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Mathematical structures that allow quantum mechanics to be explained

The mathematical formulations of quantum mechanics are those mathematical formalisms that permit a rigorous description of quantum mechanics. This mathematical formalism uses mainly a part of functional analysis, especially Hilbert spaces, which are a kind of linear space. Such are distinguished from mathematical formalisms for physics theories developed prior to the early 1900s by the use of abstract mathematical structures, such as infinite-dimensional Hilbert spaces (L2 space mainly), and operators on these spaces. In brief, values of physical observables such as energy and momentum were no longer considered as values of functions on phase space, but as eigenvalues; more precisely as spectral values of linear operators in Hilbert space.[1]

These formulations of quantum mechanics continue to be used today. At the heart of the description are ideas of quantum state and quantum observables, which are radically different from those used in previous models of physical reality. While the mathematics permits calculation of many quantities that can be measured experimentally, there is a definite theoretical limit to values that can be simultaneously measured. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically in the new formalism by the non-commutativity of operators representing quantum observables.

Prior to the development of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of formal mathematical analysis, beginning with calculus, and increasing in complexity up to differential geometry and partial differential equations. Probability theory was used in statistical mechanics. Geometric intuition played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of differential geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the development of quantum mechanics (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the SommerfeldWilsonIshiwara quantization rule, which was formulated entirely on the classical phase space.

In the 1890s, Planck was able to derive the blackbody spectrum, which was later used to avoid the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of electromagnetic radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h, is now called Planck's constant in his honor.

In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck's energy quanta were actual particles, which were later dubbed photons.

All of these developments were phenomenological and challenged the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck's constant were actually allowed. The most sophisticated version of this formalism was the so-called SommerfeldWilsonIshiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

In 1923, de Broglie proposed that waveparticle duality applied not only to photons but to electrons and every other physical system.

The situation changed rapidly in the years 19251930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrdinger, Werner Heisenberg, Max Born, Pascual Jordan, and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas. The physical interpretation of the theory was also clarified in these years after Werner Heisenberg discovered the uncertainty relations and Niels Bohr introduced the idea of complementarity.

Werner Heisenberg's matrix mechanics was the first successful attempt at replicating the observed quantization of atomic spectra. Later in the same year, Schrdinger created his wave mechanics. Schrdinger's formalism was considered easier to understand, visualize and calculate as it led to differential equations, which physicists were already familiar with solving. Within a year, it was shown that the two theories were equivalent.

Schrdinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the absolute square of the wave function of an electron should be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the interpretation of the absolute square of the wave function as the probability distribution of the position of a pointlike object. Born's idea was soon taken over by Niels Bohr in Copenhagen who then became the "father" of the Copenhagen interpretation of quantum mechanics. Schrdinger's wave function can be seen to be closely related to the classical HamiltonJacobi equation. The correspondence to classical mechanics was even more explicit, although somewhat more formal, in Heisenberg's matrix mechanics. In his PhD thesis project, Paul Dirac[2] discovered that the equation for the operators in the Heisenberg representation, as it is now called, closely translates to classical equations for the dynamics of certain quantities in the Hamiltonian formalism of classical mechanics, when one expresses them through Poisson brackets, a procedure now known as canonical quantization.

To be more precise, already before Schrdinger, the young postdoctoral fellow Werner Heisenberg invented his matrix mechanics, which was the first correct quantum mechanics the essential breakthrough. Heisenberg's matrix mechanics formulation was based on algebras of infinite matrices, a very radical formulation in light of the mathematics of classical physics, although he started from the index-terminology of the experimentalists of that time, not even aware that his "index-schemes" were matrices, as Born soon pointed out to him. In fact, in these early years, linear algebra was not generally popular with physicists in its present form.

Although Schrdinger himself after a year proved the equivalence of his wave-mechanics and Heisenberg's matrix mechanics, the reconciliation of the two approaches and their modern abstraction as motions in Hilbert space is generally attributed to Paul Dirac, who wrote a lucid account in his 1930 classic The Principles of Quantum Mechanics. He is the third, and possibly most important, pillar of that field (he soon was the only one to have discovered a relativistic generalization of the theory). In his above-mentioned account, he introduced the braket notation, together with an abstract formulation in terms of the Hilbert space used in functional analysis; he showed that Schrdinger's and Heisenberg's approaches were two different representations of the same theory, and found a third, most general one, which represented the dynamics of the system. His work was particularly fruitful in all kinds of generalizations of the field.

