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Federal State Educational Institution of Higher Professional Education Oryol State Agrarian University

Department of plant growing

Abstract on the topic:

"Heisenberg Uncertainty Principle"

Performed:

1st year student

Faculty: Economic

Specialty: Finance and Credit

Yakovleva K.V.

Checked:

Kirsanova Elena Vladimirovna

Eagle 2009

1.Introduction……………………………………………………………………………………..3

2.Main part:

2 .1 Biography………………………………………………………………………….5-8

2.3 Uncertainty principle………………………………………………9-12

2.3 Heisenberg uncertainty relation…………..13-16

2.4 Ideal measurement…………………………………………………..17-18

3.Conclusion………………………………………………………………………….19-20

4.Appendix……………………………………………………………………………..21

5. References…………………………………………………………………..22

Introduction

In everyday life, we are surrounded by material objects whose dimensions are comparable to us: cars, houses, grains of sand, etc. Our intuitive ideas about the structure of the world are formed as a result of everyday observation of the behavior of such objects. Since we all have a life behind us, the experience accumulated over the years tells us that since everything we observe over and over again behaves in a certain way, it means that in the entire Universe, on all scales, material objects should behave in a similar way. And when it turns out that somewhere something does not obey the usual rules and contradicts our intuitive concepts of the world, this not only surprises us, but shocks us.
In the first quarter of the twentieth century, this was precisely the reaction of physicists when they began to study the behavior of matter at the atomic and subatomic levels. The emergence and rapid development of quantum mechanics has opened before us a whole world, the system structure of which simply does not fit into the framework of common sense and completely contradicts our intuitive ideas. But we must remember that our intuition is based on the experience of the behavior of ordinary objects on a scale commensurate with us, and quantum mechanics describes things that happen on a microscopic and invisible level for us - no person has ever directly encountered them. If we forget about this, we will inevitably come to a state of complete confusion and bewilderment. For myself, I formulated the following approach to quantum mechanical effects: as soon as the “inner voice” begins to repeat “this cannot be!”, you need to ask yourself: “Why not? How do I know how things actually work inside an atom? Did I look there myself? By setting yourself up in this way, it will be easier for you to perceive articles that

dedicated quantum mechanics.

Main part

Heisenberg (Heisenberg) Werner (1901-1976), German theoretical physicist, one of the creators of quantum mechanics. Proposed (1925) a matrix version of quantum mechanics; formulated (1927) the uncertainty principle; introduced the concept of scattering matrix (1943). Proceedings on the structure of the atomic nucleus, relativistic quantum mechanics, unified field theory, theory of ferromagnetism, philosophy of natural sciences. Nobel Prize (1932)

Werner Karl Heisenberg was born on December 5, 1901 in the German city of Würzburg. In September 1911, Werner was sent to a prestigious gymnasium. In 1920, Heisenberg entered the University of Munich. After graduating, Werner was appointed assistant to Professor Max Born at the University of Göttingen. According to quantum theory, an atom emits light by passing from one energy state to another. And according to Einstein's theory, the intensity of light of a certain frequency depends on the number of photons. This means that it was possible to try to relate the radiation intensity to the probability of atomic transitions. Quantum oscillations of electrons, Heisenberg assured, need to be represented only with the help of mathematical relationships. It is only necessary to choose the appropriate mathematical apparatus for this. The young scientist chose matrices. The choice turned out to be successful, and soon his theory was ready. Heisenberg's work laid the foundations for the science of the movement of microscopic particles - quantum mechanics.

The mathematical apparatus used by Heisenberg and Dirac in developing the theories of the atom in the new mechanics were both unusual and complex for most physicists. Not to mention the fact that none of them, despite all the tricks, could not get used to the idea that a wave is a particle, and a particle is a wave.

In Copenhagen in September 1926, a discussion broke out between Bohr and Schrödinger, in which neither side was successful. As a result, it was recognized that none of the existing interpretations of quantum mechanics can be considered quite acceptable.

Heisenberg in February 1927 gave the necessary interpretation, formulating the uncertainty principle and not doubting its correctness. In February 1927 he submitted an article "On the Quantum Theoretical Interpretation of Kinematic and Mechanical Relations", dedicated to the uncertainty principle. According to the uncertainty principle, the simultaneous measurement of two conjugate variables, such as the position and momentum of a moving particle, inevitably leads to a limitation in accuracy. The more accurately the position of a particle is measured, the less accurate its momentum can be measured, and vice versa.

