The post Two Views on the Protein Folding Puzzle appeared first on SciTech Connect.

]]>As seen in the figure below, the protein chain is gene-encoded and initially has no structure (below left) and its intricate structure (below right), with every atom in its unique position, results from spontaneous folding.

This is as amazing as if a multicolored thread could produce a shirt itself!

The chain spontaneously finds its stable fold (Figure 1: from left to right) within minutes or faster (both in vitro and in living cells), though much more than the entire life-span of the Universe would have been required to sample all possible chain structures in search for the most stable one. This is called “the Levinthal’s paradox”. To resolve it, various models of folding were proposed during decades.

However, these models fail to overcome the Levinthal’s paradox *when* the globular structure stability is close to that of the unfolded chain and provide for no estimate of the folding rates (spaning over 11 orders of magnitude, Figure 2).

The folding rate problem was solved using *unfolding *(not folding!) as the starting point, i.e., when the free-energy barrier between the globular and unfolded state (Figure 1, middle) was viewed “from the globule side” (Figure 1: from right to left).

The trick is that, firstly, the rates of the forward and reverse reactions coincide when the globular structure stability equals to that of the unfolded chain (according to the “principle of detailed balance” well-known in physics). Secondly, it is much easier to imagine – and investigate – how the thread unfolds than how it obtains a certain fold.

The validity of this theory (proposed two decades ago, when the experimental data were scarce) has been recently confirmed by all currently available experimental data (Figure 2).

However, dissatisfaction felt because the *folding* problem has not been solved yet “from the viewpoint of the folding chain” (Figure 1: from left to right) underlay further efforts made to estimate a volume of the necessary sampling in search for the most stable chain fold.

Recently, it has been shown that this volume, when considered at the level of formation and packing of the most strongly interacting protein structure elements (helices and strands, Figure 1) is by many orders of magnitude smaller than at the Levintal-considered level of separate chain links (beads in Figure 1), and the rate of its sampling, at *folding*, become physically and biologically reasonable (and close to the unfolding-derived estimates, Figure 2).

Thus, the protein folding puzzle is solved by viewing on it from two sides: the side of unfolding *and* the side of folding.

*Protein Physics: A Course of Lectures* covers the most general problems of protein structure, folding and function; it describes key experimental facts and introduces concepts and theories. It deals with fibrous, membrane and especially water-soluble globular proteins, in both their native and denatured states. The book summarizes and presents in a systematic way the results of several decades of worldwide fundamental research on protein physics, structure and folding. It describes many simple physical models aimed to help a reader to estimate and predict of physical processes occurring in and with proteins.

The author, Alexei V. Finkelstein, has been invited to speak at the 4th International Conference on Integrative Biology on July 18-20, 2016 in Berlin, Germany. The book, *Protein Physics: A Course of Lectures *is scheduled to publish in July. If you would like to pre-order a copy at 15% off the list price, visit the Elsevier Store today. The author, Alexei V. Finkelstein, will be speaking at the 4th International Conference on Integrative Biology on July 18-20, 2016 in Berlin, Germany.

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]]>The post The Life-Changing Love of One of the 20th Century’s Greatest Physicists appeared first on SciTech Connect.

]]>One of the great short stories of the 20th century is Nobel Laureate Isaac Bashevis Singer’s The Spinoza of Market Street. It tells of an aged scholar who has devoted his life to the study of Spinoza’s great work, Ethics. Protagonist Dr Fischelson has lost his library job and, like his hero, been expelled from his religious community for his heretical views. Looking down from his garret with disdain at the crowded street below him, he devotes his days to solitary scholarship. At night he gazes up through his telescope at the heavens, where he finds verification of his master’s wisdom.

Then one day Dr Fischelson falls ill. A neighbor, an uneducated “old maid,” nurses him back to health. Eventually, though the good doctor never understands exactly how or why, they are married. On the night of the wedding, after the unlikeliest of passionate consummations, the old man gazes up at the stars and murmurs, “Divine Spinoza, forgive me. I have become a fool.” He has learned that there is more to life than the theoretical speculations that have preoccupied him for decades.

The history of modern physics boasts its own version of Fischelson. His name was Paul Dirac. I first encountered Dirac in physics courses, but was moved to revisit his life and legacy through my service on the board of the Kinsey Institute for the Study of Human Sexuality and teaching an undergraduate course on sexuality and love.

Born in Bristol, England, in 1902, Dirac became, after Einstein, the second most important theoretical physicist of the 20th century. He studied at Cambridge, where he wrote the first-ever dissertation on quantum mechanics. Shortly thereafter he produced one of physics’ most famous theories, the Dirac equation, which correctly predicted the existence of antimatter. Dirac did more than any other scientist to reconcile Einstein’s general theory of relativity to quantum mechanics. In 1933 he received the Nobel Prize in Physics, the youngest theoretical physicist ever to do so.

At the time Dirac received the Nobel Prize, he was leading a remarkably drab and, to most eyes, unappealing existence. As detailed in Graham Farmelo’s wonderful biography, The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom, on which I rely heavily in this article, Dirac was an incredibly taciturn individual. Getting him to utter even a word could prove nearly impossible, leading his mischievous colleagues to introduce a new unit of measure for the rate of human speech, the Dirac, which amounted to one word per hour.

Dirac was the kind of man who would “never utter a word when no word would do.” Farmelo describes him as a human being completely absorbed in his work, with absolutely no interest in other people or their feelings, and utterly devoid of empathy. He attributes this in part to Dirac’s tyrannical upbringing. His father ruthlessly punished him for every error in speech, and the young Dirac adopted the strategy of saying as little as possible.