The first complete mathematical formulation of this approach, known as the Diracvon Neumann axioms, is generally credited to John von Neumann's 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic paper and book. It was developed in parallel with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert's approach a generation earlier. Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the ones presented here are simple special cases.

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

Finally, some of the originators of quantum theory (notably Einstein and Schrdinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics.

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description normally consists of a Hilbert space of states, observables are self-adjoint operators on the space of states, time evolution is given by a one-parameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations. (It is possible, to map this Hilbert-space picture to a phase space formulation, invertibly. See below.)

The following summary of the mathematical framework of quantum mechanics can be partly traced back to the Diracvon Neumann axioms. The postulates are canonically presented in six statements, though there are many important points to each.[3]

Each physical system is associated with a (topologically) separable complex Hilbert space H with inner product |. Rays (that is, subspaces of complex dimension 1) in H are associated with quantum states of the system.

In other words, quantum states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with the physical interpretation that countably many observations are enough to uniquely determine the state. "A quantum mechanical state is a ray in projective Hilbert space, not a vector. Many textbooks fail to make this distinction, which could be partly a result of the fact that the Schrdinger equation itself involves Hilbert-space "vectors", with the result that the imprecise use of "state vector" rather than ray is very difficult to avoid."[4]

The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems (for instance, J. M. Jauch, Foundations of quantum mechanics, section 11.7). For a non-relativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles.

Physical observables are represented by Hermitian matrices on H. Since these operators are Hermitian, the measurement is always a real value. If the spectrum of the observable is discrete, then the possible results are quantized.

By spectral theory, we can associate a probability measure to the values of A in any state . We can also show that the possible values of the observable A in any state must belong to the spectrum of A. The expectation value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector H is A {displaystyle langle psi mid Amid psi rangle } .

Postulate III

The result of measuring a physical quantity A {displaystyle {mathcal {A}}} must be one of the eigenvalues of the corresponding observable A {displaystyle A} .

In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues. More precisely, if we represent the state in the basis formed by the eigenvectors of A, then the square of the modulus of the component attached to a given eigenvector is the probability of observing its corresponding eigenvalue.

More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator normalized to be of trace 1. The expected value of A in the state is tr ( A ) {displaystyle operatorname {tr} (Arho )} .

When a measurement is performed, only one result is obtained (according to some interpretations of quantum mechanics). This is modeled mathematically as the processing of additional information from the measurement, confining the probabilities of an immediate second measurement of the same observable. In the case of a discrete, non-degenerate spectrum, two sequential measurements of the same observable will always give the same value assuming the second immediately follows the first. Therefore the state vector must change as a result of measurement, and collapse onto the eigensubspace associated with the eigenvalue measured.

If is the orthogonal projector onto the one-dimensional subspace of H spanned by |, then tr ( A ) = A {displaystyle operatorname {tr} (Arho _{psi })=leftlangle psi mid Amid psi rightrangle } .

Though it is possible to derive the Schrdinger equation, which describes how a state vector evolves in time, most texts assert the equation as a postulate. Common derivations include using the DeBroglie hypothesis or path integrals.

Postulate VI

The time evolution of the state vector | ( t ) {displaystyle |psi (t)rangle } is governed by the Schrdinger equation, where H ( t ) {displaystyle H(t)} is the observable associated with the total energy of the system (called the Hamiltonian)

i d d t | ( t ) = H ( t ) | ( t ) {displaystyle ihbar {frac {d}{dt}}|psi (t)rangle =H(t)|psi (t)rangle }

One can in this formalism state Heisenberg's uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article.

Furthermore, to the postulates of quantum mechanics one should also add basic statements on the properties of spin and Pauli's exclusion principle, see below.

The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows:If |(t) denotes the state of the system at any one time t, the following Schrdinger equation holds:

i d d t | ( t ) = H | ( t ) {displaystyle ihbar {frac {d}{dt}}left|psi (t)rightrangle =Hleft|psi (t)rightrangle }

where H is a densely defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and is the reduced Planck constant. As an observable, H corresponds to the total energy of the system.