Heisenberg stated that as long as quantum mechanics is valid, the uncertainty principle cannot be violated.

Heisenberg's uncertainty principle entered logically closed system"Copenhagen interpretation", which Heisenberg and Born, before the meeting of the leading physicists of the world in October 1927, declared completely complete and unchangeable. This meeting, the fifth of the famous Solvay Congresses, took place only a few weeks after Heisenberg became professor of theoretical physics at the University of Leipzig. At only twenty-five years of age, he became the youngest professor in Germany.

Heisenberg was the first to present a well-articulated conclusion about the most profound consequence of the uncertainty principle related to the relation to the classical concept of causality.

It didn't take long for Heisenberg and other "Copenhageners" to convey their doctrine to those who had not attended European institutions. In the United States, Heisenberg found especially favorable environment to convert new followers. During a joint trip around the world with Dirac in 1929, Heisenberg gave a course of lectures on the "Copenhagen Doctrine" at the University of Chicago. In 1933, along with Schrödinger and Dirac, his work received the highest recognition - the Nobel Prize.

From 1941 to 1945 Heisenberg was director of the Kaiser Wilhelm Institute for Physics and professor at the University of Berlin. Repeatedly rejecting offers to emigrate, he headed the main research into the fission of uranium, in which the Third Reich was interested.

After the end of the war, the scientist was arrested and sent to England.

In 1946, Heisenberg returned to Germany. He becomes director of the Physics Institute and professor at the University of Göttingen. Since 1958, the scientist was the director of the Physics University and astrophysics, as well as a professor at the University of Munich. In recent years, Heisenberg's efforts have been aimed at creating a unified field theory. In 1958, he quantized Ivanenko's non-linear spinor equation (the Ivanenko–Heisenberg equation).

Heisenberg died at his home in Munich on February 1, 1976 from kidney and gallbladder cancer.

Uncertainty principle

The uncertainty principle fundamental position quantum theory stating that any physical system cannot be in states in which the coordinates of its center of inertia and momentum simultaneously take on quite definite, exact values. Quantitatively, the uncertainty principle is formulated as follows. If ∆x is the uncertainty of the value of the coordinate x of the center of inertia of the system, and ∆p x is the uncertainty of the projection of momentum p on the x axis, then the product of these uncertainties must be in order of magnitude not less than Planck's constant ħ. Similar inequalities must hold for any pair of so-called canonically conjugate variables, for example, for the y coordinate and the projection of the momentum p y on the y axis, the coordinate z and the projection of the momentum p z. If by the uncertainties of the position and momentum we understand the root-mean-square deviations of these physical quantities from their average values, then the uncertainty principle for them has the form:

∆p x ∆x ≥ ħ/2, ∆p y ∆y ≥ ħ/2, ∆p z ∆z ≥ ħ/2

Due to the smallness of ħ in comparison with macroscopic quantities of the same dimension, the operation of the uncertainty principle is essential mainly for phenomena of atomic (and smaller) scales and does not appear in experiments with macroscopic bodies.

It follows from the uncertainty principle that the more precisely one of the quantities included in the inequality is determined, the less certain is the value of the other. No experiment can lead to a simultaneous accurate measurement of such dynamic variables; while the uncertainty in the measurements is not related to

imperfection of experimental technique, but with the objective properties of matter.

The uncertainty principle, discovered in 1927 by the German physicist W. Heisenberg, was an important step in elucidating the patterns of intra-atomic phenomena and building quantum mechanics. An essential feature of microscopic objects is their corpuscular-wave nature. The state of a particle is completely determined by the wave function (a value that completely describes the state of a microobject (electron, proton, atom, molecule) and, in general, of any quantum system). A particle can be found at any point in space where the wave function is non-zero. Therefore, the results of experiments to determine, for example, coordinates are of a probabilistic nature. (Example: the movement of an electron is the propagation of its own wave. If you shoot an electron beam through a narrow hole in the wall: a narrow beam will pass through it. But if you make this hole even smaller, this so that its diameter is equal to the wavelength of the electron, then the electron beam will spread in all directions. And this is not a deflection caused by the nearest atoms of the wall, which can be eliminated: this is due to the wave nature of the electron. Try to predict what will happen next with the electron passing through the wall, and you will be powerless. You know exactly where it crosses the wall, but you cannot say what momentum it will acquire in the transverse direction. On the contrary, in order to determine exactly that the electron will appear with such and such a certain impulse in the original direction, you need to enlarge the hole so that the electrical This wave traveled straight, only weakly diverging in all directions due to diffraction.

