A.Yu. Sevalnikov
Quantum and time in the modern physical paradigm

The year 2000 marked the 100th anniversary of the birth of quantum mechanics. The transition across the line of centuries and centuries is an occasion to talk about time, and in this case precisely in connection with the anniversary of the quantum.

Linking the concept of time to the ideas of quantum mechanics might seem artificial and far-fetched, if not for one circumstance. We still don't understand the meaning of this theory. “It is safe to say that no one understands the meaning of quantum mechanics,” said Richard Feynman. Faced with micro-phenomena, we are faced with some mystery that we have been trying to unravel for a century. How not to remember the words of the great Heraclitus that “nature loves to hide.”

Quantum mechanics is full of paradoxes. Don't they reflect the very essence of this theory? We have a perfect mathematical apparatus, a beautiful mathematical theory, the conclusions of which are invariably confirmed by experience, and at the same time we lack any “clear and distinct” ideas about the essence of quantum phenomena. The theory here acts rather as a symbol, behind which another reality is hidden, manifested in irreducible quantum paradoxes. “The oracle does not reveal or hide, it hints,” as Heraclitus said. So what does quantum mechanics hint at?

The origins of its creation were M. Planck and A. Einstein. The focus was on the problem of emission and absorption of light, i.e. the problem of formation in a broad philosophical sense, and therefore of movement. This problem as such has not yet come into the spotlight. In discussions around quantum mechanics, the problems of probability and causality, wave-particle duality, problems of measurement, nonlocality, the participation of consciousness, and a number of others, closely related directly to the philosophy of physics, were considered first of all. However, we dare to say that it is the problem of becoming, the oldest philosophical problem, that is the main problem of quantum mechanics.

This problem has always been closely connected with the theory of quantum, from the problem of radiation and absorption of light in the works of Planck and Einstein to the latest experiments and interpretations of quantum mechanics, but always implicitly, as a kind of hidden subtext. In fact, almost all of its debatable issues are closely related to the problem of formation.

So the so-called “measurement problem”, which plays a key role in interpretations of quantum mechanics. The measurement dramatically changes the state of the quantum system, the shape of the wave function Ψ(r,t). For example, if when measuring the position of a particle we obtain a more or less accurate value of its coordinate, then the wave packet, which was the function Ψ before the measurement, is “reduced” into a less extended wave packet, which can even be point-like if the measurement is carried out very accurately. This is related to the introduction by W. Heisenberg of the concept of “reduction of a package of probabilities”, which characterizes this kind of sudden change wave function Ψ(r,t).

Reduction always leads to a new state, which cannot be foreseen in advance, since before measurement we can only predict the probabilities of various possible options.

The situation is completely different in the classics. Here, if the measurement is carried out accurately enough, then this is a statement of only the “present state”. We get the true value of a quantity that objectively exists at the moment of measurement.

The difference between classical mechanics and quantum mechanics is the difference between their objects. In the classics - it's cash existing condition, in the quantum case, is an object that arises, becomes, an object that fundamentally changes its state. Moreover, the use of the concept “object” is not entirely legitimate; we rather have the actualization of potential existence, and this act itself is not fundamentally described by the apparatus of quantum mechanics. A reduction of the wave function is always a discontinuity, a jump in the state.

Heisenberg was one of the first to argue that quantum mechanics takes us back to the Aristotelian concept of being in possibility. Such a point of view in quantum theory returns us to a two-mode ontological picture, where there is a mode of being in possibility and a mode of being actual, i.e. the world of the realized.

Heisenberg did not develop these ideas in a consistent manner. This was carried out a little later by V.A. Fok. The concepts of “potential possibility” and “actualized” that he introduced are very close to the Aristotelian concepts of “being in possibility” and “being in the stage of completion.”

According to Fock, the state of the system described by the wave function is objective in the sense that it represents an objective (independent of the observer) characteristic of the potential capabilities of a particular act of interaction of a micro-object with a device. Such an “objective state is not yet valid, in the sense that for an object in this state the indicated potential possibilities have not yet been realized; the transition from potential possibilities to realized occurs at the final stage of the experiment.” The statistical probability distribution that arises during measurement reflects the potential opportunities that objectively exist under given conditions. Actualization, “realization” according to Fock is nothing more than “becoming”, “change”, or “movement” in a broad philosophical sense. The actualization of the potential introduces irreversibility, which is closely related to the existence of the “arrow of time.”

It is interesting that Aristotle directly connects time with movement (see, for example, his “Physics” - “time does not exist without change”, 222b 30ff, book IV especially, as well as treatises - “On Heaven”, “On the Origin and destruction"). Without considering Aristotle's understanding of time in detail for now, we note that for him it is, first of all, a measure of movement, and, more broadly, a measure of the formation of being.

In this understanding, time acquires a special, distinguished status, and if quantum mechanics really points to the existence of potential being and its actualization, then in it this special character of time should be obvious.

It is precisely this special status of time in quantum mechanics that is well known and has been noted repeatedly by different authors. For example, de Broglie in the book “Heisenberg Uncertainty Relations and the Wave Interpretation of Quantum Mechanics” writes that QM “does not establish a true symmetry between spatial and temporal variables. The coordinates x, y, z of the particle are considered observable, corresponding to certain operators and having in any state (described by the wave function Ψ) some probabilistic distribution of values, while time t is still considered a completely deterministic value.

This can be clarified as follows. Let us imagine a Galilean observer carrying out measurements. He uses coordinates x, y, z, t, observing events in his macroscopic frame of reference. The variables x, y, z, t are numerical parameters, and it is these numbers that enter into the wave equation and wave function. But each particle of atomic physics corresponds to “observable quantities”, which are the coordinates of the particle. The relationship between the observed quantities x, y, z and the spatial coordinates x, y, z of the Galilean observer is statistical in nature; In the general case, each of the observed values x, y, z can correspond to a whole set of values with a certain probability distribution. As for time, in modern wave mechanics there is no observable quantity t associated with a particle. There is only a variable t, one of the space-time variables of the observer, determined by the (essentially macroscopic) clock that this observer has.”

Erwin Schrödinger says the same thing. “In CM, time is highlighted in comparison with coordinates. Unlike all other physical quantities, it does not correspond to an operator, not statistics, but only a value that is accurately read, as in good old classical mechanics, using the usual reliable clock. The distinguished nature of time makes quantum mechanics in its modern interpretation a non-relativistic theory from beginning to end. This feature of CM is not eliminated when a purely external “equality” of time and coordinates is established, i.e. formal invariance under Lorentz transformations, with the help of appropriate changes in the mathematical apparatus.

All QM statements have the following form: if now, at time t, a certain measurement is carried out, then with probability p its result will be equal to a. Quantum mechanics describes all statistics as functions of one exact time parameter... In QM it makes no sense to ask with what probability the measurement will be made in the time interval (t. t+ dt), because I can always choose the time of measurement at my discretion.”

There are other arguments showing the allocated nature of time, they are known and I will not dwell on them here. There are also attempts to overcome such isolation, even to the point where Dirac, Fock and Podolsky proposed the so-called covariance of equations. “multi-time” theory, when each particle is assigned not only its own coordinate, but also its own time.

In the book mentioned above, de Broglie shows that such a theory cannot avoid special status time, and it is very characteristic that he ends the book with the following phrase: “Thus, it seems to me impossible to eliminate the special role that a variable like time plays in quantum theory.”

