Flügge, S., 2012. Practical quantum mechanics. Springer Science & Business Media.

Prof. Azeez Abdullah Barzinjy

Thank you so much for pointing out in this thread the classical textbook by Prof. S. Flugge on practical quantum mechanics, which has served well many generations of physicists.

The book definitely, treats extensively and pedagogically the elastic scattering problem in non-relativistic QM, and also there are a couple of problems dedicated to the phase shift problem and bound states.

Best Regards.

A nice physical picture of the trouble in normal metals with the unitary scattering limit is found in the following external blog:

"Bad metals and the unitary limit"

https://condensedconcepts.blogspot.com/2016/07/bad-metals-and-unitary-limit.html

Combining nonmagnetic scattering and Zeeman splitting in a T matrix formalism is new for me.

Kind of difficult to understand how they get linked. So I post the reference:

Article Unitary Limit of Spin-orbit scattering in two-dimensional s-...

The following article related to this thread was finally published

http://www.sciencepublishinggroup.com/journal/paperinfo?journalid=290&doi=10.11648/j.ep.20210501.11

We acknowledge the support by PG Science publishing group.

Dear Prof Pedro L. Contreras E. , Excellent idea of correlations of Fermi-Dirac stats with QM - https://www.researchgate.net/post/What_are_the_major_refutations_of_classical_mechanics_to_justify_quantum_mechanics

https://www.researchgate.net/post/Whether_Quantum_Mechanics_can_be_justified_with_Quantum_Statics_Quantum_dynamics

https://www.researchgate.net/post/Partition_function_a_Quest

I am trying but not expert like you .

Thank you for the threads, Prof. Jaydip Datta

Best Regards.

Soon we will have another preprint published concerning this thread.

We just published a preprint very related to this thread. I looked for this answer for 7 years of my life with no resources at all.

I thank you Dr. Stephen A. Langford for your unique support. God Bless you and Your Family, Dear Dr.

Preprint Non-magnetic tight binding disorder effects in the γ sheet of Sr2RuO4

p wave triplet superconductors differentiate in transport properties from d wave superconductors.

Their properties will depend on the level of doping, and on the strength of the impurity field.

p wave superconductors are naturally in the unitary limit, on the other hand, d wave superconductors are in the unitary limit only for a few values of these parameters.

Research Proposal Coulomb interactions and EM statics

Thank you so much for sharing with us, your paper entitled "Research Proposal Coulomb interactions and EM statics

There are two types of unitary non-magnetic elastic scattering:

• the homogeneous one
• the inhomogeneous one

A simple kinetic explanation of the Ioffe-Regel criterion is given in the following external post:

https://www.quora.com/What-is-the-Ioffe-Regel-criterion-in-the-context-of-wave-transport.

In general, there are three limits for the l kF product according to the Ioffe-Regel criterion. If l kF >> 1, we have the metallic Born regime, for l kF ~ 1 we have the unitary regime and for l kF < 1, we have Mott - Anderson insulator-metallic limit.

We will soon have another preprint on this issue. More clear ideas are going now.

The unitary state can be investigated using a more unconventional approach following general physical principles.

We have just published a new preprint related to the subject of this thread.

Preprint Dressed behavior of the quasiparticles lifetime in the unita...

We have introduced a hypothesis of an inhomogeneous phase in triplet superconductors in the unitary limit in our last preprint.

Compelling quasiparticles (dressed electrons) disappear and only supercarriers remain in the sample.

The unitary state can be defined as the metallic state which is driven by hoping among impurity nonmagnetic static atoms with the validity of the third law of thermodynamics, and at extremely low temperatures.

P. Contreras.

This paper is related to this thread:

https://arxiv.org/pdf/1908.10145.pdf

Article What condensed matter physics and statistical physics teach ...

In the unitary limit of unconventional superconductors, the reduced scattering phase space seems to be the appropriate physical space to study the kinetic behavior of quasiparticles, since only total energy E is conserved.

(Momentum p is not conserved) .

We will be publishing a new preprint soon on this topic.

Best Regards.

The preprint of the two corresponding cases of Wigner distribution probabilities in the Unitary limit of triplet superconductors has been published:

Preprint Quasi-point versus point nodes in Sr2RuO4, the case of a fla...