Alternatively, by Stone's theorem one can state that there is a strongly continuous one-parameter unitary map U(t): H H such that

for all times s, t. The existence of a self-adjoint Hamiltonian H such that

is a consequence of Stone's theorem on one-parameter unitary groups. It is assumed that H does not depend on time and that the perturbation starts at t0 = 0; otherwise one must use the Dyson series, formally written as

where T {displaystyle {mathcal {T}}} is Dyson's time-ordering symbol.

(This symbol permutes a product of noncommuting operators of the form

into the uniquely determined re-ordered expression

The result is a causal chain, the primary cause in the past on the utmost r.h.s., and finally the present effect on the utmost l.h.s..)

It is then easily checked that the expected values of all observables are the same in both pictures

and that the time-dependent Heisenberg operators satisfy

d d t A ( t ) = i [ H , A ( t ) ] + A ( t ) t , {displaystyle {frac {d}{dt}}A(t)={frac {i}{hbar }}[H,A(t)]+{frac {partial A(t)}{partial t}},}

which is true for time-dependent A = A(t). Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.

i d d t | ( t ) = H i n t ( t ) | ( t ) {displaystyle ihbar {frac {d}{dt}}left|psi (t)rightrangle ={H}_{rm {int}}(t)left|psi (t)rightrangle }

i d d t A ( t ) = [ A ( t ) , H 0 ] . {displaystyle ihbar {d over dt}A(t)=[A(t),H_{0}].}

The interaction picture does not always exist, though. In interacting quantum field theories, Haag's theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. Moreover, even if in the Schrdinger picture the Hamiltonian does not depend on time, e.g. H = H0 + V, in the interaction picture it does, at least, if V does not commute with H0, since

So the above-mentioned Dyson-series has to be used anyhow.

The Heisenberg picture is the closest to classical Hamiltonian mechanics (for example, the commutators appearing in the above equations directly translate into the classical Poisson brackets); but this is already rather "high-browed", and the Schrdinger picture is considered easiest to visualize and understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory and many-body physics.

Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated with the symmetry (for instance, angular or linear momentum).

Summary:

The original form of the Schrdinger equation depends on choosing a particular representation of Heisenberg's canonical commutation relations. The Stonevon Neumann theorem dictates that all irreducible representations of the finite-dimensional Heisenberg commutation relations are unitarily equivalent. A systematic understanding of its consequences has led to the phase space formulation of quantum mechanics, which works in full phase space instead of Hilbert space, so then with a more intuitive link to the classical limit thereof. This picture also simplifies considerationsof quantization, the deformation extension from classical to quantum mechanics.

The quantum harmonic oscillator is an exactly solvable system where the different representations are easily compared. There, apart from the Heisenberg, or Schrdinger (position or momentum), or phase-space representations, one also encounters the Fock (number) representation and the SegalBargmann (Fock-space or coherent state) representation (named after Irving Segal and Valentine Bargmann). All four are unitarily equivalent.

The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated with a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" HE, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of HE (this requires the use of a rigged Hilbert space and a renormalization of the norm).

This is related to the quantization of constrained systems and quantization of gauge theories. Itis also possible to formulate a quantum theory of "events" where time becomes an observable (see D. Edwards).

In addition to their other properties, all particles possess a quantity called spin, an intrinsic angular momentum. Despite the name, particles do not literally spin around an axis, and quantum mechanical spin has no correspondence in classical physics. In the position representation, a spinless wavefunction has position r and time t as continuous variables, = (r, t). For spin wavefunctions the spin is an additional discrete variable: = (r, t, ), where takes the values;

That is, the state of a single particle with spin S is represented by a (2S + 1)-component spinor of complex-valued wave functions.

Two classes of particles with very different behaviour are bosons which have integer spin (S=0,1,2...), and fermions possessing half-integer spin (S=12,32,52,...).

The property of spin relates to another basic property concerning systems of N identical particles: Pauli's exclusion principle, which is a consequence of the following permutation behaviour of an N-particle wave function; again in the position representation one must postulate that for the transposition of any two of the N particles one always should have

( , r i , i , , r j , j , ) = ( 1 ) 2 S ( , r j , j , , r i , i , ) {displaystyle psi (dots ,,mathbf {r} _{i},sigma _{i},,dots ,,mathbf {r} _{j},sigma _{j},,dots )=(-1)^{2S}cdot psi (dots ,,mathbf {r} _{j},sigma _{j},,dots ,mathbf {r} _{i},sigma _{i},,dots )}

i.e., on transposition of the arguments of any two particles the wavefunction should reproduce, apart from a prefactor (1)2S which is +1 for bosons, but (1) for fermions.Electrons are fermions with S=1/2; quanta of light are bosons with S=1. In nonrelativistic quantum mechanics all particles are either bosons or fermions; in relativistic quantum theories also "supersymmetric" theories exist, where a particle is a linear combination of a bosonic and a fermionic part. Only in dimension d = 2 can one construct entities where (1)2S is replaced by an arbitrary complex number with magnitude 1, called anyons.

