Wave function and its statistical meaning. Condition for normalizing the wave function

As you know, the main task of classical mechanics is to determine the position of a macro-object at any time. To do this, a system of equations is compiled, the solution of which allows us to find out the dependence of the radius vector on time t. In classical mechanics, the state of a particle as it moves at each moment is given by two quantities: radius vector and momentum. Thus, the classical description of the motion of a particle is valid if it occurs in a region with a characteristic size much larger than the de Broglie wavelength. Otherwise (for example, near the atomic nucleus), the wave properties of microparticles should be taken into account. The limited applicability of the classical description of micro-objects having wave properties is indicated by the uncertainty relations.

Taking into account the presence of wave properties of a microparticle, its state in quantum mechanics is specified using a certain function of coordinates and time (x, y, z, t) , called wave or - function . In quantum physics it is introduced complex function, describing the pure state of the object, which is called the wave function. In the most common interpretation, this function is related to the probability of detecting an object in one of the pure states (the square of the modulus of the wave function represents the probability density).

Having abandoned the description of the motion of a particle using trajectories obtained from the laws of dynamics, and having determined instead the wave function, it is necessary to introduce an equation equivalent to Newton's laws and providing a recipe for finding solutions to particular physical problems. Such an equation is the Schrödinger equation.

The theory that describes the movement of small particles taking into account their wave properties is called quantum , or wave mechanics. Many provisions of this theory seem strange and unusual from the point of view of the ideas that have developed in the study of classical physics. It should always be remembered that the criterion for the correctness of a theory, no matter how strange it may seem at first, is the coincidence of its consequences with experimental data. Quantum mechanics in its field (the structure and properties of atoms, molecules and partly atomic nuclei) is perfectly confirmed by experience.

The wave function describes the state of a particle at all points in space and for any moment in time. To understand the physical meaning of the wave function, let us turn to experiments on electron diffraction. (Thomson and Tartakovsky's experiments on passing electrons through a thin metal foil). It turns out that clear diffraction patterns are detected even if single electrons are directed at the target, i.e. when each subsequent electron is emitted after the previous one reaches the screen. After a sufficiently long bombardment, the picture on the screen will exactly correspond to that obtained when a large number of electrons are simultaneously directed at the target.

From this we can conclude that the movement of any microparticle individually, including the location of its detection, is subject to statistical (probabilistic) laws, and when a single electron is directed at the target, the point on the screen at which it will be recorded is 100% certain in advance. -It is impossible to predict with certainty.

In Thomson's diffraction experiments, a system of dark concentric rings was formed on a photographic plate. It is safe to say that the probability of detecting (hitting) each emitted electron in various places photographic plates are not the same. In the area of ​​dark concentric rings, this probability is greater than in other areas of the screen. The distribution of electrons over the entire screen turns out to be the same as the distribution of the intensity of an electromagnetic wave in a similar diffraction experiment: where the intensity of the X-ray wave is high, many particles are recorded in Thomson’s experiment, and where the intensity is low, almost no particles appear.

From a wave point of view, the presence of a maximum number of electrons in some directions means that these directions correspond to the highest intensity of the de Broglie wave. This served as the basis for the statistical (probabilistic) interpretation of the de Broglie wave. The wave function is precisely a mathematical expression that allows us to describe the propagation of a wave in space. In particular, the probability of finding a particle in a given region of space is proportional to the square of the amplitude of the wave associated with the particle.

For one-dimensional motion (for example, in the direction of the axis Ox) probability dP detecting a particle in the gap between points x And x + dx at a point in time t equal to

dP = , (6.1)

where | (x,t)| 2 = (x,t) *(x,t) is the square of the modulus of the wave function (the * symbol denotes complex conjugation).

In general, when a particle moves in three-dimensional space, the probability dP detection of a particle at a point with coordinates (x,y,z) within an infinitesimal volume dV is given by a similar equation : dP =|(x,y,z,t)|2 dV. Born was the first to give a probabilistic interpretation of the wave function in 1926.