Based on such reasoning, we can confidently say that quantum mechanics forces us to talk about the allocation of time, about its special status.

There is another aspect of quantum mechanics that has not yet been considered by anyone.

In my opinion, it is legitimate to talk about two “times”. One of them is our ordinary time - finite, unidirectional, it is closely related to actualization and belongs to the world of the realized. The Other is what exists for the mode of being in possibility. It is difficult to characterize it in our ordinary terms, since at this level there are no concepts of “later” or “earlier”. The principle of superpositions just shows that in potency all possibilities exist simultaneously. At this level of existence, it is impossible to introduce spatial concepts “here” and “there”, since they appear only after the “unfolding” of the world, during which time plays a key role.

This statement can easily be illustrated by the famous double-slit thought experiment, which, according to Richard Feynman, contains the whole secret of quantum mechanics.

Let's direct a beam of light onto a plate with two narrow slits. Through them, light hits a screen placed behind the plate. If light consisted of ordinary “classical” particles, then we would get two bright stripes on the screen. Instead, as is known, a series of lines is observed - an interference pattern. Interference is explained by the fact that light travels not just as a stream of photon particles, but in the form of waves.

If we try to trace the path of photons and place detectors near the slits, then in this case the photons begin to pass through only one slit and the interference pattern disappears. “It appears that photons behave like waves as long as they are “allowed” to behave like waves, i.e. spread through space without occupying any specific position. However, the moment someone "asks" where exactly the photons are - either by identifying the slit they passed through, or by having them hit the screen through just one slit - they instantly become particles...

In double-slit experiments, the physicist's choice of measuring instrument causes the photon to "choose" between passing through both slits simultaneously, like a wave, or through just one slit, like a particle. However, what would happen, Wheeler asked, if the experimenter could somehow wait until the light had passed through the slits before choosing a method of observation?

Such a “delayed choice” experiment can be demonstrated more clearly using the radiation of quasars. Instead of a double-slit plate, "such an experiment would use a gravitational lens—a galaxy or other massive object that can split the quasar's radiation and then focus it toward a distant observer, creating two or more images of the quasar...

An astronomer's choice of which way to observe photons from a quasar today is determined by whether each photon traveled along both paths or just one path near the gravitational lens billions of years ago. At the moment when the photons reached the “galactic beam splitter,” they must have had something like a premonition telling them how to behave in order to correspond to the choice that would be made by unborn beings on a planet that does not yet exist.”

As Wheeler rightly points out, such speculation arises from the mistaken assumption that photons have some form before they are measured. In fact, “quantum phenomena themselves have neither a corpuscular nor a wave character; their nature is not determined until they are measured.”

Experiments conducted in the 90s confirm such “strange” conclusions from quantum theory. A quantum object truly “does not exist” until the moment of measurement, when it receives actual existence.

One aspect of such experiments has so far been virtually undiscussed by researchers, namely the time aspect. After all, quantum objects acquire their existence not only in the sense of their spatial localization, but also begin to “be” in time. Having assumed the existence of potential existence, it is necessary to draw a conclusion about the qualitatively different nature of existence at this level of existence, including the temporary one.

As follows from the principle of superposition, different quantum states exist “simultaneously”, i.e. a quantum object initially, before updating its state, exists in all admissible states at once. When the wave function is reduced from the “superposed” state, only one of them remains. Our ordinary time is closely connected with this kind of “events”, with the process of actualization of the potential. The essence of the “arrow of time” in this understanding is that objects come into existence, “come into existence”, and it is with this process that the unidirectionality of time and its irreversibility are connected. Quantum mechanics, the Schrödinger equation describes the line between the level of possible existence and actual existence, or more precisely, it gives the dynamics, the probability of the potential being realized. The potential itself is not given to us; quantum mechanics only points to it. Our knowledge is still fundamentally incomplete. We have an apparatus that describes the classical world, that is, the actual, manifest world - this is the apparatus of classical physics, including the theory of relativity. And we have the mathematical formalism of quantum mechanics that describes becoming. The formalism itself is “guessed” (here it is worth remembering how the Schrödinger equation was discovered); it is not derived from anywhere, which gives rise to the question of a more complete theory. In our opinion, quantum mechanics only brings us to the brink of manifest existence, makes it possible to slightly reveal the secrets of being and time, without revealing and without having such an opportunity to reveal it completely. We can only draw a conclusion about a more complex structure of time, about its special status.

An appeal to the philosophical tradition will also help substantiate this point of view. As you know, Plato also distinguishes between two times - time itself and eternity. For him, time and eternity are incommensurable; time is only a moving likeness of eternity. When the demiurge created the Universe, as described in the Timaeus, the demiurge “planned to create a kind of moving semblance of eternity; Having arranged the sky, he together with it creates for eternity, which resides in one, an eternal image moving from number to number, which we called time.”

Plato's concept is the first attempt to overcome, to synthesize two approaches to time and the world. One of them is the Parmenidean line, the spirit of the Eleatic school, where all movement and change were denied, where only eternal existence was recognized as truly existing, the other is associated with the philosophy of Heraclitus, who argued that the world is a continuous process, a kind of combustion or non-stop flow.

Another attempt to overcome such duality was the philosophy of Aristotle. By introducing the concept of potential being, he was able for the first time to describe movement, the doctrine of which he expounds in close connection with the doctrine of nature.

Based on Plato’s dualistic scheme of “being-non-existence,” it turns out to be impossible to describe movement; it is necessary to “find the “underlying” third, which would be a mediator between the opposites.”

Aristotle’s introduction of the concept of dynamis - “being in possibility” was caused by his rejection of Plato’s method, which proceeded from the opposites “existent-non-existent”. As a result of this approach, writes Aristotle, Plato cut off his path to comprehending change, which is the main feature of natural phenomena. “...If we take those who attribute existence and non-existence to things together, from their words it rather turns out that all things are at rest, and not in motion: in fact, there is no longer anything to change into, for all properties exist<уже>in all things." [Metaphysics, IV,5].

“So, the opposition being-non-being, says Aristotle, needs to be mediated by something third: such a mediator between them is Aristotle’s concept of “being in possibility.” Aristotle thus introduces the concept of possibility in order to explain the change, emergence and death of everything natural and thereby avoid the situation that developed in the system of Platonic thinking: emergence from non-existence is a random emergence. And indeed, everything in the world of transitory things is unknowable for Plato, because it is of a random nature. Such a reproach in relation to the great dialectician of antiquity may seem strange: after all, as is known, it is dialectics that examines objects from the point of view of change and development, which cannot be said about the formal logical method, the creator of which Aristotle is rightly considered.”

However, this reproach of Aristotle is completely justified. Indeed, paradoxically, the change that occurs with sensory things does not fall into Plato’s field of vision. His dialectics considers the subject in its change, but this, as P.P. Gaidenko rightly notes, is a special subject - logical. In Aristotle, the subject of change moved from the logical sphere to the sphere of existence, and the logical forms themselves ceased to be the subject of change. Existing in the Stagirite has a twofold character: existing in reality and existing in possibility, and since it has “a twofold character, then everything changes from existing in possibility to existing in reality... And therefore emergence can occur not only - incidentally - from non-existent , but also<можно сказать, что>everything arises from what exists, namely from that which exists in possibility, but does not exist in reality” (Metaphysics, XII, 2). The concept of dynamis has several different meanings, which Aristotle reveals in Book V of Metaphysics. The terminological distinction subsequently acquired two main meanings. Latin- potentia and possibilitas, which are often translated as “ability” and “possibility” (cf. German ability - Vermögen, and opportunity - Möglichkeit). “The name of possibility (dynamis) first of all denotes the beginning of movement or change, which is in something else or because it is another, as, for example, the art of construction is a capacity that is not in what is being built; and the art of medicine, being a certain ability, can be in the one who is being treated, but not because he is being treated” (Metaphysics, V, 12).