Dirac was socially awkward and showed no interest in the opposite sex. Some of his colleagues suspected that he might be utterly devoid of such feelings. Once, Farmelo recounts, Dirac found himself on a two-week cruise from California to Japan with the eminent physicist Werner Heisenberg. The gregarious Heisenberg made the most of the trip’s opportunities for fraternization with the opposite sex, dancing with the flapper girls. Dirac found Heisenberg’s conduct perplexing, asking him, “Why do you dance?” Heisenberg replied, “When there are nice girls, it is always a pleasure to dance.” Dirac pondered this for some minutes before responding, “But Heisenberg, how do you know beforehand that the girls are nice?”

Then one day, something remarkable entered Dirac’s life. Her name was Margit Wigner, the sister of a Hungarian physicist and recently divorced mother of two. She was visiting her brother at the Institute for Advanced Study in Princeton, New Jersey, where Dirac had just arrived.

Known to friends and family as “Manci,” one day she was dining with her brother when she observed a frail, lost-looking young man walk into the restaurant. “Who is that?” she asked. “Why that is Paul Dirac, one of last year’s Nobel laureates,” replied her brother. To which she replied, “Why don’t you ask him to join us?”

Thus began an acquaintance that eventually transformed Dirac’s life. Writes Farmelo:

*His personality could scarcely have contrasted more with hers: to the same extent that he was reticent, measured, objective, and cold, she was talkative, impulsive, subjective, and passionate.*

A self-described “scientific zero,” Manci embodied many things that were missing in Dirac’s life. After their first meeting, the two dined together occasionally, but Dirac, whose office was two doors down from Einstein, remained largely focused on his work.

After Manci returned to Europe, they maintained a lopsided correspondence. Manci wrote letters that ran to multiple pages every few days, to which Dirac responded with a few sentences every few weeks. But Manci was far more attuned than Dirac to a “universally acknowledged truth” best expressed by Jane Austen: “A single man in possession of a good fortune must be in want of a wife.”

She persisted despite stern warnings from Dirac:

*I am afraid I cannot write such nice letters to you – perhaps because my feelings are so weak and my life is mainly concerned with facts and not feelings.*

When she complained that many of her queries about his daily life and feelings were going unanswered, Dirac drew up a table, placing her questions in the left column, paired with his responses on the right. To her question, “Whom else should I love?” Dirac responded, “You should not expect me to answer this question. You would say I was cruel if I tried.” To her question, “Are there any feelings for me?” Dirac answered only, “Yes, some.”

Realizing that Dirac lacked the insight to see that many of her questions were rhetorical, she informed him that “most of them were not meant to be answered.” Eventually, exasperated by Dirac’s lack of feeling, Manci wrote to him that he should “get a second Nobel Prize in cruelty.” Dirac wrote back:

*You should know that I am not in love with you. It would be wrong for me to pretend that I am, as I have never been in love I cannot understand fine feelings.*

Yet with time, Dirac’s outlook began to change. After returning from a visit with her in Budapest, Dirac wrote, “I felt very sad leaving you and still feel that I miss you very much. I do not understand why this should be, as I do not usually miss people when I leave them.” The man whose mathematical brilliance had unlocked new truths about the fundamental nature of the universe was, through his relationship with Manci, discovering truths about human life that he had never before recognized.

Soon thereafter, when she returned for a visit, he asked her to marry him, and she accepted immediately. The couple went on two honeymoons little more than month apart. Later he wrote to her:

*Manci, my darling, you are very dear to me. You have made a wonderful alteration in my life. You have made me human… I feel that life for me is worth living if I just make you happy and do nothing else.*

A Soviet colleague of Dirac corroborated his friend’s self-assessment: “It is fun to see Dirac married, it makes him so much more human.”

In Dirac, a thoroughly theoretical existence acquired a surprisingly welcome practical dimension. A man who had been thoroughly engrossed in the life of the mind discovered the life of the heart. And a human being whose greatest contributions had been guided by the pursuit of mathematical beauty discovered something beautiful in humanity whose existence he had never before suspected.

In short, a brilliant but lonely man found something new and wonderful that had been missing his entire life: love. As my students and I discover in the course on sexuality and love, science can reveal a great deal, but there are some aspects of reality – among them, love – that remain largely outside its ambit.

Paul Dirac introduced some useful formal tools (such as his notation for integrals and operators). One of them is the Dirac delta function δ(x), an object then unknown to mathematicians, which turned out to be very useful in physics. The book *Ideas of Quantum Chemistry, 2nd Edition*, has an excellent appendix that describes the Dirac Delta Function. We are pleased to offer you a download of this sample from the book.

If you are interested in more from the *Ideas of Quantum Chemistry, 2nd Edition* you can order a print copy at up to 30% off the list price and free global shipping via the Elsevier Store. Enter discount code STC215 at checkout.

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]]>The post In Conversation with Hiroshi Amano appeared first on SciTech Connect.

]]>Upon arriving in France, he stepped out of the airport arrivals hall into a crowd of excited journalists. One of whom informed Amano that he had been jointly awarded the Nobel prize in physics with his former supervisor Isamu Akasaki and another Japanese scientist Shuji Nakamura. “It was really unexpected,” he says. In fact, he was so surprised that at first he thought it must be a joke or a mistake.

The third volume of the *Handbook of Crystal Growth* describes Amano’s and others’ recent work to improve our fundamental understanding of the growth of nitrides by MOCVD and MOVPE for a range of nitride-based devices.