But then it is impossible to say exactly where exactly the electron-particle passed through the wall: the hole is wide. How much you win in the accuracy of determining the momentum, so you lose in the accuracy with which its position is known.

This is the Heisenberg Uncertainty Principle. He played an extremely important role in the construction of a mathematical apparatus for describing the waves of particles in atoms. Its strict interpretation in experiments with electrons is that, like light waves, electrons resist any attempt to make measurements with the utmost precision. This principle also changes the picture of the Bohr atom. It is possible to determine exactly the momentum of an electron (and hence its energy level) in some of its orbits, but at the same time its location will be absolutely unknown: nothing can be said about where it is located. From this it is clear that it makes no sense to draw a clear orbit of an electron and mark it on it in the form of a circle.)

Consequently, when conducting a series of identical experiments, according to the same definition of the coordinate, in the same systems, different results are obtained each time. However, some values ​​will be more likely than others, meaning they will appear more frequently. The relative frequency of occurrence of certain values ​​of the coordinate is proportional to the square of the modulus of the wave function at the corresponding points in space. Therefore, most often those values ​​of the coordinate will be obtained that lie near the maximum of the wave function. But some scatter in the values ​​of the coordinate, some of their uncertainty (on the order of the half-width of the maximum) is inevitable. The same applies to the measurement of momentum.

Thus, the concepts of position and momentum in the classical sense cannot be applied to microscopic objects. When using these quantities to describe a microscopic system, it is necessary to introduce quantum corrections into their interpretation. Such an amendment is the uncertainty principle.

The uncertainty principle for energy ε and time t has a slightly different meaning:

∆ε ∆t ≥ ħ

If the system is in a stationary state, then it follows from the uncertainty principle that the energy of the system, even in this state, can only be measured with an accuracy not exceeding ħ/∆t, where ∆t is the duration of the measurement process. The reason for this is in the interaction of the system with the measuring device, and the uncertainty principle as applied to this case means that the energy of interaction between the measuring device and the system under study can only be taken into account with an accuracy of ħ/∆t.

Heisenberg uncertainty relation

In the early 1920s, when there was a storm of creative thought that led to the creation of quantum mechanics, this problem was first recognized by the young German theoretical physicist Werner Heisenberg. Starting with complex mathematical formulas describing the world at the subatomic level, he gradually came to a surprisingly simple formula that gives a general description of the effect of the measurement tools on the measured objects of the microworld, which we have just talked about. As a result, he formulated the uncertainty principle, now named after him:

uncertainty of x coordinate value uncertainty of speed>h/m,

whose mathematical expression is called the Heisenberg uncertainty relation:

ΔxхΔv>h/m

where Δx is the uncertainty (measurement error) of the spatial coordinate of the microparticle, Δv is the uncertainty of the particle velocity, m is the mass of the particle, and h is Planck's constant, named after the German physicist Max Planck, another of the founders of quantum mechanics. Planck's constant is approximately 6.626 x 10–34 J s, that is, it contains 33 zeros to the first significant digit after the decimal point.

Term "uncertainty of spatial coordinate" just means that we do not know the exact location of the particle. For example, if you use the global GPS to determine the location of this book, the system will calculate them with an accuracy of 2-3 meters. (GPS, Global Positioning System is a navigation system that uses 24 artificial Earth satellites. If, for example, you have a GPS receiver installed on your car, then by receiving signals from these satellites and comparing their delay time, the system determines your geographic coordinates on Earth to the nearest arc second.) However, from the point of view of the measurement made by the GPS instrument, the book could, with some probability, be anywhere within the system's specified few square meters. In this case, we are talking about the uncertainty of the spatial coordinates of the object (in this example, the book). The situation can be improved if we take a tape measure instead of GPS - in this case we can say that the book is, for example, 4 m 11 cm from one wall and 1 m 44 cm from another. But here, too, we are limited in the accuracy of measurement by the minimum division of the tape measure scale (even if it is a millimeter) and the measurement errors of the device itself - and in the best case, we will be able to determine the spatial position of the object with an accuracy of the minimum division of the scale. The more accurate instrument we use, the more accurate our results will be, the lower the measurement error and the less uncertainty. In principle, in our everyday world, it is possible to reduce uncertainty to zero and determine the exact coordinates of the book.