Dear all, we found 3 families of Wigner probabilistic distributions for the unitary limit.

These 3 families allowed us to explain the distribution of quasiparticles & Cooper pairs in unconventional superconductors.

The normal state quasiparticles appear in the superconducting state by dispersion due to non-magnetic impurities, & we called them dressed quasiparticles (dressed by the atomic interaction of non-magnetic strontium atoms in all cases).

We will write a final article where we will try to elaborate on a few pedagogical concepts starting in 2022.

In the next post I will elaborate on, the summary of the macroscopic findings, thank you all for following & recommending our work.

CC. Dianela Osorio

Eugeniy Beliayev

https://arxiv.org/pdf/1708.03872.pdf,kindly check this link.

https://www.ifsc.usp.br/~strontium/Publication/Scripts/QuantumMechanicsScript.pdf,this link may be useful.

Prof. Faraed Salman, thank you so much for your posts and the links.

Best Regards.

The following image is very related to this thread, it has also been published in our project: "Symmetry tensor field breaking effects in Sr2RuO4 and the velocity jumps due to the Ehrenfest relations"

It reflects the main vocabulary we use for our preprints and publications

CC Dianela Osorio

Eugeniy Beliayev

https://www.researchgate.net/project/Symmetry-tensor-field-breaking-effects-in-Sr2RuO4-and-the-velocity-jumps-due-to-the-Ehrenfest-relations

The unitary limit seems to be the natural limit in which non-magnetic impurities act as a breaking pairs mechanism to destroy superconductivity in certain compounds with different kinds of nodes/quasinodes in some metallic alloys, independent of the value of Tc.

The following work related to this thread has been published.

Thank you so much to the LTP Journal in Kharkiv for their Team reviewing & editing support to us.

Deleted research item The research item mentioned here has been deleted

A new preprint with the DOS effect on the unitary limit and the non-magnetic impurities effects is coming soon on Unc. Sup.

Best Regards.

We have added another preprint concerning this thread:

Preprint The two-dimensional density of states in normal and supercon...

also, we have added an article:

Deleted research item The research item mentioned here has been deleted

Best Regards.

The density of states DOS in dirty unconventional superconductors can be explained in a few cases by the following attached slide, which shows the difference when the unitary limit is present with respect to the Born limit.

All the key is the residual DOS.

We have published a work that shows that the FS γ-sheet on Sr2RuO4 is in the unitary limit for most non-magnetic impurity concentrations nimp. The result of this work was the beginning of the question thread.

We deeply acknowledged the support of Prof. Eugeniy Beliayev

Article Non-magnetic tight binding disorder effects in the γ sheet of Sr2RuO4

CC Dianela Osorio

A new version for our preprint concerning this thread has been published in arXiv and HAL. The unitary tight-binding limit is the one that helps us to see where are the nodes in Sr2RuO4 Referee comments are added.

Preprint Quasi-point versus point nodes in Sr2RuO4, the case of a fla...

The unitary limit as a function of disorder concentration results in different phases for the compound Sr2RuO4 as we showed recently:

Preprint Quasi-point versus point nodes in Sr2RuO4, the case of a fla...

Using the elusive unitary limit in a TB framework we were able to explain some experimental properties of fundamental works by Prof. Dalakova & collaborators elaborating a new phase diagram for the doped strontium substituted lanthanum cuprates.

Preprint Coalescence of Impurity Strontium Atoms in Lightly-Substitut...

Resistance does not go to zero at all ω for superconductors *. For example, the dependence on frequency (energy plays a role) even in the BCS case:

Superconductivity arises when part of the free electrons condensate in electron pairs, forming an electric superfluid carrier.

At T = 0 the occupied electronic states are separated from the free states by a narrow "gap" which is also one of the consequences of the theory of superconductivity. For this reason, at temperatures other than zero, "normal" electron quasiparticles also appear in the metal, thanks to their thermal excitation.

The density of these, according to their order of magnitude is ne -Δ/kBT.

If direct current conductivity is present, such electrons do not manifest itself in any way, since the resistance of the sample is equal to zero, and there is no voltage drop across it, which is necessary to set the normal electrons in motion.