For Aristotle, time is closely related to movement (in the broadest sense). “It is impossible for time to exist without movement.” According to Aristotle, this is obvious, since “if time exists, obviously motion must also exist, since time is a certain property of motion.” This means that there is no movement in itself, but only a changing, becoming being, and “time is the measure of movement and the presence of [a body] in a state of movement.” From here it becomes clear that time also becomes a measure of being, because “for everything else, being in time means measuring its being by time.”

There is a significant difference between the approaches of Plato and Aristotle in understanding time. For Plato, time and eternity are incommensurable; they are qualitatively different. For him, time is only a moving likeness of eternity (Timaeus, 38a), for everything that has arisen is not involved in eternity, having a beginning, and therefore an end, i.e. it was and will be, whereas eternity only exists.

Aristotle denies the eternal existence of things, and although he introduces the concept of eternity, this concept for him is rather the infinite duration of the eternal existence of the world. His logical analysis, no matter how brilliant he is, is not able to grasp the existence of something qualitatively different. Plato's approach, although it does not describe movement in the sensory world, turns out to be more far-sighted in relation to time. Subsequently, the concepts of time were developed within the framework of the Neoplatonic school and Christian metaphysics. Without being able to enter into an analysis of these teachings, we will only note the common features that unite them. They all talk about the existence of two times - ordinary time associated with our world, and eternity, the eon (αιων), associated with supersensible being.

Returning to the analysis of quantum mechanics, we note that the wave function is defined on the configuration space of the system, and the function Ψ itself is a vector of infinite-dimensional Hilbert space. If the wave function is not just an abstract mathematical construct, but has some referent in existence, then it is necessary to draw a conclusion about its “other-existence”, non-belonging to the actual four-dimensional space-time. The same thesis is demonstrated by both the well-known “unobservability” of the wave function and its quite tangible reality, for example, in the Aharonov-Bohm effect.

Along with the Aristotelian conclusion that time is a measure of being, we can conclude that quantum mechanics allows us, at least, to raise the question of the multiplicity of time. Here modern science, in the figurative expression of V.P. Vizgin, “enters into a fruitful “ideological roll call” with the ancient heritage.” Indeed, already “Einstein’s theory of relativity is closer to the ideas of the ancients about space and time as properties of being, inseparable from the order of things and the order of their movements, than to Newton’s ideas about absolute space and time, conceived as completely indifferent to things and their movements, if not dependent on them."

Time is closely related to the “event”. “In a world where there is one “reality”, where “possibility” does not exist, time does not exist either; time is a difficultly predictable creation and disappearance, a re-arrangement of the “package of possibilities” of one or another existence.” But the “package of possibilities” itself exists, as we wanted to show, in the conditions of a different time. This statement is a kind of “metaphysical hypothesis”, however, if we take into account that quantum mechanics becomes Lately“experimental metaphysics”, then we can raise the question of the experimental detection of such “supratemporal” structures associated with the wave function of the system. The presence of such other-time structures is already indirectly indicated by experiments with “delayed choice” and Wheeler’s thought experiment with a “galactic lens”, which demonstrates the possible “delay” of the experiment in time. How true this hypothesis is, time itself will tell.

Notes

Fock V.A. On the interpretation of quantum mechanics. M., 1957. P. 12.

L. de Broglie. Heisenberg uncertainty relations and the wave interpretation of quantum mechanics. M., 1986. S. 141-142.

Schrödinger E. Special theory of relativity and quantum mechanics // Einstein collection. 1982-1983. M., 1983. P. 265.

L. de Broglie. Decree. work. P. 324.

Horgan J. Quantum philosophy // In the world of science. 1992. No. 9-10. P. 73.

Horgan J. Right there. P. 73.

Right there. P. 74.

Plato. Timaeus, 38a.

Right there. 37 p.

Gaidenko P.P. Evolution of the concept of science. M., 1980. P. 280.

Right there. P. 282.

Aristotle. On creation and destruction, 337 a 23f.

Aristotle. Physics, 251b 27ff.

Ibid., 221a.

Ibid., 221a 9f.

For a description of the Neoplatonic concept, see, for example: Losev A.F. Being. Name. Space. M., 1993. S. 414-436; on the understanding of time in Christian theology: Lossky V.N. Essay on the mystical theology of the Eastern Church. M., 1991. Ch. V.

Vizgin V.P. Study of time // Philosophy. research M., 1999. No. 3. P. 149.

Right there. P. 149.

Right there. P. 157.

Horgan, John. Quanten-Philosophie // Quantenphilosophie. Heidelberg, 1996. S. 130-139.

The obvious inapplicability of classical physics, mechanics and electrodynamics to describe micro-objects, atoms, molecules, electrons and radiation. The problem of equilibrium thermal radiation. The problem of substance stability. Discretion in the microcosm. Spectral lines. Experiments of Frank and Hertz.

Discreteness in classical physics. Analogy with eigenvalue problems. String vibrations, wave equation, boundary conditions. The need for a wave description of microparticles. Experimental indications of the wave properties of micro-objects. Electron diffraction. Experiments by Davisson and Germer.

Wave and geometric optics. Description of wave fields in the limit of small wavelengths as particle flows. De Broglie's idea about the construction of quantum or wave mechanics.

Elements of classical mechanics: principle least action, Lagrange function, action as a function of coordinates, writing the principle of least action through the Hamilton function. The equation Hamilton-Jacobi. Shortened action. Action of a freely moving particle

Wave equation in classical physics. Monochromatic waves. Helmholtz equation.

Reconstruction of the wave equation for a free particle from the dispersion relation. Schrödinger equation for a free nonrelativistic particle.

2. Physical quantities in classical and quantum mechanics.

The need to introduce physical quantities as operators, using the example of momentum and Hamilton operators. Interpretation of the wave function. Probability amplitude. Superposition principle. Addition of amplitudes.

Thought experiment with two slits. Transition amplitude. Transition amplitude as the Green's function of the Schrödinger equation. Amplitude interference. Analogy with principle Huygens-Fresnel. Composition of amplitudes.

Probability distribution for position and momentum. Go to k- performance. The Fourier transform as an expansion in eigenfunctions of the momentum operator. Interpretation of operator eigenvalues as observable physical quantities.

The delta function as the kernel of the unit operator. Various views

delta functions. Calculation of Gaussian integrals. A little math. Memories of mathematical physics and a new look.

3. General theory of operators of physical quantities.

Eigenvalue problems. Quantum numbers. What does it mean “a physical quantity has a certain meaning?” Discrete and continuous spectra.

Hermitianity-definition. Validity of averages and eigenvalues. Orthogonality and normalization. Wave functions as vectors. Dot product of functions.

Decomposition of functions into operator's own functions. Basis functions and expansions. Calculation of coefficients. Operators as matrices. Continuous and discrete indices. Representations of multiplication and differentiation operators as matrices.

Dirac notation. Abstract vectors and abstract operators. Representations and transition to various bases.

4. Measurement in quantum mechanics.

Macroscopicity and classicism of the measuring device. Measurement is a “decomposition” based on the device’s own functions.

