= i ) wavefunction ( this p.21-(93), this p.2, this p.41-42 ), which has No physical reality.
(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 interactin 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 cannot 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 just copying the original successful Bohr-Sommerfeld fine structure.
(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.33(or p.27) this p.2, this p.21 ).
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 sinθeiφ ( 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 Y01 (= orbital angular momentul L = 1, z component orbital angular momentum m = 0 ) and Y11 (= 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 Jz = 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 Y01'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) Unphysical γ matrices used in (pseudo-)relativistic Dirac equation
Unphysical relativistic Dirac equation contains 4×4 γ matrices which consist of 2×2 Pauli σ matrices which have nothing to do with (imaginary) spin, contrary to the mainstream explanation.
These nonphysical γ matrices satisfy the relations like in Eq.2 and Eq.3
(Eq.3)
where I means 2 × 2 identity matrix.
(Eq.4)
And 1-3 components of γ matrices are
(Eq.5)
where j = 1, 2, 3
σj mean 2 × 2 Pauli matrices of
(Eq.6)
Multiplying Eq.1 by cγ0 from left side, and using γ0γ0 = I (= identity matrix ), we get the Dirac's Hamiltonian or total energy equation of
(Eq.7)
where the time derivative means total energy E, and the space derivative means electron's momentum based on de Broglie wave theory.
From γ matrices of Eq.3 and Eq.5, we get
(Eq.8)
From Eq.7 and Eq.8, Dirac's Hamiltonian operator ( which means the total energy ) including the Coulomb potential energy V is
(Eq.9)
The original relativistic Dirac equation is designed to satisfy the Einstein relativistic energy equation by secretly including relativistic Klein-Gordon equation in its unphysical Dirac free-particle wave solution.
But this Dirac' hydrogen does Not use the original relativistic form of Dirac's free particle solution ψ (= which must contain infinite virtual particle energies ), instead, it uses irrelevant non-relativistic Schrödinger's hydrogen solution and unrealistic imaginary radial wavefunction (= two different wavefunctions ) to incorporate the nonrelativistic Coulomb potential energy term.
So Dirac equation for hydrogen is No longer the relativistic theory (= the authentic relativistic wavefunction must be invariant under Lorentz transofrmation or seen from arbitrary moving observers, which needs infinite virtual particle energies ).
Here p means the electron's momentum operator as follows,
(Eq.10)
So, using Eq.3, Eq.5 and Eq.8, the total energy E or Hamiltonian of Eq.9 becomes
(Eq.11)
Here we divide the Dirac hydrogen's wavefunction ψ (= which is completely different from the original relativistic Dirac free-particle wavefunction ) into two 2 × 1 matrices.
(Eq.12)
The upper and lower spinors must have different angular momentums (= whose difference is 1, this p.4 ) which cause the unreal 1P1/2 and 2D3/2 states in the unphysical Dirac hydrogen solution.
Substituting Eq.12 into Eq.11, we get
(Eq.13)
As shown in Eq.13, Dirac equation originally contains the Pauli σ matrices as a part of the electron's (large) total kinetic energy (= or momentum p, this p.30 ), which has nothing to do with the tiny, tiny relativistic spin-orbit magnetic interaction. ← This is one of deceptive tricks used in (pseudo-)relativistic Dirac hydrogen.
The upper part of Eq.13 is
(Eq.14)
And the lower part of Eq.13 is
(Eq.15)
And the potential energy V of Eq.13 means Coulomb electric potential energy (= around the static central nucleus ) as follows,
(Eq.16)
First we omit 4πε, and add this later.
And we solve the simultaneous equations of Eq.14 and Eq.15 and get the hydrogen's energy E.
Of course, the energy "E" of Eq.14 and Eq.15 is the common value.
And total angular momentum j or J is also the common value.
But orbital angular momentum "l" or L is different in the upper and lower spinors ( this p.16-18, this p.4-last ).
As a result, Dirac's hydrogen wavefunction contains many contradictory states such as
(Eq.17)
1p1/2 is the energy level is n = 1, and total angular momentum is j = 1/2, which is just equal to 1s1/2 state.