The Nobel committee had awarded the prize to these three researchers for inventing efficient blue light-emitting diodes (LEDs) which enabled the development of white LEDs. Today, white LEDs are the most energy efficient and longest lasting bulbs on the market, and can be found lighting our homes and business and inside our TVs, computers and mobile phones. In 2012, more than 210 billion LEDs packages were reportedly produced worldwide – this is approximately 30 for each person on Earth.

“It was really unexpected,” said Hiroshi Amano . In fact, he was so surprised that at first he thought it must be a joke or a mistake.

To create white light, LEDs that produce all three of the primary colours of light are needed. By the end of the 1960s, red and green LEDs had been successfully made but LEDs that produced blue light were to prove elusive. Akasaki had identified that gallium nitride was the most likely candidate, but his group were struggling to grow crystals of the material of a high enough quality. Additionally, while n-type gallium nitride semiconductors were proving fairly easy to make, the p-type counterpart was not. Amano overcame both those hurdles whilst working under Akasaki’s supervision in the late 1980s. “My contribution was showing that the high quality GaN can be grown on a sapphire substrate by depositing low temperature AlN buffer layer before the growth of GaN and also that p-type GaN can be made by Mg doping followed by low energy electron beam irradiation treatment,” he explains.

Since those discoveries, Amano has gone on to set up his own successful research team currently based at Nagoya University – which is also where he did his Nobel prize-worthy research in the 1980s. His team works broadly on growing novel crystals of semiconducting group 3 nitrides with the aim of enabling the development of other sustainable devices. These crystals are grown either by MOCVD (metalorganic chemical vapor deposition) or the related technique MOVPE (metalorganic vapour phase epitaxy).

*Read more on SciTech Connect – Ice Crystals Give Up Their Secrets in Microgravity*

The design of LEDs that produce deep UV light has been one of Amano’s team’s most significant recent developments. Photons of deep UV light interact with a huge variety of different chemical and biological molecules and these types of LEDs are expected to find use in applications ranging from sensing to cleaning up pollutants. “The most exciting research carried out in my group recently was realising high efficiency deep UV LEDs by a high temperature MOVPE growth method,” Amano says. The team are also working on designing improved nitrides for powering more energy-efficient heterojunction field-effect transistors and laser diodes.

To achieve the atomic-level control needed to grow nitrides suitable for these applications; Amano’s team spend much of their time studying how the growth processes occur. They are currently developing a method to observe the growth of InGaN and related semiconductors in almost real-time inside an x-ray diffractometer. “A fundamental understanding of the growth process is essential for realizing new types of devices,” he says.

Since winning the Nobel prize, Amano says he has been inundated with invitations to give talks. “By the end of this year, I will have given more than 200 lectures since the prize was announced. Of course it is busy, but I am enjoying these unexpected encounters with researchers in the different fields to my own,” he says. “I learn a lot through discussions with researchers with different specialties.” It is also talking with others that he credits for his success so far: “I have got many of my inspirations though discussions with my colleagues.”

The third volume of the *Handbook of Crystal Growth*, published by Elsevier and available on ScienceDirect, describes Amano’s and others’ recent work to improve our fundamental understanding of the growth of nitrides by MOCVD and MOVPE for a range of nitride-based devices.

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]]>The post American Physical Society Annual Meeting: Note from Our Editor appeared first on SciTech Connect.

]]>I had the opportunity to attend this meeting and found it a great pleasure to meet our authors and editors face-to-face and to receive feedback on how we can continue to meet their needs. Elsevier was present with a booth staffed by colleagues from our books and journals groups.

One of books on display was the recently published second edition of the *Handbook of Crystal Growth* which attracted much attention. These newly revised and updated handbooks together provide a comprehensive compendium on crystal growth, documenting authoritatively in separate self-contained volumes, or as one complete publication, the fundamentals and application of the subject. The work consists of Volume 1: Fundamentals; Volume 2: Bulk Crystal Growth; and Volume 3: Thin Films and Epitaxy.

In 2015 we continue to expand our physics books list with titles that will enable scientists in academia and industry and students to perform their research and build on existing knowledge by providing specific, reliable, and authoritative information in all physics areas.

In particular I would like to bring to your attention the recently published second edition of John Morrison’s *Modern Physics for Scientists and Engineers*. This book gives a brief, focused account of what led to modern quantum theory, then discussing its underlying physics.

I am very interested in growing our physics books program with titles in all physics areas so if you are interested in authoring or editing a book I would love to hear from you.

These titles and many more are available for purchase on the Elsevier Store. Use discount code “STC215” at checkout and save up to 30% on all books and ebooks.

Read more from SciTech Connect:

Ice Crystals Give Up Their Secrets in Microgravity

Elsevier Attends the APS Conference for Women in Physics

**Anita Koch, Acquisitions Editor, Physics & Astronomy**

Anita joined Elsevier in 1991 with a master’s degree in Physics. Since then she held many positions in the Science & Technology journals and books organization. In 2011 she had the opportunity to move to books acquisition, initially in Chemical Engineering and since 2014 also in Physics & Astronomy.

She feels privileged to work with dedicated and outstanding authors and editors in developing high-quality content for students and researchers in academia and industry. Also the fast going technological developments in publishing from print books to electronically available content fascinates her.

Anita lives in the Netherlands. Her hobbies are classical music, literature, cooking, wine, and exploring long distance footpaths.

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]]>The post Ice Crystals Give Up Their Secrets in Microgravity appeared first on SciTech Connect.

]]>The post Ice Crystals Give Up Their Secrets in Microgravity appeared first on SciTech Connect.

]]>The post Elsevier Attends APS Conference for Women in Physics appeared first on SciTech Connect.