5. Schrödinger equation for a free nonrelativistic particle.

Solution by the Fourier method. Wave packet. The principle of uncertainty. Noncommutativity of the momentum and coordinate operators. What variables does the wave function depend on? The concept of a complete set. Lack of trajectory.

Commutability of operators and the existence of common eigenfunctions.

Necessity and sufficiency. Once again about the transition to different bases.

Transformations of operators and state vectors. Unitary operators are operators that preserve orthonormality.

Nonstationary Schrödinger equation. Evolution operator. Green's function. Functions from operators. Construction of the evolution operator by expansion of the stationary equation in terms of eigenfunctions. Operator of the derivative of a physical quantity with respect to time.

6. Heisenberg representation.

Heisenberg's equations. Schrödinger equation for coupled and asymptotically free systems.

7. Entangled and independent states.

Condition for the existence of a wave function for a subsystem. Pure and mixed states of the subsystem. Description of mixed states using a density matrix. Rule for calculating averages. Evolution of the density matrix. Von Neumann equation.

8. One-dimensional movement.

One-dimensional Schrödinger equation. General theorems. Continuous and discrete spectra. Solving problems with piecewise constant potentials. Boundary conditions on potential jumps. Search for discrete levels and eigenfunctions in rectangular potentials. Oscillation theorem. Variational principle. An example of a shallow hole. Existence of a bound state in a well of any depth in dimensions 1 and 2. One-dimensional scattering problem. Even potentials. Parity operator. The parity conservation law is a fundamentally quantum GS that has no analogue in the classical world.

9. Exactly solvable potentials.

Constant power. Harmonic oscillator. Morse potential. Epstein's potential. Non-reflective potentials. Mention of the inverse problem of scattering theory. Laplace's method. Hypergeometric and degenerate hypergeometric functions. Finding a solution in the form of a series. Analytical continuation. Analytical theory of differential equations. Three-dimensional Schrödinger equation. Centrally symmetrical potential. Isotropy.

10. Harmonic oscillator.

Birth and destruction operator approach. A la Feinman, "Statistical Physics". Calculation of eigenfunctions, normalizations and matrix elements. Hermite equation. Laplace's method. Finding a solution in the form of a series. Finding eigenvalues from the condition of series termination.

11. Orbital momentum operator.

Rotation transformation. Definition. Commutation relations. Eigenfunctions and numbers. Explicit expressions for orbital momentum operators in spherical coordinates. Deriving eigenvalues and operator functions. Matrix elements of orbital momentum operators. Symmetry with respect to the inversion transformation. True and pseudo scalars, vectors and tensors. Parity of various spherical harmonics. Recurrence expression for moment eigenfunctions.

12. Movement in the central field.

General properties. Centrifugal energy. Normalization and orthogonality. Free movement in spherical coordinates.

Spherical Bessel functions and their expressions through elementary functions.

Problem about a three-dimensional rectangular pit. Critical depth for the existence of a bound state. Spherical harmonic oscillator. Solution in Cartesian and spherical coordinate systems. Own functions. Degenerate hypergeometric function. The equation. Solution in the form of a power series. Quantization is a consequence of the finiteness of the series.

13. Coulomb field.

Dimensionless variables, Coulomb system of units. Solution in a spherical coordinate system. Discrete spectrum. Expression for energy eigenvalues. Relationship between principal and radial quantum numbers. Calculation of the degree of degeneracy. The presence of additional degeneracy.

14. Perturbation theory.

Stationary perturbation theory. General theory. Operator geometric progression. Stationary perturbation theory. Frequency corrections for a weakly anharmonic oscillator. Stationary perturbation theory in the case of degeneracy. Secular equation. The problem of an electron in the field of two identical nuclei. Correct zero approximation functions. Overlap integrals. Nonstationary perturbation theory. General theory. Resonant case. Golden Rule Fermi.

15. Semiclassical approximation.

Basic solutions. Local accuracy. Line layer. Airy function. VKB solution. Zwan's method. Potential well problem. Quantization rules Bora-Sommerfeld. VKB approach. Problem of under-barrier passage. The problem of above-barrier reflection.

16. Spin.

Multicomponent wave function. Analogous to the polarization of electromagnetic waves. The Stern-Gerlach experiment. Spin variable. Infinitesimal rotation transformation and spin operator.

Commutation relations. Eigenvalues and eigenfunctions of spin operators. Matrix elements. Spin 1/2. Pauli matrices. Commutation and anticommutation relations. Algebra of Pauli matrices. Calculation of an arbitrary function of a spin scalar. Finite rotation operator. Output using matrix differential equation. Conversion to Linear s form. Matrices U x,y,z . Determination of beam intensities in Stern-Gerlach experiments while rotating the analyzer.

17. Motion of an electron in a magnetic field.

Pauli equation. Gyromagnetic ratio. The role of potentials in quantum mechanics. Gauge invariance. Bohm-Aronov effect. Commutation ratios for speeds. Motion of an electron in a uniform magnetic field. Landau calibration. Solution of the equation. Landau levels. Leading center coordinate operator. Commutation relations for it.

L.D. Landau, E.M. Lifshitz, Quantum mechanics, vol. 3, Moscow, “Science”, 1989
L. Schiff, Quantum mechanics, Moscow, IL, 1967
A. Messiah, Quantum mechanics, vol. 1,2, M. Nauka, 1978
A. S. Davydov, Quantum mechanics, M. Nauka, 1973
D.I. Blokhintsev, Fundamentals of quantum mechanics, Moscow, “Science”, 1976.
V.G. Levich, Yu. A. Vdovin, V. A. Myamlin, Course of Theoretical Physics, vol.2
L.I. Mandelstam, Lectures on optics, relativity and quantum mechanics.

additional literature

R. Feynman, Leighton, Sands, Feynman Lectures on Physics (FLF), vol. 3,8,9
E. Fermi, Quantum Mechanics, M. Mir, 1968
G. Bethe, Quantum Mechanics, M. Mir, 1965
P. Dirac, Principles of quantum mechanics, M. Nauka, 1979
V. Balashov, V. Dolinov, Course of Quantum Mechanics, ed. MSU, Moscow

Problem books

A.M. Galitsky, B. M. Karnakov, V. I. Kogan, Problems in quantum mechanics. Moscow, “Science”, 1981.
M.Sh. Goldman, V.L. Krivchenkov, M. Nauka, 1968
Z. Flügge, Problems in quantum mechanics, vol. 1.2 M. Mir, 1974