But 1p1/2 state has unrealistic l = 1 angular momentum.
Because 1s1/2 has orbital angular momentum of l = j - 1/2 = 0, and 1p1/2 has l = j + 1/2 = 1.
Of course, the (unphysical) quantum mechanical hydrogen atom does NOT have orbital angular momentum l = 1 in the energy level of n = 1.
So this is self-contradiction.
Using the relations of Pauli matrices,
(Eq.18)
We obtain
(Eq.19)
Using Eq.19 and this relation, the electron's total kinetic energy term of Dirac hydrogen equation becomes
(Eq.20)
↑ The fictitious spin-orbit interaction operator (= σL ) appears in the electron's angular kinetic energy term, which is completely irrelevant to the tiny, tiny relativistic spin-orbit magnetic energy ( this p.9 ).
So Dirac's hyrogen's energy splitting is caused by the electron's kinetic energy difference (= like Bohr-Sommerfeld model without spin ), Not by (imaginary) relativistic spin-orbit magnetic energy. ← Spin or the relativistic spin-orbit interaction has No evidence theoretically or experimentally.
When physicists talk about (imaginary) relativistic spin-orbit magnetic interaction, they do Not use the original solvable relativistic Dirac hydrogen, instead, they use various versions of artificially-created ad-hoc spin-orbit relations (= which must be artificially made very tiny, multiplied by 1/c or 1/(light speed c) ) irrelevant to relativistic Dirac equation uses (fictitious) spin σ Pauli matrices as a part of electron's large kinetic energy instead of tiny spin-orbit magnetic energy.
The total angular momentum operator (J) is the sum of the angular momentum (L) and spin (S) as follows ( this p.18 ),
(Eq.21)
The square of this total angular momentum (J) of Eq.21 becomes
(Eq.22)
Here we use the relation of Pauli matrices ( σσ = I = 1, this p.10 ).
And replacing the squared angular momentum operators like J2 → J(J+1) and L2 → L(L+1)
(Eq.23)
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.24)
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 )
(Eq.25)
↑ The alleged spin-orbit operator (= σL ) is changed into the total angular momentum j which is used as the electron's large angular kinetic energy (= equal to Bohr-Sommerfeld kinetic energy quantum number nφ ) instead of tiny spin-orbit magnetic energy, which means quantum spin and relativistic spin-orbit interaction are illusion.
Seeing Eq.24 and Eq.25, we know that the pseudo-spin-orbit operator (= σL ) equals -(K+1) where the integer K = ±k = ±(J+1/2) depending on whether L = J ± 1/2 like
(Eq.26)
(Eq.27) pseudo-spin-orbit term (= σL ) → (k+1) → angular kinetic energy.
As shown in the upper figure, relativistic Dirac hydrogen's pseudo-spin-orbit term (= σL ) gives the irrelevant electron's large kinetic energy expressed as j (= about 3 eV in 2P state ) in the tangential or angular direction instead of the tiny, tiny relativistic spin-orbit magnetic energy (= only 0.000045 eV ) between the electron's spin magnetic moment and the imaginary internal magnetic field caused by the heavier nuclear orbital motion in the electron's rest frame.
Because the completely-different coefficients are used = the tiny coefficient of the spin magnetic moment equal to Bohr magneton or internal (imaginary) magnetic or electric field ( this p.20 ) is Not added to this Dirac hydrogen pseudo-spin-orbit operator which is just used as a major part of the total kinetic energy ( this p.3, this p.16-19 ).
↑ So this Dirac pseudo-spin-orbit value (= σL → k+1 or J+3/2 = integer ) constitutes the angular (kinetic energy) quantum number (= integer ) in the last principal quantum number n = n' (=radial quantun number) + |k| = n'+J+1/2 = integer ( this p.4(p.5), p.7(p.8), this-lower, this p.7-11, this p.17,29 ) with No half-integer spin left.
This shows relativistic Dirac hydrogen does Not prove the (imaginary) relativistic spin-orbit magnetic energy at all.
The so-called Dirac hydrogen's fine structure is caused by the modified electron's kinetic energy (= without spin ) like Bohr-Sommerfeld model.