Although spin and the Pauli principle can only be derived from relativistic generalizations of quantum mechanics the properties mentioned in the last two paragraphs belong to the basic postulates already in the non-relativistic limit. Especially, many important properties in natural science, e.g. the periodic system of chemistry, are consequences of the two properties.

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is, the effects of measurement.[5] The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state is the following (note, however, that von Neumann's description dates back to the 1930s and is based on experiments as performed during that time more specifically the ComptonSimon experiment; it is not applicable to most present-day measurements within the quantum domain):

For example, suppose the state space is the n-dimensional complex Hilbert space Cn and A is a Hermitian matrix with eigenvalues i, with corresponding eigenvectors i. The projection-valued measure associated with A, EA, is then

where B is a Borel set containing only the single eigenvalue i. If the system is prepared in state

Then the probability of a measurement returning the value i can be calculated by integrating the spectral measure

over Bi. This gives trivially

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate.

A more general formulation replaces the projection-valued measure with a positive-operator valued measure (POVM). To illustrate, take again the finite-dimensional case. Here we would replace the rank-1 projections

by a finite set of positive operators

whose sum is still the identity operator as before (the resolution of identity). Just as a set of possible outcomes {1...n} is associated to a projection-valued measure, the same can be said for a POVM. Suppose the measurement outcome is i. Instead of collapsing to the (unnormalized) state

after the measurement, the system now will be in the state

Since the Fi Fi* operators need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds.

The same formulation applies to general mixed states.

In von Neumann's approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is deterministic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.

In any case it seems that the above-mentioned problems can only be resolved if the time evolution included not only the quantum system, but also, and essentially, the classical measurement apparatus (see above).

An alternative interpretation of measurement is Everett's relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum physics.

Part of the folklore of the subject concerns the mathematical physics textbook Methods of Mathematical Physics put together by Richard Courant from David Hilbert's Gttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrdinger's equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of the theory was conventional at the time, whereas the physics was radically new.

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Mathematical formulation of quantum mechanics - Wikipedia

March 18, 2022

Physicists have worked and wrestled with quantum theory for more than a century now, applying it to explore and help solve the profound mysteries of Albert Einsteins theory of relativity and cosmological conundrums such as black holes, gravity and the origins of the universe.

But for Arizona State University theoretical chemist Vladimiro Mujica, there is still a vast, secret and fascinating world to explore but rather than out there in the vastness of space time, at the nexus between everyday life on Earth and the quantum world. Cellular mutations in the molecule of life, DNA, happen randomly and are governed by quantum probability rules. Download Full Image

Recently, quantum mechanics has been found to play an essential role in our understanding of chemistry and biology, and the molecular theory of evolution.

Now, Mujica will get a chance to further explore this quantum world by leading a three-year, $1 million award from the prestigious Keck Foundation. Their goal is build a foundational understanding of how the sometimes weird, exotic features of quantum physics influence the very stuff that makes life work.

To do so, Mujica will lead a multi-institutional quantum biology team that includes ASU colleague William Petuskey and leading experimentalists, including Northwestern University co-investigators Michael Wasielewski and University of California Los Angeles professors Paul Weiss and Louis Bouchard.

To be successful, we really needed to think outside of the box, with a good foundation, said Mujica, a professor in the School of Molecular Sciences. So, we put this team together of leading experimentalists, but also with a firm grasp of theory top-ranking people to take a quantum leap in this field of science.

The awards initiative, titled Chirality, Spin Coherence, and Entanglement in Quantum Biology,will explore fundamental quantumeffects in biological systems.

For example, two key processes necessary for life: photosynthesis in plants and respiration in animals, are driven by reactions that involve the transfer of electrons in molecules and across boundaries within the cell.

Electrons themselves, in addition to carrying a negative charge, have key quantum properties, including spin, that plays a fundamental role in the molecular electron transfer processes that make life possible.