Questions for control

Prove that the Schrödinger equation preserves the probability density.
Prove that the eigenfunctions of the US of infinite motion are doubly degenerate.
Prove that the eigenfunctions of the US of free motion corresponding to different impulses are orthogonal.
Prove that the eigenfunctions of the discrete spectrum are non-degenerate.
Prove that the eigenfunctions of the discrete spectrum of a US with an even well are either even or odd.
Find the eigenfunction of the USH with linear potential.
Determine the energy levels in a symmetrical rectangular well of finite depth.
Derive boundary conditions and determine the reflection coefficient from delta potential.
Write an equation for the eigenfunctions of a harmonic oscillator and reduce it to dimensionless form.
Find the eigenfunction of the ground state of the harmonic oscillator. Normalize it.
Define the creation and destruction operators. Write the Hamiltonian of the harmonic oscillator. Describe their properties.
Solving the equation in coordinate representation, find the eigenfunction of the ground state.
Using Operators a, a+ calculate the matrix elements of the operators x 2 , p 2 in the basis of the eigenfunctions of the harmonic oscillator.
How coordinates are transformed during infinitesimal (infinitesimal) rotation.
Relationship between the torque and rotation operators. Definition of the moment operator. Derive commutation relations between the components of the moment Derive commutation relations between the projections of the moment and the coordinates Derive the commutation relations between the projections of the moment and the impulses l 2 ,l_z representation.
Eigenfunctions of moment in spherical coordinates. Write the equation and its solution using the separation of variables method. Expression through adjoint Legendre polynomials.
State parity, inversion operator. Scalars and pseudoscalars, polar and axial vectors. Examples.
Inversion transformation in spherical coordinates. Relationship between parity and orbital momentum.
Reduce the two-body problem to the problem of the motion of one particle in a central field.
Separate the HS variables for the central field and write the general solution.
Write the condition for orthonormality. How many quantum numbers and which ones form a complete set.
Determine Particle Energy Levels with Momentum l, equal to 0, moving in a spherical rectangular well of finite depth. Determine the minimum hole depth required for a bound state to exist.
Determine the energy levels and wave functions of a spherical harmonic oscillator by separating the variables in Cartesian coordinates. What are quantum numbers? Determine the degree of degeneracy of the levels.
Write the SE for motion in a Coulomb field and reduce it to dimensionless form. Atomic system of units.
Determine the asymptotic behavior of the radial function of motion in a Coulomb field near the center.
What is the degree of degeneracy of levels when moving in a Coulomb field.
Derive a formula for the first correction to the wave function corresponding to the non-degenerate energy
Derive the formula for the first and second corrections to energy.
Using perturbation theory, find the first correction to the frequency of a weakly anharmonic oscillator due to the perturbation. Use the creation and destruction operators
Derive a formula for the correction to the energy in the case of m-fold degeneracy of this level. Secular equation.
Derive a formula for the correction to the energy in the case of 2-fold degeneracy of this level. Determine the correct zero-approximation wave functions.
Derive the nonstationary Schrödinger equation in the eigenfunction representation of the unperturbed Hamiltonian.
Derive a formula for the first correction to the wave function of the system under an arbitrary nonstationary disturbance
Derive a formula for the first correction to the wave function of the system under a harmonic non-resonant disturbance.
Derive a formula for the probability of transition under resonant action.
Fermi's golden rule.
Derive the formula for the leading term of the quasiclassical asymptotic expansion.
Write local conditions for the applicability of the semiclassical approximation.
Write a semi-classical solution for the US that describes the motion in a uniform field.
Write a semi-classical solution for the US that describes the motion in a uniform field to the left and right of the turning point.
Using the Zwan method, derive the boundary conditions for the transition from a semi-infinite classically forbidden region to a classically allowed one. What is the phase shift during reflection?
In the semiclassical approximation, determine the energy levels in the potential well. Quantization rule Bora-Sommerfeld.
Using the quantization rule Bora-Sommerfeld determine the energy levels of the harmonic oscillator. Compare with the exact solution.
Using the Zwan method, derive the boundary conditions for the transition from a semi-infinite classically allowed region to a classically forbidden one.
Spin concept. Spin variable. Analogous to the polarization of electromagnetic waves. The Stern-Gerlach experiment.
Infinitesimal rotation transformation and spin operator. Which variables are affected by the spin operator?
Write commutation relations for spin operators
Prove that the operator s 2 commutes with the spin projection operators.
What's happened s 2 , s z performance.
Write the Pauli matrices.
Write the matrix s 2 .
Write the eigenfunctions of the operators s x , y , z for s=1/2 in s 2 , s z representation.
Prove by direct calculation that the Pauli matrices are anticommutative.
Write the finite rotation matrices U x , y , z
A beam polarized in x is incident on a Stern-Gerlach device with its own z axis. What's the output?
A z-polarized beam is incident on the Stern-Gerlach device along the x axis. What is the output if the z" axis of the device is rotated relative to the x axis by an angle j?
Write the SE of a spinless charged particle in a magnetic field
Write US for a charged particle with spin 1/2 in a magnetic field.
Describe the relationship between spin and magnetic moment of a particle. What is gyromagnetic ratio, Bohr magneton, nuclear magneton. What is the gyromagnetic ratio of an electron?
The role of potentials in quantum mechanics. Gauge invariance.
Extended derivatives.
Write expressions for the operators of velocity components and obtain commutation relations for them at a finite magnetic field.
Write the equations of motion of an electron in a uniform magnetic field in the Landau gauge.
Bring the electron equation in a magnetic field to a dimensionless form. Magnetic length.
Derive the wave functions and energy values of an electron in a magnetic field.
What quantum numbers characterize the state? Landau levels.

The coffee gets cold, buildings collapse, eggs break, and stars go out in a universe that seems destined to settle into the gray monotony known as thermal equilibrium. Astronomer and philosopher Sir Arthur Eddington stated in 1927 that the gradual dissipation of energy was proof of the irreversibility of the “arrow of time.”

But to the bewilderment of entire generations of physicists, the concept of the arrow of time does not correspond to the basic laws of physics, which act in time both in the forward direction and in the opposite direction. According to these laws, if one knew the paths of all the particles in the universe and reversed them, energy would accumulate rather than dissipate: cold coffee would begin to heat up, buildings would rise from ruins, and sunlight would be directed back toward the Sun.

“We had difficulties in classical physics,” says Professor Sandu Popescu, who teaches physics at Britain's University of Bristol. “If I knew more, could I reverse the course of events and put all the molecules of the broken egg back together?”

Of course, he says, the arrow of time is not guided by human ignorance. And yet, since the birth of thermodynamics in the 1850s, the only known way to calculate the propagation of energy was the formula statistical distribution unknown particle trajectories and demonstrating that over time, ignorance blurs the picture of things.

Now physicists are uncovering a more fundamental source of the arrow of time. Energy dissipates and objects come into equilibrium, they say, because elementary particles become entangled when they interact. They called this strange effect “quantum mixing,” or entanglement.

“We can finally understand why a cup of coffee in a room comes into equilibrium with it,” says Bristol quantum physicist Tony Short. “There is a confusion between the state of the cup of coffee and the state of the room.”

Popescu, Short and their colleagues Noah Linden and Andreas Winter reported their discovery in the journal Physical Review E in 2009, saying that objects come to equilibrium, or a state of uniform distribution of energy, for an indefinite period of time. long time due to quantum mechanical mixing with environment. A similar discovery was made a few months earlier by Peter Reimann of the University of Bielefeld in Germany, publishing his findings in Physical Review Letters. Short and colleagues strengthened their arguments in 2012 by showing that entanglement causes equilibrium in a finite time. And in a paper published in February on the arXiv website. org, two separate groups took the next step, calculating that the majority physical systems quickly equilibrate in a time directly proportional to their size. "To show that this applies to our real physical world, processes must occur within a reasonable time frame,” says Short.

The tendency of coffee (and everything else) to come into equilibrium is “very intuitive,” says Nicolas Brunner, a quantum physicist at the University of Geneva. “But in explaining the reasons for this, for the first time we have a solid foundation taking into account microscopic theory.”

If the new line of research is correct, then the story of the arrow of time begins with the quantum mechanical idea that nature is fundamentally indeterminate. An elementary particle is devoid of specific physical properties, and it is determined only by the probabilities of being in certain states. For example, at a certain moment a particle may have a 50 percent probability of spinning clockwise and a 50 percent chance of spinning counterclockwise. An experimentally verified theorem by Northern Irish physicist John Bell states that there is no “true” state of particles; probabilities are the only thing that can be used to describe it.