Actually, this Dirac hydrogen energy solution is obtained in the rest frame of a nucleus with No relativistic (fictitious) magnetic field ( this Figure.3 ) instead of the rest frame of a (moving) electron, so there should be No relativistic spin-orbit magnetic interaction.
(Eq.28) 2P1/2 which should have spin-orbit interaction includes the paradoxical solution with No spin-orbit interaction in Dirac hydrogen.
Dirac hydrogen's solution of four component spinor must always consist of two different orbital angular momentums L (= one is L = J +1/2, the other is L = J - 1/2 in the common total angular momentum J ) in the upper and lower two-component spinors, which cause unreal states such as 1P1/2, so false.
Furthermore, each two component spinor Φ under the common total and orbital angular momentums must include the paradoxical state.
Dirac hydrogen's each two-component spinor (= angular kinetic energy part ) must always consist of two different non-relativistic Schrödinger's hydrogen spherical Harmonics (= so Dirac hydrogen is Not relativistic ) with the same orbital L but different z-component orbital angular momentum number ( this p.13, this p.21 ).
For example, the upper 2P1/2 state (= total angular momentum J = 1/2, and orbital angular momentum L = 1, z-component total angular momentum m = 1/2 = spin is parallel to z direction ) spinor consists of two different non-relativistic spherical Harmonics of Y01 (= orbital angular momentum L = 1, z-magnetic quantum number ml = 0, so this orbit is perpendicular to the spin ) and Y11 (= orbital angular momentum L = 1 and z-magnetic quantum number ml = 1 ).
↑ This Y01 causes No spin-orbit interaction (= so Not 2P1/2 state ), though it's included in 2P1/2 solution, because the spin and orbital angular momentums are perpendicular to each other (= total angular momentum = 1/2, which means the spin angular momentum points in z direction, but the orbital angular momentum of the spherical Harmonics is perpendicular to z direction ).
This paradoxical spherical Harmonics required by Dirac hydrogen is also one of evidences showing relativistic Dirac hydrogen's fine structure is illegitimate and wrong.
(Eq.29) Dirac hydrogen's solution, one of two component spinors, L = J - 1/2 case
Also when the orbital angular momentum L = J - 1/2, Dirac hydrogen solution needs two different (unreal) Schrodinger's hydrogen spherical Hamonics wavefunctions, as shown in Eq.29.
Using Eq.20 and Eq.26, the equation of Eq.14 can be expressed as
(Eq.30)
Using the artificial relation switching angular momentum wavefunctions ( this p.20, this p.7, this p.4 ),
(Eq.31)
The sperical harmonics parts (Φ) of Eq.30 becomes the common same wavefunction, and it can be eliminated as follows,
(Eq.32)
As shown in the upper Eq.32, Dirac pseudo-spin-orbit interaction operator (= σL ) changes into the irrelevant electron's kinetic energy in the angular direction ( k = J + 1/2 = an integer related to orbital angular quantized number ), which means Dirac hydrogen does Not consider the relativistic spin-orbit magnetic energy at all.
The use of angular switching operator means if there is only one angular momentum (= L ) in Dirac spinor solution, we cannot eliminate the angular wavefunction Φ, or solve it. → unreal states such as 2P1/2 are always needed.
In the same way, the other equation Eq.15 can be expressed as
(Eq.33)
where Eq.26 is used.
And using Eq.31, the angular spherical Harmonics Φ can be eliminated as follows,
(Eq.34)
As shown in Eq.26, Eq.32 and Eq.34, Dirac pseudo-spin-orbit interaction σL turns into the angular component of electron's kinetic energy (= about 3 eV in 2P ) expressed as the integer k or j+1/2 ( this p.20 ), Not the tiny, tiny relativistic spin-orbit magnetic energy (= only 0.000045 eV ).
↑ To change this large angular kinetic energy into the tiny, tiny relativistic spin-orbit magnetic energy, the very small magnetic coefficient of magnetic permeability and Bohr magneton must be added to it, which is Not the case in Dirac hydrogen's pseudo-spin-orbit term used as a large kinetic energy.