Vladimiro Mujica. Photo courtesy Mary Zhu

Chiral is the Greek word for hand. No matter how hard one tries, a left hand and right hand are non-superimposable mirror images of each other. Ever try to shake a persons hand with the opposite hand? That awkward encounter simply because the thumbs are in different positions is an everyday demonstration of chirality.

It turns out molecules, and life, have the same chiral properties. But how does that help their biological function?

We're trying to decipher in a way, a mystery of nature and evolution, Mujica said. Because it turns out that biological systems use these chiral molecules in proteins, DNA and RNA. These are some of the most important molecules in biology. For example, DNA is a double-helix ladder that is intrinsically chiral. And so are the proteins encoded by these fundamental biological molecules, which are the bricks and mortars of the cell, doing all the work that makes us alive.

Quantum mechanics is all-across biology: Photosynthesis. Cellular respirationc. Oxygen transport.Cellular mutations.

Are all governed by quantum effects.

These happen randomly and are governed by quantum probability rules.

One can zoom in further on life, under the skin all the way to the molecules at the atomic level and clouds of electrons in quantum states. In everyday life, we are used to electrons being transported through copper wires to deliver electricity to our homes.

But what are the wires that deliver electrons in living system, a process that involves substantial amounts of energy and heat? And how do they avoid frying life, or by proxy, us?

In living systems, how electrons are transferred or transported depends on organic molecules, Mujica said. Now, organic molecules are far less efficient than copper wires or anything like that to transport or transfer electrons. But nevertheless, evolution chose this in a way.

Mujica refers to this as a real mystery as to why Mother Nature chose these lousy molecules for transferring electrons.

Yet, as Jeff Goldblums quirky scientist character in "Jurassic Park" famously once said: "Life finds a way.

It turns out electrons are transported in organic molecules primarily by tunneling, not diffusion as in copper wires.

The mechanism electrons going through organic molecules is to a large extent a quantum phenomenon, Mujica said Its a mechanism called tunneling, and what it implies is that electrons can go from one region of the molecule to the other, even if they do not have enough energy to overcome intrinsic barriers.

The research team wants to investigate why and how electrons use this tunneling mechanism for biological function essential to life. First, they have designed a series of experiments using synthetic pairs of right or left-handed DNA structures. Next, they will custom tailor electron donors andacceptors as part of their structures to probe this chirality-dependentelectron transfer. All this experimental effort is guided by a predictive theoretical and computational effort.

Some of themodelsystems tweaks they will examine are the effect of the electron donor-acceptordistance, the temperature, redox properties and the coupling to their surrounding environment.

An electron transfer process with the electron-vibration (phonon) interaction. The process is essential to understanding and controlling charge and energy flow in various electronic, photonic and energy conversion devices or, in this case, a biomolecule. The "IN" and "OUT" have either the same or distorted phase, depending on whether the transport is coherent or incoherent.

A fundamental quantum electron property is spin. Electrons can be like spinning tops, rotating on their own axis.

Mujica explains that because electrons are charged particles, "this rotation creates a magnetic moment, which only has two components; one component aligns in the direction of transport and the other component is aligned in the opposite direction to transport.

"As they tunnel through chiral organic molecules, they have a preferential orientation due to the spin orbit interaction and the loss of time-inversion symmetry.

This is known as spin polarization.

It turns out, when electron spin is polarized, electrons can tunnel much easier and farther because one of the two spin components has a larger transmission probability.

Mujica likens it to a bullet going through the barrel of a gun. The first guns that were ever made all had smooth, hollowed-out barrels. But when grooves were etched, it gave the bullet a spin that allowed it to travel straighter and farther. Also, it is easy to understand with this simple analogy that bullets rotating clockwise will not go through counter-clockwise designed barrels, and vice versa. A classical analogy to what happens with electron spins.

And so, for their second set of experiments, they willuse magnetic substrates, nanoscale chemical patterning, andmultimodalspin-polarized scanningtunneling microscopyand spectroscopieswith orientedenantiomeric pairs of DNAandintercalated metalstoelucidate and to quantifythe molecular and interface contributionstochirality-induced spin selectivity.

Since most biological molecules, including amino acids inproteins and nucleotides in RNA and DNA, are chiral, thecriticalroles of spin polarization inelectron transport within and between biological molecules will be determined.