]]>This panel will provide students with insight into career tracks they may be interested in once they’ve received their degrees. She has shared her professional experiences before in the form of blog posts she wrote three years ago about her time working at the Large Hadron Collider at CERN in Geneva:

Just before the panel, Senior Acquisitions Editor Katey Birtcher and Editorial Project Manager Marisa LaFleur will host a booth at the conference’s Graduate School and Career Fair, providing more information to students interested in the field of academic publishing.

We are excited to be a part of encouraging young women to consider career options involving academic publishing, whether as authors, editors, or journals managers. The options available to them are wide and diverse, and we look forward to welcoming them to the work force in the coming years.

For more information on STEM careers for women, see our blog posts: Success Strategies from Women in STEM “Book in Press” Party and Women In STEM: Q&A with Dr. Denise Faustman of MGH

To celebrate this event, we are offering **up to 25% off Physics titles** when you use discount code “PHYSICS215” at checkout. Be sure to take a look at some of our top Physics books written by leading women in the field:

*Tensors, Relativity, and Cosmology*by Mirjana Dalarsson*Physics in the Arts*by P.U.P.A Gilbert*Ultrarelativistic Heavy-Ion Collisions*by Ramona Vogt*Linear Ray and Wave Optics in Phase Space*by Amalia Torre*Nanophysics: Coherence and Transport*by Hélène Bouchiat

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]]>The post New Challenge for Quantum Physics appeared first on SciTech Connect.

]]>At present, the general problem of anharmonic vibrations commands a significant place among scientific investigations in various branches of experimental and theoretical physics. The phenomena of anharmonicity are displayed in the vibrations of molecules and crystals, in the mechanics of molecular rotations and librations, in the resonance interaction of vibrational levels, in electro-optical effects, in non-linear spin interaction, and so on.

Behind the scenes of various phenomena, a unique essence is hidden. Through the absence of a perfect harmony, the various physical effects become possible. A harmony fails to be a tendency to a simple ideal, but it is a capacity of the nature to order an anharmonicity. The understanding of the natural beauty leads us to the anharmonic world. Under the constraint of physical laws, the anharmonicity bears the harmony…

The simplest choice to describe the vibrations of an arbitrary system in quantum mechanics is a model of harmonic oscillators. A harmonic case is certainly only an idealization of vibrations of a real system. The potential energy of vibrations, as in classical physics, is generally written in a form of an expansion in terms of normal coordinates; in this expansion, the linear terms disappear through the fact that the first derivative of the potential equals zero at the equilibrium condition. A set of harmonic oscillators corresponds to a first approximation. This model is only qualitatively correct: vibrations, failing to conform to a harmonic law, are anharmonic. To describe correctly the vibrations, apart from the quadratic part of the potential energy, one must therefore take into account the normal coordinates to greater than quadratic powers in an expansion of the potential. These terms additional to the harmonic Hamiltonian are defined by anharmonicity coefficients and characterize the interactions among various vibrational modes. The calculation of the corresponding corrections is generally performed with a perturbation theory for stationary states of a perturbed Hamiltonian, in which the perturbation function represents an expansion in powers of a small parameter. A formalism in terms of the polynomials of quantum numbers might serve as one example of the perturbation methods. In the book ‘Uncommon Paths in Quantum Physics’, we consider the polynomial formalism in detail.

First, the polynomials form, with the required accuracy, all necessary physical observables of the anharmonicity problem. The desired quantities are obtained immediately on solving or opening the recurrence equations or relations avoiding conventional intermediate manipulations. We compare two schemes to construct the stationary perturbation theory:

1) Schrödinger equation → eigenfunctions and eigenvalues →

→ matrix elements;

2) recurrence equations → eigenvalues and matrix elements.

The first scheme is conventional, whereas we proposed the second scheme. The main disadvantage of the conventional scheme is that, at each stage, one must return virtually to the beginning — to the Schrödinger equation — to improve the eigenfunctions by increasing the order of the perturbation calculation. Only after these calculations is one in a position to evaluate the matrix elements. In our method, intermediate calculations are performed on an equal footing; i.e., the procedures to calculate the eigenvalues and arbitrary matrix elements are performed simultaneously.

Second, the proposed theory automatically keeps track of non-zero contributions of the total perturbation to the result sought, and takes into account the history of the calculations, i.e., the intermediate calculations. This advantage is achieved on expanding, in a small parameter, the derivatives of the energies and their wave functions, rather than by expanding the eigenfunctions and eigenvalues as is done traditionally. In this sense, the expansion in exact eigenvectors plays a principal role [K. V. Kazakov, Electro-optics of molecules, *Opt. Spectrosc.*, **97**, 725—734, 2004],

because it ensures a full use of the history of the calculations and, consequently, significantly simplifies the general solution algorithm. If the expansion is performed in terms of the exact eigenvectors, rather than in terms of zero-order basis functions, it is assumed that the former functions exist and are expressible algebraically, for example, with recurrence relations. In addition, one might avoid the renormalization of the function; this problem presents considerable difficulties in the traditional approach in which the function should be renormalized upon passing from one perturbation order to the next.

Other advantages of this method appear in various applications of this perturbation theory [K. V. Kazakov, *Quantum Theory of Anharmonic Effects in Molecules*, 2012, Elsevier]. For example, in a framework of the polynomial formalism, one might consider the problem of electro-optical anharmonicity; this problem involves an electric dipolar-moment function in a non-linear form, and its solution requires evaluation of matrix elements. The absolute values of dipolar-moment derivatives might be unknown beforehand, which complicates the problem. In the traditional formalism, the consideration proceeds, as a rule, from the wave function of a definite order, which leads to the loss of significant contributions. In the polynomial formalism, we consider separately each term in an expansion of the dipolar-moment function, and, consequently, calculate the entire matrix element in a given order in a small parameter.