But the issue turns completely different if we examine two phenomena: the conductivity at high frequencies or the electronic thermal conductivity.

The study of the behavior of a superconductor in an alternate electric field is to some extent analogous to the study of the viscosity of 4He isotope according to torsional oscillations. The pairs do not experience friction in the crystal lattice but do possess inertia.

Therefore, if the electric field outside the superconductor varies rapidly, inside the metal (near its surface) the electric field E remains. A similar situation is observed in the magnetic coils: due to their inertia, the current cannot be established instantaneously, and for this reason, a voltage proportional to the current arises between the ends of the coils. The variation of the magnetic flux in the coils with the normal electrons gives their energy.

That is why in an alternate field, superconductors have higher resistance, the higher the frequency (figure from Prof. Pippard's 1947 paper).

But the lower the temperature, the less is its resistance thanks to the sharp reduction in the density of normal electrons. Such behavior is observed when the frequency is not too large, i.e., when the photon energy hω < 2Δ But when the radiation energy hω > 2Δ, the quantum energy of the electromagnetic wave is enough to destroy the Cooper pair, and even at T = 0 K, superconductivity practically does not manifest itself.

Figure explanation: Variation of T) of the relationship between R of tin at different ω, and the corresponding resistance R0 when T > TI, In the latter case the resistance does not depend on the temperature at all frequencies.

It decreases in jumps to zero when performing measurements with direct current (plot 1), it varies more smoothly at frequencies of 1.2 GHz (plot 2) and 9.4 GHz (plot 3) and practically does not depend on the temperature at 2,000 GHz (4)

(From Prof. A.B. Pippard. 1947).

* see for example: "Вблизи абсолютного нуля" 2001, Series Kvant by Prof. V. S. Edelman

CC. Thank you for correcting the mistake Prof. Belayiev.

Pedro L. Contreras E. ,"Superconductivity arises when part of the free electrons condensate in electron pairs, forming an electric superfluid carrier."

See similarity in: Research Proposal Initialization of plasma processes

About "The DC component...(ICCM) You write:

Thank you for the post, Prof. Jaroslav Patúc I will read the paper.

Best Regards.

Our RG thread on the unitary limit in QM has been positioned in Google search as the first lecture on the topic.

Thanks to all community RG members and external researchers for reading it. As a scientist, I follow Prof. Richard Feynman's advice, "the only way to learn is by teaching others as much as possible".

https://twitter.com/proffeynman/status/1452134181039837188

@Mindset_Machine replying to @ProfFeynman

https://twitter.com/Mindset_Machine/status/1452134366709108736

The Feynman Learning Technique:

• Pretend to teach a concept you want to learn about to a student in the sixth grade.
• Identify gaps in your explanation. Go back to the source material to better understand it.
• Organize and simplify.
• Transmit.

Best Regards.

• A sketch of the evolution of the density of states (DOS) in unconventional superconductors, includes singlet and triplet behavior in the reduced phase scattering space

A new research article related to this thread is coming soon, this time there was no preprint in repositories.

Here, I show an image of several types of DOS for superconductors in the case we include the reduced phase scattering space which is relevant in the unitary resonance limit.

It is not complete because it does not include in the most general way the triplet point group DOS behavior case, but it is helpful and very instructive.

I add a reference to compare with, I just included more cases than Prof Brison.

Probably some additional explanation is needed in some cases, the residual DOS is not included, but we have published a pedagogical preprint with it.

Preprint The two-dimensional density of states in normal and supercon...

Best Regards

Let us have some posts including the non-magnetic dirt by defining the type of magnetic/non-magnetic atoms in a physics illustrative way.

Basic Magnetism in crystal cells

Slide adapted from a book: Electrons. Phonons. Magnons, by Moisey Isaakovich Kaganov, Mir Moscu, 1981.

Now lets us add the strontium single atom atomic configuration adapted from:

https://your-online.ru/en/electronic-formulas/Sr

1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2

Strontium is a diamagnetic atom

Several posts on metal-insulator transitions:

First, an interesting picture of electrical conductivity in a solid arises when an electron is multiply scattered by impurities so that it diffuses through the solid, as proposed by Prof. P. Anderson *.