The probability of detecting a particle in the entire infinite space is equal to one. This implies the condition for normalizing the wave function:

. (6.2)

The value is probability density , or, which is the same thing, the density distribution of particle coordinates. In the simplest case of one-dimensional particle motion along the axis OX the average value of its coordinate is calculated by the following relation:

<x(t)>= . (6.3)

For the wave function to be an objective characteristic of the state of a microparticle, it must satisfy a number of restrictive conditions. The function Ψ, which characterizes the probability of detecting a microparticle in a volume element, must be finite (the probability cannot be greater than one), unambiguous (the probability cannot be an ambiguous value), continuous (the probability cannot change abruptly) and smooth (without kinks) throughout the entire space .

The wave function satisfies the principle of superposition: if the system can be in different states described by the wave functions Ψ1, Ψ2, Ψ n, then it can be in a state described by a linear combination of these functions:

, (6.4)

Where Cn(n= 1, 2, 3) are arbitrary, generally speaking, complex numbers.

The addition of wave functions (probability amplitudes determined by the squared moduli of wave functions) fundamentally distinguishes quantum theory from classical statistical theory, in which the addition of probabilities theorem is valid for independent events.

The wave function Ψ is the main characteristic of the state of microobjects.

For example, the average distance<r> electron of the nucleus is calculated by the formula:

,

where the calculations are carried out as in case (6.3). Thus, it is impossible to accurately predict in diffraction experiments where a particular electron will be recorded on the screen, even knowing its wave function in advance. One can only assume with a certain probability that the electron will be fixed in a certain place. This is the difference between the behavior of quantum objects and classical ones. In classical mechanics, when describing the movement of macrobodies, we knew in advance with 100% probability where in space the material point(For example, space station) at any time.

De Broglie used the concept of phase waves (matter waves or de Broglie waves) to visually interpret Bohr's rule for quantizing electron orbits in an atom in the case of a single-electron atom. He examined a phase wave traveling around the nucleus in a circular orbit of an electron. If an integer number of these waves fits along the length of the orbit, then the wave, when going around the nucleus, will each time return to the starting point with the same phase and amplitude. In this case, the orbit becomes stationary and no radiation occurs. De Broglie wrote down the condition for stationary orbit or the quantization rule in the form:

Where R- radius of the circular orbit, P- integer (principal quantum number). Believing here and given that L=RP is the angular momentum of the electron, we get:

which coincides with the rule of quantization of electron orbits in a hydrogen atom according to Bohr.

Subsequently, condition (6.5) was generalized to the case of elliptical orbits, when the wavelength varies along the electron trajectory. However, in de Broglie's reasoning it was assumed that the wave does not propagate in space, but along a line - along the stationary orbit of the electron. This approximation can be used in the limiting case, when the wavelength is negligible compared to the radius of the electron's orbit.

Bohr's postulates

The planetary model of the atom made it possible to explain the results of experiments on the scattering of alpha particles of matter, but fundamental difficulties arose in justifying the stability of atoms.
The first attempt to construct a qualitatively new – quantum – theory of the atom was made in 1913 by Niels Bohr. He set the goal of linking into a single whole the empirical laws of line spectra, the Rutherford nuclear model of the atom, and the quantum nature of the emission and absorption of light. Bohr based his theory on Rutherford's nuclear model. He suggested that electrons move around the nucleus in circular orbits. Circular motion, even at constant speed, has acceleration. Such accelerated charge movement is equivalent alternating current, which creates an alternating electromagnetic field in space. Energy is consumed to create this field. The field energy can be created due to the energy of the Coulomb interaction of the electron with the nucleus. As a result, the electron must move in a spiral and fall onto the nucleus. However, experience shows that atoms are very stable formations. It follows from this that the results of classical electrodynamics, based on Maxwell’s equations, are not applicable to intra-atomic processes. It is necessary to find new patterns. Bohr based his theory of the atom on the following postulates.
Bohr's first postulate (postulate of stationary states): in an atom there are stationary (not changing with time) states in which it does not emit energy. Stationary states of an atom correspond to stationary orbits along which electrons move. The movement of electrons in stationary orbits is not accompanied by the emission of electromagnetic waves.
This postulate is in conflict with classical theory. In the stationary state of an atom, an electron, moving in a circular orbit, must have discrete quantum values ​​of angular momentum.
Bohr's second postulate (frequency rule): when an electron moves from one stationary orbit to another, one photon with energy is emitted (absorbed)