** Beyond the Predictable Trend**

A prospectively fruitful direction for further investigation is to proceed beyond solutions with perturbation theory. We assume that the effective internuclear potential is a real function that is represented as an expansion in a power series in terms of the normal coordinates. In this case, the procedure of quantization, i.e., the calculation of matrix elements of an arbitrary coordinate function, taking into account the influence of anharmonicity, is reduced to the sum of polynomials multiplied by factor √g [K. V. Kazakov, *Uncommon Paths in Quantum Physics*, 2014, Elsevier]:

Expanding here the polynomials in terms of quantum numbers, we obtain this intriguing formula,

for the one-dimensional case, and this one,

for the many-dimensional case.

The derived expansions in terms of quantum numbers hold for the matrix elements of an arbitrary physical function that is represented as an expansion in a power series in terms of creation and destruction operators. This consequence of perturbation-theory calculations is trivial. The values of energy *E _{n}* are expressible also from the formula for (

in which Ω-coefficients_{ }are the mechanical anharmonicity parameters. To generalize our theory, we assume that quantity *E _{n}* is a function of quantum numbers

We can determine heuristically a function Φ for the matrix element of a particular physical quantity *ƒ*(*ξ*), for instance, the dipolar moment, as a dependence on quantum number *n+k*/2+1/2 :

Functions Φ* _{k}* are here arbitrarily expressible, for example,

with parameters θ_{k }_{ }and ϕ* _{k }*determined from experiment. In the present formalism, one might also construct phenomenologically function Φ for a system with

Konstantin’s books *Uncommon Paths in Quantum Physics* and *Quantum Theory of Anharmonic Effects in Molecules* are both available on the Elsevier Store. Use discount code “STC215” and **save up to 30%** on your very own copies!

**About the Author**

Konstantin V. Kazakov obtained a Dr. Sc. in Physics and Mathematics at the St. Petersburg State University. He has published papers in internationally scientific journals, communications at scientific symposia and congresses, as well as 3 books.

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]]>The post A Fresh Look at the Structure and Concepts of Quantum Theories appeared first on SciTech Connect.

]]>**On The Theory of Anharmonic Effects in Quantum Physics**

Vibrational phenomena have always fascinated scientists and engineers. A molecule constitutes a vibrational system of an important class that is mainly the subject of our present concern. High-resolution infrared absorption spectra provide information about the distribution of vibration-rotational energy levels and the transition probabilities of real molecules. Spectral lines command physical interest through their interpretation with the aid of physical models, i.e., the relation of frequencies and intensities of spectral lines to molecular motions of various types. As the precision of measurements made with various experimental techniques increases relentlessly, the interpretation of observed spectra becomes correspondingly challenging. This condition stimulates the search for, and development of, innovative methods to investigate vibrational systems for which a conventional description fails.

Intuitively, the most natural model of intramolecular motions involves interacting anharmonic oscillations of atomic centers, but this simple physical model lacks a mathematically exact solution. The use of perturbation theory, however, solves the problem. This classical method is simple and clear, but its application is generally limited to the first few orders of theory that any textbook on quantum mechanics describes. The determination of corrections of higher orders becomes complicated through the sheer bulk of the calculations. The calculation of frequencies and intensities of spectral lines with an accuracy defined by experiment hence becomes difficult. A real spectrum of a sample containing even diatomic molecules of a particular chemical compound can comprise lines numbering a few thousands. Despite these difficulties, some success in developing an adequate method of calculation has been achieved, embracing perturbation theory. In what follows in the book ‘Quantum Theory of Anharmonic Effects in Molecules’, we consider the development of techniques of perturbation theory applied to problems of molecular spectroscopy to calculate the frequencies and intensities of vibration-rotational transitions.

Historically, a quantum-mechanical consideration of the anharmonicity of diatomic molecules began with Dunham’s work; deriving matrix elements for vibrational transitions up to the third derivative of the dipolar moment in terms of perturbation theory, he determined a numerical value for the second derivative of the dipolar moment function of a HCl molecule from the experimental distribution of intensities in the infrared spectrum of a gaseous sample. Using various computational methods and varied initial assumptions about functions for potential energy and dipolar moment, other authors have subsequently tried to improve the techniques of calculations. In this regard, we mention specifically the hypervirial theorem, the method of Feynman diagrams and the canonical or contact transformations. The objective of the respective authors was typically the eventual results; the procedure of the calculations was thus afforded little attention.

Although for diatomic molecules an application of the hypervirial theorem was fruitful in calculations of matrix elements of a one-dimensional anharmonic oscillator through recurrence relations, this method is inefficient for polyatomic molecules.

The method of Feynman graphs enables one to eliminate the recurrence scheme of perturbation theory. Circumventing calculations of preceding orders, one might work directly with expressions for wave functions and energy of arbitrary order; this capability is a great advantage of this method. A characteristic of problems in molecular spectroscopy is, however, that one must initially calculate corrections of low order and only then proceed to approximations of higher order. The stated advantage for calculations of low order is rapidly lost in corrections of higher order. For example, the conversion of a diagram of twentieth order into an algebraic expression becomes a complicated procedure, for which one must be concerned about the risk of error.