In his approach, the diffusivity of the electron depends strongly on its mean free path, and the average distance it travels between scattering centers.

The Ohmic conductivity becomes directly proportional to the mean free path. In addition, if the quantum nature of the electron is taken into account, the conductivity is also proportional to the wavenumber at the Fermi surface.

An important issue for the case of diffusion is whether any increase of disorder would lead only to a shorter mean free path and the reduction of electrical conductivity, or another behavior would develop.

Prof. Anderson found that there was a critical level of scattering by impurities beyond which the diffusive motion of the electron comes to a complete stop. The electron is then said to be localized.

Later work predicted that the change in behavior occurs when the electron’s mean free path becomes smaller than its De Broglie wavelength. This diffusive localization suppresses conductivity and the material becomes an insulator

cc Wiki:

* https://en.wikipedia.org/wiki/Anderson_localization.

Second post, other mechanism for a metallic-dielectric transition.

The Peierls distortion *

The Peierls distortion or Peierls transition is a dimerization, where at low temperatures a one-dimensional monovalent crystal changes its metallic behavior to insulating behavior. The atoms in the lattice are slightly arranged, going from a crystal with regular and equal spacing to staggered spacing; that is, the atoms form pairs.

This costs the system elastic energy since for identical atoms the lowest state must be one with equal spacings for any reasonable potential. However, the electrons are able to move to a lower energy state by this maneuver.

Consider a one-dimensional metal with a gas of electrons filling all conduction band orbitals up to the wave vector kF, at absolute zero temperatures.

Prof. Rudolf Peierls suggested that such a linear metal is unstable with respect to a static deformation of the wave vector lattice G=2kF.

This deformation creates a bandgap at the Fermi surface, thereby lowering the energy of the electrons below the bandgap. The deformation continues until it is limited by the increase in elastic energy.

Quantum Theory of Solids by Rudolf Peierls, 1955, Clarendon, Oxford University Press.

* More Surprises in Theoretical Physics by Rudolf Peierls, 1991, Princeton Series in Physics.

A confirmation of Prof. Peirerls distortion in certain periodic tables elements which I consider important is discussed on March 5, 2014, in a lecture by Professor B. Johanson titled "Structural and electronic bonds between lanthanide and actinide elements"

https://misis.ru/university/news/education/2014-03/1582/

I unquote part of the abstract: "...The behavior of the crystal structure of lanthanides and heavier actinides is determined by the number of 5d (or 6d) electrons in the metallic state since the f-electrons of these elements are localized and nonbonding. Electronic structure calculations for earlier metallic actinides (where the 5f orbitals are treated as part of the valence band) explain very well the observed ground-state crystal structures. The distorted structures can be considered as the result of the Peierls distortion in symmetric BCC structures, caused by the fact that strongly binding 5f electrons occupy relatively narrow 5f states...".

The third transition metal-dielectric mechanism is the Mott transition:

It is a sharp change in the electrical conductivity of a solid with an increase in the concentration of charge carriers, due to the screening of the Coulomb interaction between electron and hole normal state quasiparticles.

The properties of a significant number of transition metal compounds are poorly described by the band theory in connection with the interelectronic interaction.

For some of these materials, a sharp increase in electrical conductivity with increasing temperature is characteristic.

A possible explanation for this phenomenon is that an increase in temperature leads to a change in the crystal lattice period, at which the localized states of electrons become delocalized.

Prof. Neville Francis Mott proposed an explanation for this phenomenon, which is related to the screening of the Coulomb interaction. At a low concentration of charge carriers, an electron and a hole quasiparticles are excited in it to form an exciton quasiparticle with a binding energy

V=-e2/ ( 2 ε a0 )

where e is the electron charge, ε is the permittivity, and a0 is the Bohr radius for the exciton quasiparticle. In addition, it is important to state that an exciton quasiparticle is neutral with no charge; therefore, excitation does not lead to the appearance of electron/hole conduction.