And here we come to the most fundamental difference between the microworld and our everyday physical world. In the ordinary world, when measuring the position and speed of a body in space, we practically do not influence it. Thus, ideally, we can simultaneously measure both the speed and the coordinates of the object absolutely accurately (in other words, with zero uncertainty).

In the world quantum phenomena, however, any measurement affects the system. The very fact that we measure, for example, the location of a particle, leads to a change in its speed, and unpredictable at that (and vice versa). That is why the right side of the Heisenberg relation is not zero, but a positive value. The smaller the uncertainty about one variable (for example, Δx), the more uncertain the other variable (Δv) becomes, since the product of two errors on the left side of the ratio cannot be less than the constant on its right side. In fact, if we manage to determine one of the measured quantities with zero error (absolutely accurately), the uncertainty of the other quantity will be equal to infinity, and we will know nothing about it at all. In other words, if we were able to absolutely accurately establish the coordinates of a quantum particle, we would not have the slightest idea about its speed; if we could accurately fix the speed of a particle, we would have no idea where it is. In practice, of course, experimental physicists always have to find some kind of compromise between these two extremes and select measurement methods that make it possible to judge both the velocity and the spatial position of particles with a reasonable error.

In fact, the uncertainty principle connects not only spatial coordinates and speed - in this example, it simply manifests itself most clearly; the uncertainty also connects other pairs of mutually related characteristics of microparticles to an equal extent. By analogous reasoning, we come to the conclusion that it is impossible to accurately measure the energy of a quantum system and determine the moment of time at which it has this energy. That is, if we measure the state of a quantum system in order to determine its energy, this measurement will take a certain period of time - let's call it Δt. During this period of time, the energy of the system randomly changes - its fluctuations occur - and we cannot reveal it. Let us denote the energy measurement error as ΔE. By reasoning similar to the above, we will come to a similar relationship for ΔE and the uncertainty of time that a quantum particle of this energy possessed:

ΔЕΔt>h

Two more important remarks need to be made regarding the uncertainty principle:

It does not imply that any one of the two characteristics of a particle - spatial location or speed - cannot be measured with arbitrariness;

The uncertainty principle operates objectively and does not depend on the presence of a reasonable subject making measurements.

Ideal measurements

The uncertainty principle in quantum mechanics is sometimes explained in such a way that the measurement of the coordinate necessarily affects the momentum of the particle. It appears that Heisenberg himself offered this explanation, at least initially. That the influence of the measurement on the momentum is insignificant can be shown as follows: consider an ensemble of (non-interacting) particles prepared in the same state; for each particle in the ensemble, we measure either momentum or position, but not both. As a result of the measurement, we get that the values ​​are distributed with some probability, and the uncertainty relation is true for the variances dp and dq.

The Heisenberg uncertainty ratio is the theoretical limit to the accuracy of any measurement. They are valid for the so-called ideal measurements, sometimes called von Neumann measurements. They are even more valid for nonideal or Landau measurements.

Accordingly, any particle (in the general sense, for example, carrying a discrete electric charge) cannot be described simultaneously as a "classical point particle" and as a wave. (The very fact that any of these descriptions can be true, at least in some cases, is called wave-particle duality).

The uncertainty principle, as originally proposed by Heisenberg, is true when neither of these two descriptions is wholly and exclusively appropriate, for example particle in a box with a certain value of energy; that is

for systems that are not characterized by any specific "position" (any specific value of the distance from the potential wall), neither a certain value of momentum (including its direction).

There is a precise, quantitative analogy between the Heisenberg uncertainty relations and the properties of waves or signals. Consider a time-varying signal, such as a sound wave. It makes no sense to talk about the frequency spectrum of a signal at any point in time. To accurately determine the frequency, it is necessary to observe the signal for some time, thus losing the accuracy of timing. In other words, a sound cannot have both an exact time value, such as a short pulse, and an exact frequency value, such as a continuous pure tone. The temporal position and frequency of a wave in time are like the position and momentum of a particle in space

Conclusion

It would not be an exaggeration to say that since its inception, physics has always operated with illustrative and, if possible, simple models - at first these were systems of classical material points, and then an electromagnetic field was added to them, which, in essence, also used representations from the arsenal of continuum mechanics . The discussions between Bohr and Heisenberg led to the realization of the need to revise those images, those concepts that the theory operates in order to single out from them really only those that appear in experience. What is, for example, the orbit of an electron, can it be observed? If we take into account the dual, corpuscular-wave nature of the electron, is it possible to speak of its trajectory at all? Is it possible to construct a theory that considers only quantities actually observed in experiment?

