Wave function and its statistical meaning. Types of wave function and its collapse

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. This accelerated movement of charge is equivalent to 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 the 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. Therefore, direct experimental proof of the validity of Bohr’s postulates, especially the first one – about the existence of stationary states – was of great importance. 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, with the help of quantum mechanics, it is possible to answer many questions regarding the structure and properties of atoms of any elements.

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 consider only stationary states. The y-function is a probabilistic characteristic of the state of a 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.

The 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 are often called 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 in time, the wave function must be a continuous function of 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 only in rare cases of problems with idealized force fields can suffer a discontinuity at those points in space where the potential energy describing 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 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.

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.

3. ELEMENTS OF QUANTUM MECHANICS

3.1.Wave function

Every microparticle is a special kind of formation, combining the properties of both particles and waves. The difference between a microparticle and a wave is that it is detected as an indivisible whole. For example, no one has observed a half-electron. At the same time, the wave can be divided into parts and then each part can be perceived separately.

The difference between a microparticle in quantum mechanics and an ordinary microparticle is that it does not simultaneously have certain values ​​of coordinates and momentum, so the concept of a trajectory for a microparticle loses its meaning.

The probability distribution of finding a particle at a given time in a certain region of space will be described by the wave function (x, y, z , t) (psi function). Probability dP that the particle is located in a volume element dV, proportional
and volume element dV:

dP=
dV.

It is not the function itself that has physical meaning
, and the square of its modulus is the probability density. It determines the probability of a particle being at a given point in space.

Wave function
is the main characteristic of the state of microobjects (microparticles). With its help, in quantum mechanics, the average values ​​of physical quantities that characterize a given object in a state described by the wave function can be calculated
.

3.2. Uncertainty principle

In classical mechanics, the state of a particle is specified by coordinates, momentum, energy, etc. These are dynamic variables. A microparticle cannot be described by such dynamic variables. The peculiarity of microparticles is that not all variables obtain certain values ​​during measurements. For example, a particle cannot simultaneously have exact coordinate values X and impulse components R X. Uncertainty of values X And R X satisfies the relation:

(3.1)

– the smaller the uncertainty of the coordinate Δ X, the greater the uncertainty of the pulse Δ R X, and vice versa.

Relation (3.1) is called the Heisenberg uncertainty relation and was obtained in 1927.

Δ values X and Δ R X are called canonically conjugate. The same canonically conjugate are Δ at and Δ R at, and so on.

The Heisenberg Uncertainty Principle states that the product of the uncertainties of two conjugate variables cannot be less than Planck's constant in order of magnitude. ħ.

Energy and time are also canonically conjugate, therefore
. This means that the determination of energy with an accuracy of Δ E should take time interval:

Δ t ~ ħ/ Δ E.

Let's determine the coordinate value X freely flying microparticle, placing in its path a gap of width Δ X, located perpendicular to the direction of particle motion. Before the particle passes through the slit, its momentum component is R X has the exact meaning R X= 0 (the gap is perpendicular to the momentum vector), so the uncertainty of the momentum is zero, Δ R X= 0, but the coordinate X particles is completely uncertain (Fig. 3.1).

IN the moment the particle passes through the slit, the position changes. Instead of complete uncertainty of coordinates X uncertainty appears Δ X, and momentum uncertainty Δ appears R X .

Indeed, due to diffraction, there is some probability that the particle will move within an angle of 2 φ , Where φ – the angle corresponding to the first diffraction minimum (we neglect the maxima of higher orders, since their intensity is small compared to the intensity of the central maximum).

Thus, uncertainty arises:

Δ R X =R sin φ ,

But sin φ = λ / Δ X– this is the condition of the first minimum. Then

Δ R X ~рλ/Δ X,

Δ XΔ R X ~рλ= 2πħ ≥ħ/ 2.

The uncertainty relationship indicates to what extent the concepts of classical mechanics can be used in relation to microparticles, in particular, with what degree of accuracy we can talk about the trajectory of microparticles.