A systematic investigation of vibration-rotational spectra of polyatomic molecules has been conducted mainly with the method of contact transformations, which allowed the retention of the *q*-number approach and eliminated a problem of superfluous summation over the matrix elements. Although corrections in canonical transformation theory are considered to be equivalent to approximations of the common perturbation theory, this point of view is inaccurate; rather, this method can be used to choose an effective hamiltonian. For instance, Watson proposed an hypothesis that there exist many rotational hamiltonians, which all describe experimental data equally validly. Choosing an initial hamiltonian, by means of a convenient canonical transformation we obtain another hamiltonian that yields the same eigenvalues and has a simple parameterization for the interpretation of experimental data. The principal deficiency of this method is that it lacks a clear form of all expressions; as a result, formulae become much too bulky, which impede a clear understanding by experimenters.

Dunham’s practice of standard perturbation theory can be extrapolated to polyatomic molecules, but alternative algorithms of perturbation theory for the pure vibrational problem have been developed. According to the book ‘Uncommon Paths in Quantum Physics’, a novel method within a formulation of quantum theory is based on differentiation with respect to coupling parameters; it produces simple and clear equations for matrix elements. Essentially a recurrence scheme, it represents a form of solution involving polynomials of quantum numbers. This formalism allows one to generate rules to calculate observable matrix elements, which determine the frequencies and intensities of vibrational transitions. This approach is reminiscent of Feynman diagrams: we calculate all desired polynomials, make convenient tables and then express physical quantities in terms of the polynomial quantities.

The principal objective of this formalism is to simplify the traditional perturbation theory. According to this polynomial method, we accrue all advantages and avoid all shortcomings of the preceding techniques. Efficient for both diatomic and polyatomic molecules, this method is free from the problem of superfluous summation. A convenient recurrence scheme implemented with contemporary computers allows one to optimize all calculations and to decrease greatly the duration of calculations of vibrational frequencies and intensities. When we allude here to approximations of higher order, we have in mind perturbation theory in the tenth or twentieth orders.

Introducing this formalism certainly does not solve all problems: many specific questions, such as those concerned with the effects of vibration-rotational interaction, remain. For instance, a theorem of extraneous quantum numbers has been formulated; with its help an exact solution for coefficients of the Herman—Wallis factor has been obtained — this method is highly original. As a result, we greatly simplify the calculation of intensities for diatomic molecules. For arbitrary linear polyatomic molecules, a comparable success is foreseen, but the possibility of extending this theorem to describe the vibration-rotational spectra of nonlinear molecules has yet to be investigated.

**The Creation and Destruction of States and Quanta **

To solve most equations of quantum mechanics, one generally applies the powerful apparatus of mathematical physics, which is based on traditional methods of the theory of integro-differential equations. Many problems might otherwise be solved in a purely algebraic manner. For instance, to describe a vibrational system in quantum mechanics, one uses the model of an anharmonic oscillator. This simple model might be regarded as founded on an exact solution of the problem for a harmonic case that substantially represents the description of some physical system in an approximation of zero order.

As the necessity to take into account the influence of anharmonicity increases, this solution becomes improved through the pertinent methods of perturbation theory. The non-zero matrix elements between the corresponding states of a perturbed system determine the observable quantities. In the case of the first few orders of the theory, the matrix elements are readily calculated in an algebraic manner, for instance, in the framework of a formalism of creation and destruction operators that follows from the classical work by Fock and Dirac. The calculations of higher orders are performed with the aid of special methods; the recurrence formalism of the perturbation theory in terms of the polynomials of quantum numbers might serve as an example of one among these special devices.

In some applications it is convenient to use, instead of the harmonic oscillator, the Morse oscillator as a zero-order approximation. Applying in this case the recurrence formalism of the perturbation theory, one might, in a manner analogous to the solution in the form of the polynomials of quantum numbers, evaluate the influence on the energy levels and the matrix elements of a term additional to the anharmonic field of the Morse potential that plays the role of a perturbation. Similar conclusions are applicable to another important case in which a non-perturbed system is described through the states in the form of the solutions of Schrödinger’s equation for the Pöschl—Teller potential. Moreover, in a search for a solution of each such problem, the methods of non-commutative algebra according to the language of so-called ladder operators, which are substantially the same of those as the creation and destruction operators, become applicable.

For a concrete physical problem, the terminology of ladder operators might be introduced in not just one way. For instance, a traditional analysis involves appropriate recurrence relations for special functions corresponding to the exact solutions of the Schrödinger’s equation, in which in the role of the potential appear the well known functions of type Morse and Pöschl—Teller, and also a series of other simple potential functions. The purely algebraic methods of factorization are applied less commonly. Among the latter algebraic enunciations is a technique of factorization described by Green that is simple and elegant: in the books ‘Quantum Theory of Anharmonic Effects in Molecules’ and ‘Uncommon Paths in Quantum Physics’, we consider it in detail.

**‘Dirac’s Theory Resolves the Paradoxes of Quantum Electrodynamics**

The anomalous magnetic moment of an electron and the Lamb shift of atomic levels of hydrogen have played a key role in the development of the contemporary quantum electrodynamics. The theoretical investigations were, however, far from immediately able to confirm the experimental facts unique at that moment of time. For instance, the calculation of the self-energy of an electron, as in classical electrodynamics, yielded an infinite value. The situation was analogous to the calculation of the electromagnetic shift of atomic levels. As Bethe said, “one might overlook the Lamb shift, because the latter was infinite in all the then existing theories”. This scenario was generally applicable to all infinities, which were simply discarded. Do the infinities lack a physical meaning?

