Top page (correct Bohr model including helium )

Derivation of Fermi golden rule and statistics.

Spintronics of condensed matter physics is illusion.

- Failure of "spin" and quantum field theory.
- Superfluid, Cooper pair = "imaginary" phonon ?
- Kondo effect ( Anderson model ) doesn't prove "spin".

*(Fig.1) Spinning speed of an electron is much faster than light !*

According to actual experiments, a single electrons is a very small **point**-like particle.

But quantum mechanics insists this **point**-like electron has **1/2 ħ** angular momentum.

As you know, the angular momentum means mv× r ( m= electron mass, v=rotation speed, r is electron radius ).

Considering electron's very **small** mass and radius, this rotating speed "v" must be **more than 100** times **light speed** to reach 1/2 ħ ! ( See also this page. )

This means quantum mechanical spin is **unrealistic** and just a **fantasy**.

*(Fig.2) Pauli exclusion principle is a force ? → "Shut up and calculate !"*

If you see some condensed matter physics textbooks, you would find those books are **filled** with unrealistic "spins".

Main reason is "Pauli **exclusion** principle", in which two identical fermions **cannot** enter in the **same** place (= state ).

( But I think, this definition of the "same" place is very **vague**. How can we define these **boundary** lines ?? )

For example, in 1s energy state, only spin "up" and "down" electrons can exist.

They introduced very artificial "colors" (= red, blue, green ) in quarks to satisfy Pauli exclusion principle by force.

So, the force involved in this **mysterious** Pauli exclusion principle is **much stronger** than all other fundamental forces !

The problem is physicists **NEVER** try to ask what this Pauli exclusion principle **really** is, even in 21th century now.

Instead of **wasting** a great deal of money in useless accelerator, they should investigate the **true** form of this Pauli exclusion, first, I think.

*(Fig.3) How "mathematical" operator can describe various phenomena, rich in variety ??*

In the present quantum field theories including condensed matter physics, standard model, and string theory, they completely **rely** on "mathematical" ( **NOT** physical ) operators to describe various phenomena.

As you see this page, these creation and annihilation operators do **NOT** have the power to express various **dynamic** and complicated phenomena in the actual world.

Because, when a particle is created ( or annihilated ), these operators **cannot** designate various differernt **places** and **times** in each different dynamic particle.

So the **main** obstructions to the development of sicence are these very **abstract** oparators (and "spin ").

*(Fig.4) Real variable world is NOT mathematical and simple operators !*

As you see various physical phenomena and **complicated** mechanism in our human bodies, actual particles such as electrons and protons are **moving around
**, interacting with each other at various **different** places and times.

For example, when a particle A is approching particle B, other another C is getting away from particle D ....

And various interactive "**timing**" and positional **relationship** determine various biological functions, which is **rich** in variety.

Unfortunately, the present mathematical operators such as "a^{†}" and "a" **cannot** describe these **variable** functions at all.

*(Fig.5) Spin - spin interaction is unrealistically weak.*

It is said that ferromagnetism is caused by **spin-spin** interactions in metals.

But this spin magnetic moment is **too weak** to cause ferromagnetism in room temperature !

The lattice constant of iron is 2.84 Å.

In this distance, the energy difference between parallel and antiparallel spin interactions are **too small**, as shown in Fig.6.

Using Boltzmann constant k, this energy difference is equal to only **0.22** kelvin. ( See this page. )

So if spin-spn interaction is caused by their magnetic moments, a magnet is easily **broken** in room temperature !

*(Fig.6) Weak "spin" can always turn other spins toward the same ( opposite ) directions ??*

According to this book ( Story of Spin by S.Tomonaga ), the magnetic moment of electron spin is too weak (= fine structure level, **0.00005 eV** ).

So these spins **don't** have the **initiative** at all, overcoming various Coulomb interactions.

But they try to give **imaginary** "leadership" to this **very weak** spin, by force. Why ?

Because as I said, mysterious Pauli exclusion principle is the **strongest** in all fundamental forces.

And they like various **artifical** rules in condensed matter physics.