Finally, electrons have a dual particle-wave quantum nature; they have particle-like properties such as mass and charge, but their dynamics and propagation follows the rules of wave quantum mechanics.

In biology, as the electrons encounter other molecules or molecular barriers like cell membranes, they are scattered, and their wave properties are modified. Two wave sources arecoherentif their frequency and waveform are identical. If not, the waves can be canceled or enhanced due to interference. This interference can be destructive and leads to noise, which can also be due to thermal interactions.

Spin coherence can coexist with spin polarization Mujica said. What it means is that you have in-phase transport, so you're not reducing the intensity of the wave, and we're not changing the phase of a wave associated to that transfer.

Spin coherence is intimately associated to another quantum process, entanglement, that is of fundamental importance in quantum information and quantum computing.

Mujica says this is a high-risk, high-reward project that may upset the current conventional wisdom in quantum biology.

I mean, the common knowledge was that you couldn't have coherence in a quantum biological system, because the environmental effects would destroy coherence in a very short time.

They will try to put it all together by determining how chirality influences theelectronic, vibrational and spin-polarized electron transferfrom electrondonors to acceptor sites as spin-coherent electron pairs are generated in photo-induced electron transfer reactions.

Essentially, the grant focuses on the role of spin-polarized electrons and how it influences the behavior of biological systems, especially the length and temperature dependence, and how spin polarization and spin coherence can coexist, Mujica said. These are key unsolved issues in biological electron-transfer reactions.

In addition tostudying the unexplored roles of spin coherence in quantum biology, Mujicas team will study how it can coexist with spinpolarization and how, or if, it can create what is referred to as the spooky "action at a distance," or quantum entangled states.

The overarching Keck grant goal is to answer these questions, and the contributions of three key ingredients: tunneling, spin and coherence. These are central to discovering the underpinnings of the emerging field of quantumbiology.

By exploring these questions, Mujicas team ultimately hopes to use the Keck grant as a catalyst to create an ASU center for quantum biology, and further down the road, practical applications, such as quantum information and computing. All this could help position ASU in quantum technologies and information efforts, which are of strategic importance for the U.S.

If we can provide enough evidence, we hope to unveil some very important questions that will be crucial for an ASU effort in quantum information sciences, and this is something that we are starting with efforts in engineering and physics, Mujica said.

We want to weigh in on the roadmap to be able to use molecules for quantum information. From our perspective, we really think of this as a step in the direction of defining our capabilities of using quantum biology in molecular quantum information sciences, a field that is experiencing a true renaissance.

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Keck award will help scientists take quantum leap to explore the mysteries of life - ASU News Now

Fragments of the interior of a proton have been shown by scientists from Mexico andPoland to exhibit maximum quantum entanglement.

The discovery, already confronted with experimental data, allows us to suppose that in some respects thephysics of the inside of a proton may have much in common not only with well-known thermodynamic phenomena, but even with the physics of... black holes.

Various fragments of the inside of a proton must be maximally entangled with each other, otherwise theoretical predictions would not agree with the data collected in experiments, it was shown in European Physical Journal C. The theoretical model (which extends the original proposal by physicists Dimitri Kharzeev and Eugene Levin) makes it possible to suppose that, contrary to current belief, the physics operating inside protons may be related to such concepts as entropy or temperature, which in turn may relate it to such exotic objects as black holes. The authors of the discovery are Dr. Martin Hentschinski from the Universidad de las Americas Puebla in Mexico and Dr. Krzysztof Kutak from the Institute of Nuclear Physics of the Polish Academy of Sciences (IFJ PAN) in Cracow, Poland.

The Mexican-Polish theorists analysed the situation in which electrons are fired at protons. When an incoming electron carrying a negative electric charge approaches a positively charged proton, it interacts with it electromagnetically and deflects its path. Electromagnetic interaction means that a photon has been exchanged between the electron and the proton. The stronger the interaction, the greater the change in momentum of the photon and therefore the shorter the associated electromagnetic wave.

"If a photon is 'short' enough to 'fits' inside a proton, it begins to 'resolve' details of its internal structure. The result of interacting with this sort of photon can be the decay of the proton into particles. We have shown that there is entanglement between the two situations. If the observation by the photon of the interior part of the proton leads to its decay into a number of particles, let's say three, then the number of particles originating from the unobserved part of the proton is determined by the number of particles seen in the observed part of the proton," explains Dr. Kutak.