As the concentration of charge carriers increases, the Coulomb interaction is screened. When the screening radius becomes smaller than the exciton Bohr radius, the electron and hole quasiparticles are no longer bound, so they can move freely, contributing to the conductivity. According to Mott, such a transition occurs at

n1/3 a0 > 1/4 π /31/3 where n is the electron concentration. This equality is equivalent to rs

The fourth is Mott- Ioffe- Regel criterium, please attached post and a nice reference to introduce the topic:

Article Ioffe-Regel criterion and resistivity of metals

CC Eugeniy Beliayev

So finally we have several criteria for the transition metal dielectric.

• Peirerls distortion.
• Anderson Localization.
• Mott transition
• Mott-Ioffe- Regel criterium

In addition, we have an issue concerning the Lifshitz localization.

Where is the unitary limit, for those that still find it difficult to understand and read the elastic cross-section, imaginary, and real parts:

a few remarks:

• Metals are quantum objects, I attach a report from a previous post in the thread about physical kinetics. They are naturally within the unitary limit.
• |kF | ℓ >> 1 criterium does not suffix in any case for metals, metals are not semiconductors with impurities, and localization becomes a resonance at zero frequencies.
• Another post that classifies non-relativistic quantum mechanics and the type of problems was given by Prof. Moisey Isaakovich Kaganov in his popularization book in 1981.

CC Eugeniy Beliayev

Dianela Osorio

Fermi liquid and the unitary limit, let us write some posts about it:

We are going to explore the strange metal behavior.

Evolution of the RPS for HTSC with substituted strontium.

A new preprint related to this thread is available at:

Contreras, Pedro, Osorio, Dianela and Beliayev, Eugeniy, The Tight-Binding Flat Band Case in the Strontium-Substituted Lanthanum Cuprate. Available at SSRN: https://ssrn.com/abstract=4143305 or http://dx.doi.org/10.2139/ssrn.4143305

Best Regards.

A new important preprint substituting an old one & very related to this thread will appear tomorrow here and on other open access platforms.

Best Regards and let the world decide, not only two people.

Here is the preprint to all, open access in arXiv

Preprint Coalescence of Impurity Strontium Atoms in Lightly-Substitut...

Best Regards to all & thanks for the great support.

There are differences in studying superconductivity theory using the Green function formalism:

• Superconductors at T = 0 K, among properties studied at this level are the following: Gorkov formalism, the existence of an external electromagnetic field, BCS theory, the thermodynamics of BCS superconductors, and several other properties.
• Superconductors at Finite temperature T: The Gorkov formalism at finite T, superconductors in a weak electromagnetic field, superconductors in a variable magnetic field. Gap equation, among others.

See, for example, the book pointed in the following link:

Article Quantum Field Theoretical Methods in Statistical Physics

Eventually studying the self-consistent elastic scattering cross-section, several macroscopic properties of unconventional superconductors such as the differentiation of several phases can be achieved.

Best Regards.

CC. Dianela Osorio

New preprints will appear soon concerning the unitary limit in an unconventional numerical superconductors, Dear Fellows!

A new work, concerning this thread, has been posted in the SSRN Electronic Journal:

Article The Effect of Nonmagnetic Disorder in the Energy Gap at Zero...

Any wishing to read it has to go to the SSRN platform and read it there from the link that provides the DOI in the publication mentioned.

It concerns the unitary limit (c = 0) and also the intermedium limit ( c = 0.4) using the zero temperature GF technique in the triplet superconducting compound Sr2RuO4.

Thank you all.

CC Anjna Devi

A new paper has been appear concerning this thread, inquiries about the manuscript should be made to the journal directly.

Please follow the DOI.

Article The effect of nonmagnetic disorder in the superconducting en...

A new paper has been written in our group concerning this thread:

Article Quasi-point versus point nodes in Sr2RuO4, the case of a fla...

A new preprint concerning this thread on unconventional superconductors and its DoS using the reduced phase space is available in the following arXiv link link and in RG platform:

Preprint The two-dimensional density of states in normal and supercon...

Kind Regards.

Hi.

There is a new publication by our group, concerning this thread subject.

Article Self-Consistent Study of the Superconducting Gap in the Stro...

Until today we keep the project related to this thread.

Thank to all.

Dear Readers and followers

A new article related to this thread has been published

Article The collision frequency in two unconventional superconductors

Best Regards