In L-S coupling, when there is **little** electrons, total angular momentum becomes J = **L-S**, but at some point, this J **suddenly** becomes **L+S**, they insist. ( very "convenient" spin. )

Of course, it is **impossible** that we confirm these mathematical orbitals and ( rotating )"spin" directly.

*(Fig.7) Real spin state ? Point particle ?*

In addition to artifial rules, "spin" is precessing, they insist.

So according to quantum mechanics, "real" spin angular momentum is **NOT** 1/2 ħ !

How **point**-like spin can precess ?

And why this angle of inclination (= θ ) is **constant** even in various different magnetic fields ?

This is strange.

*(Fig.8) "Mathematical" Phonons = viscosity ??*

Superfluidity is a state of matter in which the matter behaves like a fluid with **zero viscosity** at extremely low temperature.

The problem is that the present quantum theory tries to describe this phenomena using "mathematical" **phonons**.

"Phonons" are quasiparticles, so they are **NOT** real.

But the present theories are **restricted** strictly to very abstract "mathematical" **symbols**, so they **cannot** explain what these phonons ( and "spin" ) really are, using **clearer** objects.

This is the main reason why the present condensed matter physics **cannot** be developed to express more **dynamic** nano-phenomena.

*(Eq.1) Symmetry breaking ? *

According to the present quantum theory, Higgs potential in non-relativistic field is given by Eq.1.

*(Eq.2)*

When φ is determined by a fixed value of "v" (= Eq.2 ), this is ground state, and "symmetry" is **broken**, they insist.

*(Eq.3)*

They like to introduce very abstact concept such as "**phase**".

( Though they **NEVER** try to explain what this "phase" really is. )

When this phase χ(x) includes the position variable "x", energy of Eq.1 becomes **higher** than the ground state, by

*(Eq.4)*

So this **ununiformity** in the phase χ(x) causes superfluid, they insist.

But as most people feel, it is **impossible** to describe various **variable** phenomena and interactions using ONLY these abstract mathematical **symbols**.

In short, these theories completely **lack** realistic useful tools.

They are just bound with some mathematical **spells**.

*(Eq.5)*

They **suddenly** introduced Hamiltonian including imaginary "phonons", like Eq.5.

They insist these creation or annihilation operators (= ζ ) mean "phonons", which can cause superconductivity, as if by **magic**.

The first term means electron's Hamiltonian (= kinetic energy ), and the second term is **phonon**'s energy.

And the third term means the **interactive** Hamiltonian among electrons and "phonon", they insist.

*(Eq.6)*

When some electrons have higher energies than a single phonon, they can emit phonon and cause **viscosity** with other electrons, they insist.

So when each electron has **lower** energy than a single phonon at extremely low temperature, they cannot interact with each other, which is "superfluid".

Of course, considering these phonons are **unreal** quasiparticles, they should **clarify** what these phonons really are, as soon as possible.

*(Eq.7)*

According to the BCS theory, two **negative** electrons are **attracted** toward each other, and form stable Cooper pair, they insist.

Again, they don't try to explain **concrete** mechanism of this mysterious **attraction**.

When "g" of the third term of Eq.7 is positive ( g > 0 ), spin up and down electrons can form Cooper pair by attractive interaction, they insist.

*(Eq.8)*

So Eq.8 is Cooper pair.

But the **concrete** mechanisms remain **unknown** even in 21th century now.

This is strange.

Instead of wasting great deal of money in **useless** accelerator, they should **clarify** the true form of this phonons as soon as possible, I think.

*(Fig.9) Kondo effect doesn't prove "spin" at all.*

Usually, as temperature becomes lower, the electrical resistance becomes weaker.

But in metals with magnetic impurities ( ex. Cu + 0.1 % Fe ), this resistance becomes stronger at **much lower** temperature again.

They insist these effects are caused by **spin-spin** interaction among s and 3d electrons.

But as I say in this section, these theories **cannot** prove existence of "spin" at all.

And "spin" proves to be only "mathematical" ( **NOT** physical ) **symbols**.

*(Eq.9) Anderson Hamiltonian.*

In Eq.9, c^{†} and c mean creation and annihilation operators of conduction electrons (= s orbital ).

And d^{†} and d mean **3d** orbital electrons of a magnetic impurity such as Fe.