Quantum uncertainty inevitably leads to confusion, the supposed source of the arrow of time.

When two particles interact, they can no longer be described by separate, independently evolving probabilities called “pure states.” Instead, they become entangled components of a more complex probability distribution that describes the two particles together. They can, for example, indicate that particles are spinning in opposite directions. The system as a whole is in a pure state, but the state of each particle is “mixed” with the state of another particle. Both particles may be moving light years apart, but the rotation of one particle will be correlated with the other. Albert Einstein described it well as “spooky action at a distance.”

“Entanglement is in some sense the essence of quantum mechanics,” or the laws governing interactions on the subatomic scale, Brunner says. This phenomenon underlies quantum computing, quantum cryptography and quantum teleportation.

The idea that mixing could explain the arrow of time first occurred to Seth Lloyd 30 years ago, when he was a 23-year-old philosophy graduate from Cambridge University with a Harvard degree in physics. Lloyd realized that quantum uncertainty, and its spread as particles become increasingly entangled, could replace human uncertainty (or ignorance) in the old classical proofs and become the true source of the arrow of time.

Using a little-known quantum mechanical approach in which units of information are the basic building blocks, Lloyd spent several years studying the evolution of particles in terms of the shuffling of ones and zeros. He found that as the particles became increasingly intermingled with each other, the information that described them (for example, 1 for clockwise rotation, and 0 for counterclockwise rotation) would be transferred to describe the system of entangled particles as a whole. The particles seemed to gradually lose their independence and became pawns of the collective state. Over time, all the information goes into these collective clusters, and individual particles have none left at all. At this point, Lloyd discovered, the particles reach a state of equilibrium and their states stop changing, like a cup of coffee cooling to room temperature.

“What's really going on? Things become more connected. The arrow of time is the arrow of increasing correlations.”

This idea, outlined in Lloyd's doctoral thesis in 1988, fell on deaf ears. When the scientist sent an article about this to the editors of the journal, he was told that “there is no physics in this work.” Quantum information theory “was deeply unpopular” at the time, Lloyd says, and questions about the arrow of time “were the preserve of loonies and demented Nobel laureates.”

“I was damn close to becoming a taxi driver,” he said.

Since then, advances in quantum computing have made quantum information theory one of the most active areas of physics. Currently a professor at MIT, Lloyd is recognized as one of the founders of the discipline, and his forgotten ideas are being revived by physicists at Bristol. The new evidence is more general, scientists say, and applies to any quantum system.

“When Lloyd came up with the idea in his dissertation, the world was not ready for it,” says Renato Renner, head of the Institute for Theoretical Physics at ETH Zurich. - Nobody understood him. Sometimes you need ideas to come at the right time.”

In 2009, the evidence of a team of Bristol physicists resonated with quantum information theorists, who discovered new ways to apply their methods. They showed that as objects interact with their environment—the way particles in a cup of coffee interact with air—information about their properties “leaks and spreads throughout that environment,” Popescu explains. This local loss of information causes the state of the coffee to remain the same even as the net state of the entire room continues to change. With the exception of rare random fluctuations, the scientist says, “its state ceases to change over time.”

It turns out that a cold cup of coffee cannot spontaneously warm up. Basically, as the pure state of the room evolves, coffee can suddenly be released from the air of the room and return to the pure state. But there are many more mixed states than pure ones, and practically coffee can never return to its pure state. To see this, we will have to live longer than the universe. This statistical improbability makes the arrow of time irreversible. “Essentially, mixing opens up a huge space for us,” Popescu says. — Imagine that you are in a park, in front of you is a gate. As soon as you enter them, you are thrown out of balance, you find yourself in a huge space and get lost in it. You will never return to the gate."

IN new history arrows of time, information is lost through the process of quantum entanglement, not through a subjective lack of human knowledge of what brings a cup of coffee and a room into balance. The room eventually balances out with external environment, and the environment moves even more slowly towards equilibrium with the rest of the universe. The giants of 19th-century thermodynamics viewed this process as a gradual dissipation of energy that increases the overall entropy, or chaos, of the universe. Today, Lloyd, Popescu and others in the field view the arrow of time differently. In their opinion, information becomes increasingly diffuse, but never completely disappears. Although entropy increases locally, the overall entropy of the universe remains constant and zero.

“The universe as a whole is in a pure state,” says Lloyd. “But its individual parts, intertwined with the rest of the universe, come to a mixed state.”

But one mystery of the arrow of time remains unsolved. “There's nothing in these works that explains why you start at the gate,” Popescu says, returning to the park analogy. “In other words, they do not explain why the original state of the universe was far from equilibrium.” The scientist hints that this question relates to the nature of the Big Bang.

Despite recent advances in calculating equilibration times, the new approach still cannot be used to calculate the thermodynamic properties of specific things like coffee, glass, or unusual states of matter. (Some traditional thermodynamicists say they know very little about the new approach.) “The point is that you need to find criteria for which things behave like a window pane and which things behave like a cup of tea,” Renner says. “I think I will see more work in this direction, but there is still a lot to be done.”

Some researchers have expressed doubt that this abstract approach to thermodynamics will ever be able to accurately explain how specific observable objects behave. But conceptual advances and a new set of mathematical formulas are already helping researchers ask theoretical questions about thermodynamics, such as the fundamental limitations of quantum computers and even the ultimate fate of the universe.

“We're thinking more and more about what we can do with quantum machines,” says Paul Skrzypczyk of the Institute of Photonic Sciences in Barcelona. - Let's say the system is not yet in a state of equilibrium, and we want to make it work. How much useful work can we extract? How can I intervene to do something interesting?”

Context

Quantum computer in human brain?

Futura-Sciences 01/29/2014

How a nanosatellite can reach a star

Wired Magazine 04/17/2016

Beauty as the secret weapon of physics

Nautilus 01/25/2016
Caltech cosmology theorist Sean Carroll applies new formulas in his latest work on the arrow of time in cosmology. “I'm interested in the long-term fate of cosmological spacetime,” says Carroll, who wrote From Eternity to Here: The Quest. for the Ultimate Theory of Time (From infinity to here. Search for the ultimate theory of time). “In this situation, we do not yet know all the necessary laws of physics, so it makes sense to turn to the abstract level, and here, it seems to me, this quantum mechanical approach will help us.”

Twenty-six years after the failure of Lloyd's grand idea for the arrow of time, he enjoys watching its revival and trying to apply the ideas of his last work to the paradox of information falling into a black hole. “I think now people will still talk about the fact that there is physics in this idea,” he says.

And philosophy even more so.

Our ability to remember the past but not the future, a confusing manifestation of the arrow of time, can also be seen as increasing correlations between interacting particles, scientists say. When you read a note on a piece of paper, your brain correlates the information through the photons that hit your eyes. Only from this moment can you remember what is written on paper. As Lloyd notes, “The present can be characterized as a process of establishing correlations with our surroundings.”

The background for the steady growth of interweaving throughout the universe is, of course, time itself. Physicists emphasize that despite great strides in understanding how changes occur in time, they are no closer to understanding the nature of time itself or why it is different from the other three dimensions of space (in conceptual terms and in the equations of quantum mechanics) . Popescu calls this mystery "one of the greatest unknowns in physics."