equal to the difference between the energies of the corresponding stationary states (En and Em are, respectively, the energies of the stationary states of the atom before and after radiation/absorption).
The transition of an electron from a stationary orbit number m to a stationary orbit number n corresponds to the transition of an atom from a state with energy Em into a state with energy En (Fig. 4.1).

Rice. 4.1. To an explanation of Bohr's postulates

At En > Em, photon emission occurs (the transition of an atom from a state with higher energy to a state with lower energy, i.e., the transition of an electron from an orbit more distant from the nucleus to a closer one), at En< Еm – его поглощение (переход атома в состояние с большей энергией, т. е, переход электрона на более удаленную от ядра орбиту). Набор возможных дискретных частот

quantum transitions and determines the line spectrum of an atom.
Bohr's theory brilliantly explained the experimentally observed line spectrum of hydrogen.
The successes of the theory of the hydrogen atom were achieved at the cost of abandoning the fundamental principles of classical mechanics, which has remained unconditionally valid for more than 200 years. That's why great importance had direct experimental proof of the validity of Bohr's postulates, especially the first - on the existence of stationary states. The second postulate can be considered as a consequence of the law of conservation of energy and the hypothesis about the existence of photons.
German physicists D. Frank and G. Hertz, studying the collision of electrons with gas atoms using the retarding potential method (1913), experimentally confirmed the existence of stationary states and the discreteness of atomic energy values.
Despite the undoubted success of Bohr's concept in relation to the hydrogen atom, for which it turned out to be possible to construct a quantitative theory of the spectrum, it was not possible to create a similar theory for the helium atom next to hydrogen based on Bohr's ideas. Regarding the helium atom and more complex atoms, Bohr's theory allowed us to draw only qualitative (albeit very important) conclusions. The idea of ​​certain orbits along which an electron moves in a Bohr atom turned out to be very conditional. In fact, the movement of electrons in an atom has little in common with the movement of planets in orbit.
Currently using quantum mechanics You can answer many questions regarding the structure and properties of atoms of any element.

5. basic principles of quantum mechanics:

Wave function and its physical meaning.

From the content of the previous two paragraphs it follows that a wave process is associated with a microparticle, which corresponds to its movement, therefore the state of a particle in quantum mechanics is described wave function, which depends on coordinates and time y(x,y,z,t). Specific view y-function is determined by the state of the particle and the nature of the forces acting on it. If the force field acting on the particle is stationary, i.e. independent of time, then y-function can be represented as a product of two factors, one of which depends on time, and the other on coordinates:

In what follows we will only consider stationary states. The y-function is a probabilistic characteristic of the state of the particle. To explain this, let us mentally select a sufficiently small volume within which the values ​​of the y-function will be considered the same. Then the probability of finding dW particles in a given volume is proportional to it and depends on the squared modulus of the y-function (the squared modulus of the de Broglie wave amplitude):

This implies the physical meaning of the wave function:

The squared modulus of the wave function has the meaning of probability density, i.e. determines the probability of finding a particle in a unit volume in the vicinity of a point with coordinates x, y, z.

By integrating expression (3.2) over the volume, we determine the probability of finding a particle in this volume under stationary field conditions:

If the particle is known to be within the volume V, then the integral of expression (3.4), taken over the volume V, must be equal to one:

– normalization condition for the y-function.