We can speak of quantum entanglement of various quantum objects, if certain characteristic of the objects are related to each other in a particular way. The classical analogy of the phenomenon can be represented by the toss of a coin. Let's assume that one object is one side of the coin, and the other object is its other side. When we flip a coin, there is the same probability that the coin will land heads or tails facing up. If it lands heads up, we know for sure that the other side is tails. We can then speak of maximum entanglement since the probability which determines the value of an object's characteristic does not favour any possible value: we have a 50% chance of heads and the same for tails. A smaller than maximum entanglement occurs when the probability starts to favour one of the possible outcomes to a greater or lesser extent.

"Our study shows that the interior of a proton seen by a passing photon must be entangled with the unseen part in just this maximal manner, as suggested by Kharzeev and Levin. In practice, this means that we have no chance of predicting whether, due to interaction with the photon, the proton will decay into three, four or any other number of particles," explains Dr. Hentschinski.

The new theoretical predictions have already been verified. If entanglement inside the proton were not maximal, there would be discrepancies between theoretical calculations and the results of the H1 experiment at the HERA accelerator at the DESY centre in Hamburg, where positrons (i.e. antiparticles of the electrons) were collided with protons until 2007. Such discrepancies were not observed.

The success of the Polish-Mexican tandem is due to the fact that the researchers managed to correctly identify the factors responsible for the maximum entanglement of the proton interior.

In the naive schoolbook view, the proton is a system of three elementary particles: two up quarks and one down quark. However, the strong interactions between these quarks, carried by gluons, can be so strong that they lead to the creation of virtual particle-antiparticle pairs. These can be not only pairs of virtual gluons (which are their own antiparticles), but also pairs made up of any quark and its corresponding antiparticle (even one as massive as charm). All this means that inside the proton, apart from three valence quarks, there are constantly 'boiling' seas of virtual gluons and virtual quarks and antiquarks.

"In earlier publications, physicists dealing with the subject assumed that the source of entanglement should be a sea of gluons. Later, attempts were made to show that quarks and antiquarks are the dominant source of entanglement, but even here the proposed methods of description did not stand the test of time. Meanwhile, according to our model, verified by confrontation with experimental data, the sea of virtual gluons is responsible for about 80% of the entanglement, while the sea of virtual quarks and antiquarks is responsible for the remaining 20%," emphasizes Dr. Kutak.

Most recently, quantum physicists have been associating entropy with the state inside a proton. This is a quantity well known from classical thermodynamics, where it is used to measure the degree of disordered motion of particles in an analysed system. It is assumed that when a system is disordered, it has high entropy, whereas an ordered system has low entropy. It has recently been shown that in the case of the proton, we can successfully talk about entanglement entropy. However, many physicists have considered the proton to be a pure quantum state in which one should not speak of entropy at all. The consistency of the Mexican-Polish model with experiment is a strong argument for the fact that the concept of entanglement inside the proton as proposed by Kharzeev and Levin has a point. Last but not least, since entanglement entropy is also related to concepts such as the surface area of black holes, the latest result opens an interesting field for further research.

Source:https://www.ifj.edu.pl/en/

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Fragments of the Inside of a Proton Exhibit Maximum Quantum Entanglement - AZoQuantum

What does a six-dimensional figure look like? Theoretical physicist Richard Nally cant show you exactly, but he does have a sculpture a pink shape the size of a grapefruit that can help you imagine a piece of one.

Its called a K3 surface, said Nally, a Klarman Fellow in physics in the College of Arts and Sciences (A&S). Of course, we cant make sculptures of things that live in six dimensions, but you can take little slices of them to see what they look like. This is a slice of a four-dimensional shape that is really important to the history and practice of string theory.

Researchers have known about the shapes in string theory for decades, Nally said. But in the past few years, he and others have started to take the shapes seriously as number theoretic objects and to study them in that framework. Nally will spend his three-year Klarman Postdoctoral Fellowship seeking to understand the mathematical structures at the root of gravity and quantum mechanics.

We want to find a nice shape that lets us keep the solution to quantum gravity, while getting the features such as an expanding universe and only having four dimensions that we see in the world around us, he said.

Read the full story on the College of Arts and Sciences website.

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Klarman fellow blends physics and math to explore string theory | Cornell Chronicle - Cornell Chronicle