( As I said in Fig.3, these very **abstract** operators **cannot** express more **dynamic** electron's behavior. )

The first and second terms mean each independent energy, and the third term means **repulsive** Coulomb interaction between 3d electrons.

And the fourth term is **interactive** Hamiltonian between s (= c ) and 3d electrons, they insist.

*(Eq.10)*

As shown in this page, effective Hamiltonian of second order ( with respect to "V" ) becomes like Eq.10.

"f", "m" and "i" mean "final", "middle" and "initial" states.

*(Fig.10) Conduction electron is scatterd by 3d electron into k' state.*

Fig.10 shows the process in which a conduction electron moves from its fermi energy state into 3d orbital, and then, other conductive state (= k' ) by some scattering.

Unfortunately, all they can do is to express these physical process using **only** very abstract creation and annihilation **operators**, as shonw in Eq.11.

*(Eq.11)*

A conduction electron is first annihilated (= c ) in the fermi sphere, and then created (= d^{†} ) in 3d orbital.

After that, this electron is annihilated (= "d" ), and moves into k' state (= created, c^{†} )

Only interactive Hamiltonian "V" of Eq.9 is involved in the numerator.

And when two electrons exist in 3d orbital, the repulsive interaction (= "U" ) appear.

Of course, two electrons in 3d orbital satisfy Pauli exclusion principle.

( But these states are only **speculations**, and cannot be seen directly, )

*(Eq.12)*

In Eq.11, **anticommutation** relation of Eq.12 is used.

So as I said above, this anticommulation relation (= Pauli exclusion principle ) has **absolute** power, in any situation !

Though spin magnetic moment is extremely **weak**.

*(Eq.13)*

In Eq.13, the final k' electron has "up" spin instead of "down" spin in Eq.11.

In this way, there are total 8 different patterns in this process.

Summing up **all** patterns of scattering, they get s-d model Hamiltonian of

*(Eq.14)*

In Eq.14, they change various interactive V_{kd} into **one** common form of "V", approximately.

But the most important part is the **diversity** of this interactive V.

As long as they don't try to investigate **concrete** dynamic interactive **motions** of electrons, the present quantum theory **cannot** escape blind alley.

*(Fig.11) "Spin" = Math ?? NO physical reality.*

In conclusion, "spin" in various papers in condensed matter physcis and spintronics is just a "**mathematical**" thing, and has **NO** physical reality at all.

If they want to describe various scattering, they should use more **realistic** and **concrete** models to express them, **NOT** relying on "abstract" operators.

*(Eq.15)*

They change 3d electron's operators into spin operator "S".

As you see, they **believe** "mathematical" spin (= symbol ) **blindly**.

So in various papers in condensed matter physics, the word of "spin" **always** appears.

Though they **don't** try to explain what this "spin" really is.

*(Eq.16)*

Here they define repulsive Coulomb energy U as **infinity**.

So the total 3d electron's number in final state becomes "1" ( not "2" ).

Using Eq.15 and Eq.16,

*(Eq.17)*

They suppose the energy of conduction electron is much smaller than repulsive U.

And the special energy relation of "particle hole symmetry" is defined, like Eq.18.

*(Eq.18)*

Here, the following Pauli matrices are used.

*(Eq.19)*

The important point is that "x" (= 1 ) and "y" (= 2 ) components of Pauli matrices **contradict** quantum mechanics itself.

Because there are only two states (= "up" and "down" spins in "z" ) according to Stern-Gerlach.

So these σ_{x} and σ_{y} are only **mathematical** products.

Using Pauli matrices, they change conduction electron's operators, as follows,

*(Eq.20)*

As a result, s-d Hamiltonian of Eq.14 becomes

*(Eq.21)*

In Eq.21, J_{eff} is negative, so when the spins of s (= σ ) and d (= S ) electrons are antiparallel to each other,

*(Eq.22)*

Hamiltonian (= H_{s-d} ) becomes negative and stable (= Kondo singlet ).

So they insist conduction and 3d electrons have **diamagnetism** relationships.

Of course, spin magnetic moment itself do **NOT** have power overcoming various Coulomb forces.

So the relation of Eq.22 is completely **unrealistic**.