So the mainstream claim that relativistic quantum field theory based on Dirac hydrogen shows the fine structure energy splitting caused by the tiny, tiny relativistic spin-orbit (fictitious) magnetic interaction is completely wrong and baseless.
We replace the radial functions of f(r) and g(r) by
(Eq.35)
Substituting Eq.35 into Eq.32, we have ( this p.2, this p.19, this p.22~ )
(Eq.36)
Coulomb potential V of Eq.16 is used here.
In the same way, substituting Eq.35 into Eq.34, we have
(Eq.37)
We expand u(r) and v(r) as follows,
(Eq.38)
Substituting Eq.38 into Eq.36 and Eq.37, and seeing the power of γ-1, we have
(Eq.39)
To avoid the solution of a0= b0 = 0 in Eq.39, the following relation must be satisfied.
(Eq.40)
To avoid the divergence at the origin, the γ must be plus as follows,
(Eq.41)
To converge at r → ∞, we suppose u(r) and v(r) satisfiy
(Eq.42)
And at r → ∞, Eq.36 and Eq.37 become
(Eq.43)
Substituting Eq.42 into Eq.43, we have
(Eq.44)
From Eq.44, λ (= plus) becomes
(Eq.45)
Considering Eq.38 and Eq.42, we can express u(r) and v(r) as follows,
(Eq.46)
As you may notice, the replacement of Eq.46 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.16) to γ of Eq.41, it can be expressed by the fine structure constant (=α)
(Eq.47)
where this fine structure constant α is approximately
(Eq.48)
Substituting Eq.46 into Eq.36, we can get the following relations ( using Eq.45 and the green part of Eq.44 ).
(Eq.49)
and Substituting Eq.46 into Eq.37, we have
(Eq.50)
Here we define (using Eq.41 and Eq.45)
(Eq.51)
and replace r by x as follows,
(Eq.52)
Using Eq.51 and Eq.52, Eq.50 become
(Eq.53)
And Eq.49 becomes
(Eq.54)
Here we expand ω1 and ω2 as follows,
(Eq.55)
Substituting Eq.55 into Eq.53, coefficient of x to the power of n - 1 ( = xn-1 ) becomes,
(Eq.56)
In the same way, substituting Eq.55 into Eq.54, the coefficient of xn-1 is
(Eq.57)
Summing Eq.56 and Eq.57, we obtain
(Eq.58)
Substituting cn and cn-1 of Eq.58 into a and b of Eq.56, and using Eq.51, we get the relation of
(Eq.59)
When we use the replacements of
(Eq.60)
Eq.59 beomes
(Eq.61)
From the relation of Eq.61, we can define the function F(x) as follows,
(Eq.62)
And using Eq.58, ω1 and ω2 (= Eq.55) can be expressed using the common cn,
(Eq.63)
Eq.63 means if we decide the power of cn, the upper and lower spinors become the same as cn.
The power of cn 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 2s1/2, L = 1 in 2p1/2 ).
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.64)
To make F(x) a finite series, μ of Eq.61 must satisfy
(Eq.65)
where F(x) becomes the n' th degrees with resprct to x.
From Eq.60,
(Eq.66)
Substituting Eq.51 into Eq.66,
(Eq.67)
Here we return 4πε.
Using the fine structure constant α of Eq.48, Eq.67 becomes
(Eq.68)
They difine the new integer n (= energy level or principal quantum number ) as
(Eq.69) 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.68 becomes ( using Eq.47 and |k| = j + 1/2 )
(Eq.70) 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.70 ( this p.9, this p.7 ) are completely consistent with those of Bohr Sommerfeld model of Eq.71 ( this p.12 ).
(Eq.71) 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,
2p1/2 (n=2, j=1/2) -------- elliptic, (nr=1, nφ=1)
2p3/2 (n=2, j=3/2) -------- circular, (nr=0, nφ=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, eiφ ( 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 r0 is Bohr radius.
Eq.76 means the exponential part of Dirac's hydrogen 1S is equal to R10 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, a0 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 r0 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 σ3 does not commute with α1 and α2 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.
.PNG){kind=link}
.PNG){kind=link}