This problem was solved in 1925 by the twenty-four-year-old Heisenberg, who proposed the so-called matrix mechanics (Nobel Prize 1932). Shortly thereafter, Erwin Schrödinger proposed another, "wave" version of quantum theory, equivalent to the "matrix" one. Quantum theory had a new mathematical base, but the physical and epistemological side of the matter still needed to be analyzed.

The result of this analysis was the Heisenberg uncertainty relations and Bohr's principle of complementarity. After analyzing the procedures for measuring coordinates and momenta, Heisenberg came to the conclusion that it is fundamentally impossible to obtain for them simultaneously and precisely defined values ​​of coordinates and momentum.

If the x-coordinate is determined with a spread x, and the projection of the momentum on the x-axis - with a spread  R x, then these spreads (or “uncertainties”) are related by the relation х р x  h / 2  , where h is Planck's constant.

Application

Bibliography

Encyclopedia of Cyril and Methodius. (2008)

http://www.elementy.ru

http:// www. bestreferat. en

Checking Bell's inequalities Photoelectric effect Compton effect

Heisenberg uncertainty principle(or Heisenberg) in quantum mechanics - a fundamental inequality (uncertainty relation), which establishes the limit of accuracy of the simultaneous determination of a pair of physical observables characterizing a quantum system (see physical quantity), described by non-commuting operators (for example, coordinate and momentum, current and voltage, electric and magnetic field). The uncertainty relation sets a lower limit for the product of the standard deviations of a pair of quantum observables. The uncertainty principle, discovered by Werner Heisenberg in Germany, is one of the cornerstones of quantum mechanics.

Short review

The Heisenberg uncertainty relations are the theoretical limit to the accuracy of simultaneous measurements of two noncommuting observables. They are valid both for ideal measurements, sometimes called von Neumann measurements, and for non-ideal or Landau measurements.

According to the uncertainty principle, a particle cannot be described as a classical particle, that is, for example, its position and velocity (momentum) cannot be accurately measured at the same time, just like an ordinary classical wave and like a wave. (The very fact that any of these descriptions can be true, at least in some cases, is called wave-particle duality). The uncertainty principle, as originally proposed by Heisenberg, also applies when none of these two descriptions is not completely and exclusively suitable, for example, a particle with a certain energy value, located in a box with perfectly reflective walls; that is, for systems that are not characterized neither some specific “position” or spatial coordinate (the wave function of the particle is delocalized to the entire space of the box, that is, its coordinates do not have a specific value, the localization of the particle is not carried out more precisely than the dimensions of the box), neither a certain value of the momentum (including its direction; in the particle-in-the-box example, the momentum modulus is defined, but its direction is not defined).

Uncertainty relations do not limit the accuracy of a single measurement of any quantity (for multidimensional quantities, in the general case, only one component is meant here). If its operator commutes with itself at different moments of time, then the accuracy of multiple (or continuous) measurements of one quantity is not limited. For example, the uncertainty relation for a free particle does not prevent the exact measurement of its momentum, but it does not allow the exact measurement of its position (this limitation is called the standard quantum limit for the position).

The uncertainty relation in quantum mechanics is, in the mathematical sense, a direct consequence of some property of the Fourier transform.

There is a precise quantitative analogy between the Heisenberg uncertainty relations and the properties of waves or signals. Consider a time-varying signal, such as a sound wave. It makes no sense to talk about the frequency spectrum of a signal at any point in time. To accurately determine the frequency, it is necessary to observe the signal for some time, thus losing the accuracy of timing. In other words, the sound cannot simultaneously have both the exact value of its fixation time, as it has a very short impulse, and the exact value of the frequency, as is the case for a continuous (and, in principle, infinitely long) pure tone (pure sinusoid). The time position and frequency of the wave are mathematically completely analogous to the coordinate and (quantum mechanical) momentum of the particle. Which is not at all surprising, considering that (or p x = k x in the system of units), that is, the momentum in quantum mechanics is the spatial frequency along the corresponding coordinate.