It subsequently became clear that the reply to the appearance of infinities was hidden directly within them. On quantizing an electromagnetic field, we discard an infinite -number, associating it with the zero-point vibrations of the field or, as one says, with an electromagnetic vacuum. One might ignore the vacuum, but only when we consider the transitions of an electron between excited states. Moreover, there exists an electron-positron vacuum that is otherwise called the Dirac sea. To calculate the radiative corrections, we must hence take into account the interaction of electrons with the virtual electron-positron pairs of the Dirac sea, not only with the zero-point vibrations of an electromagnetic field. As a result of the interaction with the «vacuum», there arise the remarkable effects of quantum electrodynamics — the anomalous magnetic moment and the Lamb shift of atomic levels.

Discussing the fields, we imply real physical objects, for instance, the electrons and photons. At present, a field formalism extends far beyond the relativistic quantum mechanics: it is applicable in the physics of solids, in the theory of an atomic nucleus and in the theory of a plasma, and also in many other branches of physics, to describe the principal characteristics of quantum systems. It is well known that, to the vibrations of a crystalline lattice, one might ascribe a field of phonons, and, to describe ferromagnetic phenomena, one should apply a concept of spin waves that represent a field of magnons. One might continue this series, but it is already clear that the method of quantum fields is firmly entrenched in a prime place in all quantum physics. Even for some problems of the simplest atoms such as hydrogen and helium, it becomes simpler to apply a many-particle description than a clear traditional representation of a system in a form of several interacting particles. Through the fields we comprehend the particles.

In the book ‘Uncommon Paths in Quantum Physics’, our aim was to generate a first acquaintance with the field point of view that allows one to enter the general range of questions of quantum electrodynamics. In the Heisenberg picture, we have obtained the information about an electron when considering the temporal variation of its creation operator. One might perform something similar also for a creation operator of a photon, and further, for example, for an operator of creation of a muon, under a condition, of course, that it becomes possible to introduce a muon field into the theory. We fail, however, to derive thereby special dividends. For instance, the self-energy of a photon becomes infinite. Yes, we might apply the procedure of a regularization to ensure that this energy becomes large but finite; we cannot, however, eliminate it entirely. If a Hamiltonian comprises a mass parameter, we can manipulate it, suggesting that there exist an electromagnetic part of a mass, a meson part and other. For a photon, this scenario is questionable.

Beginning with the Bethe formula, there appeared many works aiming at seeking practical rules of a game with expressions that generally diverge, but not at a general comprehensive theory. We concentrated our attention on the theory that was developed by Dirac, which is, in our opinion, suitable for a first acquaintance with the principles of quantum electrodynamics. Despite the fact that there was a common preference to work in Schrödinger’s formalism, Dirac believed that it was much more logical to use Heisenberg’s equations. In this case, the theory retains the natural harmony and a reasonable sequence of conclusions.

This theory certainly does not lack shortcomings. Firstly, we failed to cope with ultraviolet divergences; as a result of the procedure of regularization of divergent integrals, the theory lost relativistic invariance. Secondly, in the calculations of observables, we neglected some quantities that were supposed to be small; this condition is not, however, entirely obvious, and many approximations are questionable in a strict sense. Thirdly, a gauge-invariant Hamiltonian was formally separated into two parts that are related to the free fields and their interaction respectively; the second part was considered a perturbation. We have thereby put both parts on unequal footings and lost partly the sense of gauge invariance.

In other respects, the theory is satisfactory. Solving Heisenberg’s equations, Dirac obtained general quantities that describe the interaction of fields according to perturbation theory. A regularization organically enters a computation, simply excluding the region of high energies from consideration. The approximations of various kinds are necessary to achieve an agreement with experiment. How should we understand such an approach? Likely, the theory cannot be perfect, such that subsidiary rules of a game become necessary. We arrive ultimately at the results in a form of expansions with respect to a coupling constant. This form is convenient for a comparison with experiment, but at the same time it forces us to work in a framework of the method of perturbation theory. One might think that, in the course of the calculations, we neglect instinctively some quantities only to convert our solution into a form resembling a reasonable expansion with respect to a coupling constant. Our actions are justifiable because this constant is small. If, further, it becomes possible to represent a solution in a form of some function of coupling constant without applying perturbation theory, a question regarding a gauge invariance of separate parts of a Hamiltonian becomes removed. One might thus conclude that quantum electrodynamics is based upon not only the equations and the methods to solve them but also the definite rules of the game to operate with the field variables.

Konstantin’s books *Uncommon Paths in Quantum Physics* and *Quantum Theory of Anharmonic Effects in Molecules* are both available on the Elsevier Store. Use discount code “STC3014” and **save up to 30%** on your very own copies!

**About the Author**

Konstantin V. Kazakov obtained a Dr. Sc. in Physics and Mathematics at the St. Petersburg State University.He published papers in internationally scientific journals, communications at scientific symposia and congresses, and 3 books.

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**1) What is the name of the first electricity detective?**

*Answer: Sherlock Ohms*

**2) A neutron walked into a bar and asked, “How much for a drink?” The bartender replied, “For you, no charge.”**

**3) Where does bad light end up?**

*Answer: In a prism.*

**4) A Higgs Boson walks into a Church. The priest says, “We don’t allow Higgs Bosons in here. The particle responds by saying: “But without me, how can you have Mass?”**

**5) How many theoretical physicists specializing in general relativity does it take to change a light bulb?**

*Answer: Two. One to hold the bulb and one to rotate the universe.*

* *

**6) Why can’t you trust an atom?**

*Answer: They make up everything.*

**7) What do you get when you mix sulfur, tungsten, and silver?**

*Answer: SWAG*

**8) A photon checks into a hotel. The bellhop asks, “Can I help you with your luggage?” The photon replies, “I don’t have any. I’m traveling light!”**