“We can discuss how an hour ago our brain was in a state that correlated with fewer things,” he says. “But our perception that time is passing is a completely different matter. Most likely, we will need a new revolution in physics to tell us about this.”

InoSMI materials contain assessments exclusively of foreign media and do not reflect the position of the InoSMI editorial staff.

In quantum mechanics, each dynamic variable - coordinate, momentum, angular momentum, energy - is associated with a linear self-adjoint (Hermitian) operator.

All functional relations between quantities known from classical mechanics are replaced in quantum theory by similar relations between operators. The correspondence between dynamic variables (physical quantities) and quantum mechanical operators is postulated in quantum mechanics and is a generalization of vast experimental material.

1.3.1. Coordinate operator:

As is known, in classical mechanics, the position of a particle (system N- particles) in space in this moment time is determined by a set of coordinates - vector or scalar quantities. Vector mechanics is based on Newton's laws, the main ones here are vector quantities - speed, momentum, force, angular momentum (angular momentum), torque, etc. Here, the position of the material point is specified by the radius vector, which determines its position in space relative to the selected reference body and the coordinate system associated with it, i.e.

If all vectors of forces acting on the particle are determined, then the equations of motion can be solved and a trajectory can be constructed. If movement is considered N- particles, then it is more expedient (regardless of whether the movement of bound particles is considered or the particles are free in their movements from all kinds of connections) to operate not with vector, but with scalar quantities - the so-called generalized coordinates, velocities, impulses and forces. This analytical approach is based on the principle of least action, which in analytical mechanics plays the role of Newton’s second law. Characteristic feature analytical approach is the absence of a rigid connection with any specific coordinate system. In quantum mechanics, each observable dynamic variable (physical quantity) is associated with a linear self-adjoint operator. Then, obviously, the classical set of coordinates will correspond to a set of operators of the form: , the action of which on a function (vector) will be reduced to multiplying it by corresponding coordinates, i.e.

from which it follows that:

1.3.2. Momentum operator:

The classic expression for momentum by definition is:

considering that:

we will have accordingly:

Since any dynamic variable in quantum mechanics is associated with a linear self-adjoint operator:

then, accordingly, the expression for the momentum, expressed through its projection onto three non-equivalent directions in space, is transformed to the form:

The value of the momentum operator and its components can be obtained by solving the operator eigenvalue problem:

To do this, we will use the analytical expression for a plane de Broglie wave, obtained earlier:

considering also that:

we have thus:

Using the de Broglie plane wave equation, we now solve the problem of eigenvalues of the momentum operator (its components):

because the:

and the function is on both sides of the operator equation:

then the wave amplitude will decrease, therefore:

thus we have:

since the momentum component operator (similarly and ) is a differential operator, its action on the wave function (vector) will obviously be reduced to calculating the partial derivative of a function of the form:

Solving the operator eigenvalue problem, we arrive at the expression:

Thus, in the course of the above calculations, we came to an expression of the form:

then accordingly:

considering that:

after substitution we get an expression of the form:

In a similar way, one can obtain expressions for other components of the momentum operator, i.e. we have:

Given the expression for the total momentum operator:

and its component:

we have accordingly:

Thus, the total momentum operator is a vector operator and the result of its action on the function (vector) will be an expression of the form:

1.3.3. Angular momentum (angular momentum) operator:

Let's consider the classic case absolutely solid, rotating around fixed axis OO passing through it. Let's divide this body into small volumes with elementary masses: located at distances: from the axis of rotation OO. When a rigid body rotates relative to the fixed axis OO, its individual elementary volumes with masses will obviously describe circles of different radii and will have different linear velocities: . From the kinematics of rotational motion it is known that:

If a material point performs a rotational motion, describing a circle with radius , then after a short period of time it will rotate through an angle from its original position.

The linear velocity of the material point, in this case, will be equal to, respectively:

because the:

Obviously, the angular velocity of elementary volumes of a solid body rotating around a fixed axis OO at distances from it will be equal to, respectively:

When studying the rotation of a rigid body, they use the concept of moment of inertia, which is a physical quantity equal to the amount products of masses - material points of the system by the squares of their distances to the considered axis of rotation OO, relative to which rotational motion occurs:

then we find the kinetic energy of the rotating body as the sum of the kinetic energies of its elementary volumes:

because the:

then accordingly:

Comparison of formulas for kinetic energy translational and rotational movements:

shows that the moment of inertia of a body (system) characterizes the measure of inertia of this body. Obviously, the greater the moment of inertia, the more energy must be expended to achieve a given speed of rotation of the body (system) in question around the fixed axis of rotation OO. An equally important concept in solid mechanics is the torque vector, so by definition the work of moving a body over a distance is equal to:

since, as already stated above, during rotational motion:

then accordingly we will have:

considering the fact that:

then the expression for the work of rotational motion, expressed in terms of the moment of force, can be rewritten as:

because in general:

then, therefore:

Differentiating the right and left sides of the resulting expression with respect to , we will have, respectively:

considering that:

we get:

Moment of power ( torque) acting on the body, equal to the product its moment of inertia by angular acceleration. The resulting equation is an equation for the dynamics of rotational motion, similar to the equation of Newton's second law:

here, instead of force, the moment of force plays the role, the role of mass is played by the moment of inertia. Based on the above analogy between the equations for translational and rotational motions, the analogue of impulse (amount of motion) will be the angular momentum of the body (angular momentum). The angular momentum of a material point by mass is the vector product of the distance from the axis of rotation to this point and its momentum (amount of motion); we then have:

Considering that a vector is determined not only by a triple of components:

but also by explicit expansion in unit vectors of the coordinate axes:

we will have accordingly:

The components of the total angular momentum can be represented as algebraic complements of the determinant, in which the first line is unit vectors (unit vectors), the second line is Cartesian coordinates and the third line – the components of the impulse, then, accordingly, we will have an expression of the form:

from which it follows that:

From the formula for angular momentum as a vector product, an expression of the form also follows:

or for a particle system:

taking into account relations of the form:

we obtain an expression for the angular momentum of a system of material points:

Thus, the angular momentum of a rigid body relative to a fixed axis of rotation is equal to the product of the moment of inertia of the body by angular velocity. The angular momentum is a vector directed along the axis of rotation in such a way that from its end one can see the rotation occurring clockwise. Differentiating the resulting expression with respect to time gives another expression for the dynamics of rotational motion, equivalent to the equation of Newton’s second law:

similar to the equation of Newton's second law:

“The product of the angular momentum of a rigid body relative to the axis of rotation OO is equal to the moment of force relative to the same axis of rotation.” If we are dealing with a closed system, then the moment of external forces is zero, then, therefore:

The equation obtained above for a closed system is an analytical expression of the law of conservation of momentum. “The angular momentum of a closed system is a constant quantity, i.e. does not change over time." So, in the course of the above calculations, we came to the expressions we need in further discussions:

and thus we have accordingly:

Since in quantum mechanics any physical quantity (dynamic variable) is associated with a linear self-adjoint operator:

then the corresponding expressions are:

are transformed to the form:

because by definition:

and also considering that:

Then, accordingly, for each of the components of angular momentum we will have an expression of the form:

based on an expression of the form:

1.3.4. Squared angular momentum operator:

In classical mechanics, the square of angular momentum is determined by an expression of the form:

Therefore, the corresponding operator will have the form:

whence it follows accordingly that:

1.3.5. Kinetic energy operator:

The classic expression for kinetic energy is:

Considering that the expression for momentum is:

we have accordingly:

expressing impulse through its components:

we will have accordingly:

Since each dynamic variable (physical quantity) in quantum mechanics corresponds to a linear self-adjoint operator, i.e.

then, therefore:

taking into account expressions of the form:

and thus we arrive at an expression for the kinetic energy operator of the form:

1.3.6. Potential energy operator:

The potential energy operator when describing the Coulomb interaction of particles with charges has the form:

It coincides with a similar expression for the corresponding dynamic variable (physical quantity) - potential energy.