For the wave function to be an objective characteristic of the state of microparticles, it must be finite, unambiguous, continuous, since the probability cannot be greater than one, cannot be an ambiguous value and cannot change in jumps. Thus, the state of the microparticle is completely determined by the wave function. A particle can be detected at any point in space at which the wave function is nonzero.

· Quantum observable · Wave function· Quantum superposition · Quantum entanglement · Mixed state · Measurement · Uncertainty · Pauli principle · Dualism · Decoherence · Ehrenfest's theorem · Tunnel effect

Wave function, or psi function $\backslash psi$ is a complex-valued function used in quantum mechanics to describe the pure state of a system. Is the coefficient of expansion of the state vector over a basis (usually a coordinate one):

$\backslash left|\backslash psi(t)\backslash right\backslash rangle=\backslash int\; \backslash Psi(x,t)\backslash left|x\backslash right\backslash rangle\; dx$

Where $\backslash left|x\backslash right\backslash rangle\; =\; \backslash left|x\_1,\; x\_2,\; \backslash ldots\; ,\; x\_n\backslash right\backslash rangle$ is the coordinate basis vector, and $\backslash Psi(x,t)=\; \backslash langle\; x\backslash left|\backslash psi(t)\backslash right\backslash rangle$- wave function in coordinate representation.

Normalization of the wave function

Wave function $\backslash Psi$ in its meaning must satisfy the so-called normalization condition, for example, in the coordinate representation having the form:

$(\backslash int\backslash limits\_(V)(\backslash Psi^\backslash ast\backslash Psi)dV)=1$

This condition expresses the fact that the probability of finding a particle with a given wave function anywhere in space is equal to one. In the general case, integration must be carried out over all variables on which the wave function in a given representation depends.

Principle of superposition of quantum states

For wave functions, the principle of superposition is valid, which is that if a system can be in states described by wave functions $\backslash Psi\_1$ And $\backslash Psi\_2$, then it can also be in a state described by the wave function

$\backslash Psi\_\backslash Sigma\; =\; c\_1\; \backslash Psi\_1\; +\; c\_2\; \backslash Psi\_2$ for any complex $c\_1$ And $c\_2$.

Obviously, we can talk about the superposition (imposition) of any number of quantum states, that is, about the existence of a quantum state of the system, which is described by the wave function $\backslash Psi\_\backslash Sigma\; =\; c\_1\; \backslash Psi\_1\; +\; c\_2\; \backslash Psi\_2\; +\; \backslash ldots\; +\; (c)\_N(\backslash Psi)\_N=\backslash sum\_(n=1)^(N)\; (c)\_n(\backslash Psi)\_n$.

In this state, the square of the modulus of the coefficient $(c)\_n$ determines the probability that, when measured, the system will be detected in a state described by the wave function $(\backslash Psi)\_n$.

Therefore, for normalized wave functions $\backslash sum\_(n=1)^(N)\backslash left|c\_(n)\backslash right|^2=1$.

Conditions for the regularity of the wave function

The probabilistic meaning of the wave function imposes certain restrictions, or conditions, on wave functions in problems of quantum mechanics. These standard conditions often call conditions for the regularity of the wave function.

1. Condition for the finiteness of the wave function. The wave function cannot take infinite values ​​such that the integral $(1)$ will become divergent. Consequently, this condition requires that the wave function be a quadratically integrable function, that is, belong to Hilbert space $L^2$. In particular, in problems with a normalized wave function, the squared modulus of the wave function must tend to zero at infinity.
2. Condition for the uniqueness of the wave function. The wave function must be an unambiguous function of coordinates and time, since the probability density of detecting a particle must be determined uniquely in each problem. In problems using a cylindrical or spherical coordinate system, the uniqueness condition leads to the periodicity of wave functions in angular variables.
3. Condition for the continuity of the wave function. At any moment of time the wave function must be continuous function spatial coordinates. In addition, the partial derivatives of the wave function must also be continuous $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; x)$, $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; y)$, $\backslash frac(\backslash partial\; \backslash Psi)(\backslash partial\; z)$. These partial derivatives of functions are only in rare cases of problems with idealized force fields can suffer a gap at those points in space where potential energy, which describes the force field in which the particle moves, experiences a discontinuity of the second kind.