**9) An electron and a positron go into a bar.**

*Positron: “You’re round.”**Electron: “Are you sure?”**Positron: “I’m positive.”*

**10) Why did Erwin Schrödinger, Paul Dirac and Wolfgang Pauli work in very small garages?**

*Answer: Because they were quantum mechanics.*

**11) What is a physicist’s favourite food?**

*Answer: Fission chips*

**12) What did one uranium-238 nucleus say to the other?**

*Answer: “Gotta split!”*

Didn’t understand the jokes? Not to worry, brush up on your physics with Elsevier’s Physics and Astronomy books! Get up to **30% off** using discount code **STC314**

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]]>Many quantum learning algorithms rely on Grover’s search (Grover, 1996), an algorithm to find elements in an unordered set quadratically faster than by any classical variant. Mostly unsupervised learning methods use this approach: *K*-medians, hierarchical clustering, and quantum manifold embedding. In addition, quantum associative memory and quantum neural networks often rely on this search, and also an early version of quantum support vector machines. Adding it up, about half of all the methods proposed for learning in a quantum setting use Grover’s.

As Grover’s search has a quadratic speedup, this sets the limit to how much faster those learning methods can get that rely on it. Exponential speedup is possible in scenarios where both the input and output are also quantum: listing out class membership or reading the classical data once would already imply at least linear time complexity. In turn, this would limit the speedup to polynomial order. Examples that use quantum data include quantum principal component analysis, quantum *K*-means, and a different flavor of quantum support vector machines. Regression based on quantum process tomography requires an optimal input state, and, in this regard, it needs a quantum input. At a high level, it is also possible to define an abstract class of problems that can only be learned in polynomial time by quantum algorithms using quantum input (Gavinsky, 2012).

Generalization performance estimates how well a learning algorithm will perform on data it has not seen during training. Naturally we would like to have algorithms which are not only fast to learn, but that also generalize well. Curiously, few authors were interested in the generalization performance of quantum learning algorithms. Analytical investigations are especially sparse, with quantum boosting by adiabatic quantum computing being a notable exception, along with a form of quantum support vector machines. Numerical comparisons favor quantum methods in the case of quantum neural networks and quantum nearest neighbors.

We are far from developing scalable universal quantum computers. Learning methods, however, do not require universal hardware: special cases of quantum computing are attainable with current technology. A controversial example is adiabatic quantum optimization in large-scale learning problems, most notably, in boosting. More gradual and well-founded are small-scale implementations of quantum perceptrons and neural networks. As theoretical work and advances in quantum technology continue, we can expect even more quantum learning algorithms in the next few years.

Peter Wittek received his PhD in Computer Science from the National University of Singapore, and he also holds an MSc in Mathematics. He is interested in interdisciplinary synergies, such as scalable learning algorithms on supercomputers, computational methods in quantum simulations, and quantum machine learning. He has been involved in major EU research projects, and obtained several academic and industry grants.

Affiliated with the University of Borås, he works location-independently, and did research stints at several institutions, including the Indian Institute of Science, Barcelona Supercomputing Center, Bangor University, Tsinghua University, the Centre for Quantum Technologies, and the Institute of Photonic Sciences.

He is the author of * Quantum Machine Learning: What Quantum Computing Means to Data Mining,* available on the Elsevier Store at a 25% discount. This book bridges the gap between abstract developments in quantum computing and the applied research on machine learning. It captures a broad array of highly specialized content in an accessible and up-to-date review of the growing academic field of quantum machine learning and its applications in industry.

**References**

Anguita, D., Ridella, S., Rivieccio, F. and Zunino, R. (2003). Quantum optimization for training support vector machines, Neural Networks, vol. 16, iss. 5, pp. 763-770.

Aïmeur, E., Brassard, G. and Gambs, S. (2013). Quantum speed-up for unsupervised learning, Machine Learning, vol. 90, iss. 2, pp. 261-287.

Bisio, A., D’Ariano, G. M., Perinotti, P. and Sedlák, M. (2011). Quantum learning algorithms for quantum measurements, Physics Letters A, vol. 375, pp. 3425-3434.

Gavinsky, D. (2012). Quantum Predictive Learning and Communication Complexity with Single Input, Quantum Information & Computation, vol. 12, iss. 7-8, pp. 575-588.

Grover, L. K. (1996). A Fast Quantum Mechanical Algorithm for Database Search, in Proceedings of STOC0-96, 28th Annual ACM Symposium on Theory of Computing, pp. 212-219.

Lloyd, S., Mohseni, M. and Rebentrost, P. (2013a). Quantum algorithms for supervised and unsupervised machine learning, arXiv:1307.0411.

Lloyd, S., Mohseni, M. and Rebentrost, P. (2013b). Quantum principal component analysis, arXiv:1307.0401.

Narayanan, A. and Menneer, T. (2000). Quantum artificial neural network architectures and components, Information Sciences, vol. 128, iss. 3-4, pp. 231-255.

Rebentrost, P., Mohseni, M. and Lloyd, S. (2013). Quantum support vector machine for big feature and big data classification, arXiv:1307.0471.

Trugenberger, C. A. (2001). Probabilistic Quantum Memories, Physical Review Letters, vol. 87, p. 67901

Ventura, D. and Martinez, T. (2000). Quantum associative memory, Information Sciences, vol. 124, iss. 1, pp. 273-296.

Wiebe, N., Kapoor, A. and Svore, K. M. (2014). Quantum Nearest Neighbor Algorithms for Machine Learning, arXiv:1401.2142.

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