1.3.7. System total energy operator:

The classical expression for the Hamiltonian, known from Hamilton's analytical mechanics, has the form:

based on the correspondence between quantum mechanical operators and dynamic variables:

we arrive at the expression for the operator of the total energy of the system – the Hamilton operator:

taking into account the expressions for the potential and kinetic energy operators:

we arrive at an expression of the form:

Operators of physical quantities (dynamic variables) - coordinates, momentum, angular momentum, energy - are linear self-adjoint (Hermitian) operators, therefore, based on the corresponding theorem, their eigenvalues are real (real) numbers. It was this circumstance that served as the basis for the use of operators in quantum mechanics, since as a result of a physical experiment we obtain precisely real quantities. In this case, the eigenfunctions of the operator corresponding to different eigenvalues are orthogonal. If we have two different operators, then their eigenfunctions will be different. However, if operators commute with each other, then the eigenfunctions of one operator will also be the eigenfunctions of another operator, i.e. the systems of eigenfunctions of operators commuting with each other will coincide.

Using the well-known quantum mechanical approach, in which units of information are the basic building blocks, Lloyd spent several years studying the evolution of particles in terms of the shuffling of ones (1s) and zeros (0s). He found that as particles become increasingly entangled with each other, the information that described them (1 for clockwise spin, and 0 for counterclockwise spin, for example) would be transferred to describe the system of entangled particles as a whole. It is as if the particles gradually lost their individual autonomy and became pawns of a collective state. At this point, Lloyd discovered, the particles enter a state of equilibrium, their states stop changing, like a cup of coffee cooling to room temperature.

“What's really going on? Things are becoming more connected. The arrow of time is the arrow of increasing correlations.”

The idea presented in a 1988 doctoral dissertation fell on deaf ears. When the scientist submitted it to a journal, he was told that “there is no physics in this work.” Quantum information theory “was deeply unpopular” at the time, Lloyd says, and questions about the arrow of time “were left to the loonies and Nobel laureates who have retired."

“I was damn close to becoming a taxi driver,” Lloyd said.

Since then, advances in quantum computing have made quantum information theory one of the most active areas of physics. Today Lloyd remains a professor at MIT, recognized as one of the founders of the discipline, and his forgotten ideas are resurfacing in a more confident form in the minds of physicists at Bristol. The new evidence is more general, scientists say, and applies to any quantum system.

“When Lloyd proposed the idea in his thesis, the world was not ready,” says Renato Renner, head of the Institute for Theoretical Physics at ETH Zurich. - Nobody understood him. Sometimes you need ideas to come at the right time.”

In 2009, a proof by a team of Bristol physicists struck a chord with quantum information theorists, opening up new ways to apply their methods. It showed that as objects interact with their environment—the way particles in a cup of coffee interact with air, for example—information about their properties “leaks out and gets smeared with the environment,” Popescu explains. This local loss of information causes the coffee's state to stagnate, even as the net state of the entire room continues to evolve. With the exception of rare random fluctuations, the scientist says, “his condition ceases to change over time.”

It turns out that a cold cup of coffee cannot spontaneously heat up. Basically, as the pure state of the room evolves, the coffee may suddenly "become unmixed" with the air and enter the pure state. But there are so many more mixed states available in coffee than pure ones that this will almost never happen—the universe will end sooner than we can witness it. This statistical improbability makes the arrow of time irreversible.

“Essentially, entanglement opens up a huge space for you,” Popescu comments. - Imagine that you are in a park, in front of you is a gate. As soon as you enter them, you will find yourself in a huge space and get lost in it. You’ll never return to the gate either.”

In the new story of the arrow of time, information is lost through the process of quantum entanglement, rather than due to a subjective lack of human knowledge, resulting in the equilibration of a cup of coffee and a room. The room eventually equilibrates with the outside environment, and the environment - even more slowly - drifts toward equilibrium with the rest of the universe. The thermodynamic giants of the 19th century viewed this process as a gradual dissipation of energy that increases the overall entropy, or chaos, of the universe. Today, Lloyd, Popescu and others in the field see the arrow of time differently. In their opinion, information becomes increasingly diffuse, but never completely disappears. Although entropy increases locally, the overall entropy of the universe remains constant and zero.

“The universe as a whole is in a pure state,” says Lloyd. “But its individual parts, being entangled with the rest of the universe, remain mixed.”

One aspect of the arrow of time remains unresolved.

“There's nothing in these works that explains why you start at the gate,” Popescu says, returning to the park analogy. “In other words, they do not explain why the original state of the universe was far from equilibrium.” The scientist hints that this question applies.

Despite recent progress in calculating equilibration times, the new approach still cannot become a tool for calculating the thermodynamic properties of specific things like coffee, glass or exotic states of matter.

“The point is to find the criteria under which things behave like a window pane or a cup of tea,” Renner says. “I think I will see more work in this direction, but there is still a lot of work ahead.”

Some researchers have expressed doubt that this abstract approach to thermodynamics will ever be able to accurately explain how specific observable objects behave. But conceptual advances and new mathematical formalism are already helping researchers ask theoretical questions in thermodynamics, such as the fundamental limits of quantum computers and even the ultimate fate of the universe.

“We're thinking more and more about what we can do with quantum machines,” says Paul Skrzypczyk of the Institute of Photonic Sciences in Barcelona. - Let's say the system is not yet in a state of equilibrium and we want to make it work. How much useful work can we extract? How can I intervene to do something interesting?

Sean Carroll, a theoretical cosmologist at the California Institute of Technology, applies a new formalism in his latest work on the arrow of time in cosmology. “I am interested in the very long-term fate of cosmological space-time. In this situation, we do not yet know all the necessary laws of physics, so it makes sense to turn to the abstract level, and here, I think, this quantum mechanical approach will help me.”

Twenty-six years after the epic failure of Lloyd's idea for the arrow of time, he is delighted to witness its rise and tries to apply the ideas of his last work to the paradox of information falling into a black hole.

“I think now people will still talk about the fact that there is physics in this idea.”

And philosophy - even more so.

According to scientists, our ability to remember the past but not the future, another manifestation of the arrow of time, can also be seen as increasing correlations between interacting particles. When you read something from a piece of paper, the brain correlates the information through photons that reach the eyes. Only from this moment will you be able to remember what is written on paper. As Lloyd notes:

“The present can be defined as the process of connecting (or making correlations) with our surroundings.”

The backdrop for the steady growth of entanglements throughout the universe is, of course, time itself. Physicists stress that despite great strides in understanding how changes occur in time, they are not one iota closer to understanding the nature of time itself or why it is different from the other three dimensions of space. Popescu calls this mystery “one of the greatest mysteries in physics.”

“We can discuss the fact that an hour ago our brain was in a state that correlated with fewer things,” he says. “But our perception that time is passing is a completely different matter. Most likely, we will need a revolution in physics that will reveal this secret to us.”