Actually, they **cannot** show concrete state of " spin ". They only show "mathematical" matrices.

Unfortunately, this is the **true** figure of "spin" in various spintronics papers.

*(Eq.23)*

The electric conductivity (= σ(T) ) is defined as J = σ(T) E.

( J = current, E = external electric field ).

In this section, we show this conductivity can be expressed like Eq.23.

In Eq.23, τ means **relaxation time**.

After this τ, electron's state almost returns to its original state.

"f(ε)" is electron's distribution (= Fermi-Dirac statistics ).

*(Eq.24)*

We define electric current j as shown in Eq.25.

"ħk" is momentum "p", and Ω is resistance.

*(Eq.25)*

" f_{↑↓} " and " f_{0} " represent fermi distributions with or without electric currents, respectively.

They suppose after the relaxation time of τ, "f_{↑↓}" returns to static "f_{0}".

The derivative of f_{↑↓} with respective to "t" is

*(Eq.26)*

They define current (= state ) as almot **constant**, so

*(Eq.27)*

According to Newton law,

*(Eq.28)*

From Eq.26 to Eq.28, the change of state "f(ε)" during the time of τ is

*(Eq.29)*

where

*(Eq.30)*

From Eq.25, Eq.29, and Eq.30, the current j becomes

*(Eq.31)*

where

*(Eq.32)*

In Eq.31, they change Σ (= sum of "momentums" ) into integration of ε, using probability density of ρ

As a result, they can get conductivity σ of Eq.23.

The transition probability is given by Fermi golden rule, as follows,

*(Eq.33)*

According ot perturbation theory, the transition matrix "T" can be defined as

*(Eq.34)*

First we think about first-order perturbation with respect to H_{sd} ( not "V" ) in Eq.34.
( = the first term in Eq.34 right. )

Spin operators satisfy

*(Eq.35)*

And the relaxation time τ can be defined as

*(Eq.36)*

From Eq.21, Eq.33-Eq.36,

*(Eq.37)*

The result of Eq.37 means first-order perturbation (= Born approximation ).

It **cannot** show the relation between the temperature "T" and conductivity ( or resistance ).

Because temperture "T" doesn't appear.

As shown in the second term of Eq.34 right, the second-order Hamiltonian includes **reciprocal** of ( E-H ).

This mathematical definition by quantum mechanics leads to the **serious** divergence in Kondo effects.

*(Eq.38)*

The second term of Eq.34 right include two H_{sd}, so the number of operators becomes **twice**.

So the conduction electron is **more scatterd**, to and fro.

Much more scattering patterns have to be considered in this second-order.

*(Fig.12) High order = weak interaction ↓ → scattering ↑↑ ?*

Strange to say, in these perturbation methods, as **higher** order interaction is considered, each particle is **more scattered**.

This interpretation is strange.

Because higher order interactions are much **weaker**, due to **small** λ.

So in addition to **lack** of clear pictures, this method includes self-contradiction.

*(Eq.39)*

Like Ee.39, anticommutator is used.

c^{†}c is number operator (= f(ε) ).

Summing up all patterns, Eq.38 includes the following equations

In Eq.40, Σ is changed into integration.

As a result, the second order perturbation with respect to H_{sd} gives the **temperature** (= T ).

The electric conductivity is

So the electric resistance becomes

*(Eq.42)*

As shown in Eq.42, as the temperature becomes lower, the resistance becomes **stronger** (= Kondo effect ).

The problem is when the temperature T is equal to 0 (K), this resistance becomes divergent to **infinity**.

*(Eq.43)*

So they try to solve this serious problem using various mathematical tricks such as **renormalization**.

But as you feel, these unitary definitions of quantum mechanics always cause **serious flaws** such as **infinity**.

As you know, the **final** form of this quantum mechanics is unrealistic **10** dimensional string theory.

Considering all these problems, Bohr model with **NO** divergence and singularity is more **reasonable** than strange quantum mechanics.

We can naturally think that **de Broglie wave** effects by impurities become **dominant** at very low temperature, which causes resistance to conduction electron.

So Kondo effect can be described by de Broglie waves, in the same way as quantized magnetic flux and ferromegnetism.

2013/9/15 updated. Feel free to link to this site.