In everyday life, we usually do not observe quantum uncertainty because the value is extremely small, and therefore the uncertainty relations impose such weak restrictions on measurement errors that are obviously imperceptible against the background of real practical errors of our instruments or sense organs.

Definition

If there are several identical copies of the system in a given state, then the measured values ​​of position and momentum will obey a certain probability distribution - this is a fundamental postulate of quantum mechanics. By measuring the standard deviation Δ x coordinates and standard deviation Δ p momentum, we find that:

where is the reduced Planck constant.

Note that this inequality gives several possibilities - the state can be such that x can be measured with high accuracy, but then p will be known only approximately, or vice versa p can be determined exactly, while x- No. In all other states, and x and p can be measured with "reasonable" (but not arbitrarily high) accuracy.

Variants and examples

Generalized uncertainty principle

The uncertainty principle does not apply only to position and momentum (as it was first proposed by Heisenberg). In its general form, it applies to every pair conjugate variables. In general, and in contrast to the case of position and momentum discussed above, the lower bound on the product of the "uncertainties" of two conjugate variables depends on the state of the system. The uncertainty principle then becomes a theorem in operator theory, which we present here.

Therefore, the following general form is true uncertainty principle, first bred in the city by Howard Percy Robertson and (independently) Erwin Schrödinger:

This inequality is called Robertson-Schrödinger ratio.

Operator AB − BA called a switch A and B and denoted as [ A,B] . It is for those x, for which both ABx and BAx .

From the Robertson-Schrödinger relation it immediately follows Heisenberg uncertainty relation:

Suppose A and B- two physical quantities, which are associated with self-adjoint operators. If a ABψ and BAψ are defined, then:

Mean value of magnitude operator X in the state ψ of the system, and

It is also possible that there are two noncommuting self-adjoint operators A and B, which have the same eigenvector ψ . In this case, ψ is a pure state that is simultaneously measurable for A and B .

General observable variables that obey the uncertainty principle

The previous mathematical results show how to find the uncertainty relations between physical variables, namely, to determine the values ​​of pairs of variables A and B, whose commutator has certain analytic properties.

• The best-known uncertainty relation is between the position and momentum of a particle in space:

• the uncertainty relation between two orthogonal components of the operator of the total angular momentum of a particle:

• The following uncertainty relation between energy and time is often presented in physics textbooks, although its interpretation requires care since there is no operator representing time:

An expression for the finite amount of Fisher information available

The uncertainty principle is alternatively derived as an expression of the Cramer-Rao inequality in classical measurement theory when the position of a particle is measured. The root-mean-square momentum of the particle enters the inequality as the Fisher information. See also full physical information.

Interpretations

Einstein was convinced that this interpretation was wrong. His reasoning was based on the fact that all already known probability distributions were the result of deterministic events. The distribution of a coin toss or a rolling die can be described by a probability distribution (50% heads, 50% tails). But this does not mean that they physical movements unpredictable. Ordinary mechanics can calculate exactly how each coin will land if the forces acting on it are known and heads/tails are still distributed randomly (with random initial forces).

Einstein suggested that there are hidden variables in quantum mechanics that underlie observed probabilities.

Neither Einstein nor anyone else since has been able to construct a satisfactory theory of hidden variables, and Bell's inequality illustrates some very thorny paths in trying to do so. Although the behavior of an individual particle is random, it is also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process, then it turns out that particles at large distances must immediately transmit information to each other in order to guarantee correlations in their behavior.

The uncertainty principle in popular culture

The uncertainty principle is often misunderstood or misrepresented in the popular press. One common misstatement is that observing an event changes the event itself. Generally speaking, this has nothing to do with the uncertainty principle. Almost any linear operator changes the vector on which it acts (that is, almost any observation changes state), but for commutative operators there are no restrictions on the possible spread of values ​​(). For example, the projections of momentum on the axes c and y can be measured together arbitrarily accurately, although each measurement changes the state of the system. In addition, the uncertainty principle is about the parallel measurement of quantities for several systems that are in the same state, and not about sequential interactions with the same system.

Other (also misleading) analogies with macroscopic effects have been proposed to explain the uncertainty principle: one of them involves pressing a watermelon seed with a finger. The effect is known - it is impossible to predict how fast or where the seed will disappear. This random result is based entirely on randomness, which can be explained in simple classical terms.