Wave function in various representations

The set of coordinates that act as function arguments represents a complete system of commuting observables. In quantum mechanics it is possible to select several complete sets of observables, so the wave function of the same state can be written in terms of different arguments. The complete set of quantities chosen to record the wave function determines wave function representation. Thus, a coordinate representation, a momentum representation are possible; in quantum field theory, secondary quantization and the representation of occupation numbers or the Fock representation, etc., are used.

If the wave function, for example, of an electron in an atom, is given in coordinate representation, then the squared modulus of the wave function represents the probability density of detecting an electron at a particular point in space. If the same wave function is given in impulse representation, then the square of its module represents the probability density of detecting a particular impulse.

Matrix and vector formulations

The wave function of the same state in different representations will correspond to the expression of the same vector in different systems coordinates Other operations with wave functions will also have analogues in the language of vectors. In wave mechanics, a representation is used where the arguments of the psi function are complete system continuous commuting observables, and the matrix representation uses a representation where the arguments of the psi function are the complete system discrete commuting observables. Therefore, the functional (wave) and matrix formulations are obviously mathematically equivalent.

Philosophical meaning of the wave function

The wave function is a method of describing the pure state of a quantum mechanical system. Mixed quantum states (in quantum statistics) should be described by an operator like a density matrix. That is, some generalized function of two arguments must describe the correlation between the location of a particle at two points.

It should be understood that the problem that quantum mechanics solves is the problem of the very essence of the scientific method of knowing the world.

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Literature

• Physical encyclopedic Dictionary/ Ch. ed. A. M. Prokhorov. Ed. count D. M. Alekseev, A. M. Bonch-Bruevich, A. S. Borovik-Romanov and others - M.: Sov. Encyclopedia, 1984. - 944 p.

Links

• Quantum mechanics- article from the Great Soviet Encyclopedia.

The discovery of the wave properties of microparticles indicated that classical mechanics cannot give correct description behavior of such particles. A theory covering all properties elementary particles, must take into account not only their corpuscular properties, but also their wave properties. From the experiments discussed earlier, it follows that a beam of elementary particles has the properties of a plane wave propagating in the direction of the particle speed. In the case of propagation along the axis, this wave process can be described by the de Broglie wave equation (7.43.5):

(7.44.1)

where is the energy and is the momentum of the particle. When propagating in any direction:

(7.44.2)

Let's call the function a wave function and find out its physical meaning by comparing the diffraction of light waves and microparticles.

According to wave concepts of the nature of light, the intensity of the diffraction pattern is proportional to the square of the amplitude of the light wave. According to views photon theory, the intensity is determined by the number of photons hitting a given point in the diffraction pattern. Consequently, the number of photons at a given point in the diffraction pattern is given by the square of the amplitude of the light wave, while for one photon the square of the amplitude determines the probability of the photon hitting a particular point.

The diffraction pattern observed for microparticles is also characterized by an unequal distribution of microparticle fluxes. From the point of view of wave theory, the presence of maxima in the diffraction pattern means that these directions correspond to the highest intensity of de Broglie waves. The intensity is greater where larger number particles. Thus, the diffraction pattern for microparticles is a manifestation of a statistical pattern and we can say that knowledge of the type of de Broglie wave, i.e. Ψ -function allows one to judge the probability of one or another of the possible processes.

So, in quantum mechanics, the state of microparticles is described in a fundamentally new way - using the wave function, which is the main carrier of information about their corpuscular and wave properties. The probability of finding a particle in an element with volume is

(7.44.3)

Magnitude

(7.44.4)

has the meaning of probability density, i.e. determines the probability of finding a particle in a unit volume in the vicinity given point. Thus, it is not the function itself that has a physical meaning, but the square of its module, which sets the intensity of de Broglie waves. The probability of finding a particle at a moment in time in a finite volume, according to the theorem of addition of probabilities, is equal to

(7.44.5)

Since a particle exists, it is sure to be found somewhere in space. The probability of a reliable event is equal to one, then

. (7.44.6)

Expression (7.44.6) is called the probability normalization condition. The wave function characterizing the probability of detecting the action of a microparticle in a volume element must be finite (the probability cannot be greater than one), unambiguous (the probability cannot be an ambiguous value) and continuous (the probability cannot change abruptly).

Wave function, or psi function ψ (\displaystyle \psi )- a complex-valued function used in quantum mechanics to describe the pure state of a system. Is the coefficient of expansion of the state vector over a basis (usually a coordinate one):

| ψ (t) ⟩ = ∫ Ψ (x, t) | x ⟩ d x (\displaystyle \left|\psi (t)\right\rangle =\int \Psi (x,t)\left|x\right\rangle dx)

Where | x⟩ = | x 1 , x 2 , … , x n ⟩ (\displaystyle \left|x\right\rangle =\left|x_(1),x_(2),\ldots ,x_(n)\right\rangle ) is the coordinate basis vector, and Ψ(x, t) = ⟨x | ψ (t) ⟩ (\displaystyle \Psi (x,t)=\langle x\left|\psi (t)\right\rangle )- wave function in coordinate representation.

Normalization of the wave function

Wave function Ψ (\displaystyle \Psi ) in its meaning must satisfy the so-called normalization condition, for example, in the coordinate representation having the form:

This condition expresses the fact that the probability of finding a particle with a given wave function anywhere in space is equal to one. In the general case, integration must be carried out over all variables on which the wave function in a given representation depends.

Principle of superposition of quantum states

For wave functions, the principle of superposition is valid, which consists in the fact that if a system can be in states described by wave functions Ψ 1 (\displaystyle \Psi _(1)) And Ψ 2 (\displaystyle \Psi _(2)), then it can also be in a state described by the wave function

Ψ Σ = c 1 Ψ 1 + c 2 Ψ 2 (\displaystyle \Psi _(\Sigma )=c_(1)\Psi _(1)+c_(2)\Psi _(2)) for any complex c 1 (\displaystyle c_(1)) And c 2 (\displaystyle c_(2)).

Obviously, we can talk about the superposition (addition) of any number of quantum states, that is, about the existence of a quantum state of the system, which is described by the wave function Ψ Σ = c 1 Ψ 1 + c 2 Ψ 2 + … + c N Ψ N = ∑ n = 1 N c n Ψ n (\displaystyle \Psi _(\Sigma )=c_(1)\Psi _(1)+ c_(2)\Psi _(2)+\ldots +(c)_(N)(\Psi )_(N)=\sum _(n=1)^(N)(c)_(n)( \Psi )_(n)).

In this state, the square of the modulus of the coefficient c n (\displaystyle (c)_(n)) determines the probability that, when measured, the system will be detected in a state described by the wave function Ψ n (\displaystyle (\Psi )_(n)).

Therefore, for normalized wave functions ∑ n = 1 N | c n | 2 = 1 (\displaystyle \sum _(n=1)^(N)\left|c_(n)\right|^(2)=1).

Conditions for the regularity of the wave function

The probabilistic meaning of the wave function imposes certain restrictions, or conditions, on wave functions in problems of quantum mechanics. These standard conditions are often called conditions for the regularity of the wave function.

Wave function in various representations states are used in different representations - will correspond to the expression of the same vector in different coordinate systems. Other operations with wave functions will also have analogues in the language of vectors. In wave mechanics, a representation is used where the arguments of the psi function are the complete system continuous commuting observables, and the matrix representation uses a representation where the arguments of the psi function are the complete system discrete commuting observables. Therefore, the functional (wave) and matrix formulations are obviously mathematically equivalent.