Movement along a trajectory is characterized by certain values ​​of the particle’s speed and its coordinates at each moment of time. Substituting into the uncertainty relation instead R X expression for momentum
, we have:

–

The greater the mass of the particle, the less uncertainty in its coordinates and speed, the more accurately the concepts of trajectory are applicable to it.

For example, for a microparticle with a size of 1·10 -6 m, the uncertainties Δх and Δ go beyond the accuracy of measuring these quantities, and the movement of the particle is inseparable from the movement along the trajectory.

The uncertainty relation is a fundamental proposition of quantum mechanics. For example, it helps explain the fact that an electron does not fall on the nucleus of an atom. If an electron fell on a point nucleus, its coordinates and momentum would take on certain (zero) values, which is incompatible with the uncertainty principle. This principle requires that the uncertainty of the electron coordinate Δ r and momentum uncertainty Δ R satisfied the relation

Δ rΔ p ≥ ħ/ 2,

and meaning r= 0 is impossible.

The energy of an electron in an atom will be minimal at r= 0 and R= 0, so to estimate the lowest possible energy we set Δ r ≈ r, Δ p ≈ p. Then Δ rΔ p ≥ ħ/ 2, and for the smallest uncertainty value we have:

we are only interested in the order of magnitudes included in this relation, so the factor can be discarded. In this case we have
, from here р = ħ/r. Electron energy in a hydrogen atom

(3.2)

We'll find r, at which energy E minimal. Let us differentiate (3.2) and equate the derivative to zero:

,

We discarded the numerical factors in this expression. From here
- radius of the atom (radius of the first Bohr orbit). For energy we have

One might think that with the help of a microscope it would be possible to determine the position of a particle and thereby overthrow the uncertainty principle. However, a microscope will allow one to determine the position of a particle, at best, with an accuracy of up to the wavelength of the light used, i.e. Δ x ≈ λ, but because Δ R= 0, then Δ RΔ X= 0 and the uncertainty principle is not satisfied?! Is it so?

We use light, and light, according to quantum theory, consists of photons with momentum p =k/λ . To detect a particle, at least one of the photons of the light beam must be scattered or absorbed by it. Consequently, momentum will be transferred to the particle, at least reaching h/λ . Thus, at the moment of observation of a particle with coordinate uncertainty Δ x ≈ λ the momentum uncertainty must be Δ p ≥h/λ .

Multiplying these uncertainties, we get:

–

the uncertainty principle is satisfied.

The process of interaction of the device with the object being studied is called measurement. This process takes place in space and time. There is an important difference between the interaction of a device with macro- and micro-objects. The interaction of a device with a macro-object is the interaction of two macro-objects, which is quite accurately described by the laws of classical physics. In this case, we can assume that the device does not have any influence on the measured object, or that the influence is small. When the device interacts with microobjects, a different situation arises. The process of fixing a certain position of a microparticle introduces a change in its momentum that cannot be made equal to zero:

Δ R X ≥ ħ/ Δ X.

Therefore, the impact of the device on the microparticle cannot be considered small and insignificant; the device changes the state of the microobject - as a result of the measurement, certain classical characteristics of the particle (momentum, etc.) turn out to be specified only within the framework limited by the uncertainty relation.

3.3. Schrödinger equation

In 1926, Schrödinger obtained his famous equation. This is the fundamental equation of quantum mechanics, the basic assumption on which all quantum mechanics is based. All the consequences arising from this equation are consistent with experience - this is its confirmation.

The probabilistic (statistical) interpretation of de Broglie waves and the uncertainty relation indicate that the equation of motion in quantum mechanics must be such that it allows us to explain the experimentally observed wave properties of particles. The position of a particle in space at a given moment in time is determined in quantum mechanics by specifying the wave function
(x, y, z, t), or rather the square of the modulus of this quantity.
is the probability of finding a particle at a point x, y, z at a point in time t. The fundamental equation of quantum mechanics must be an equation with respect to the function
(x, y, z, t). Further, this equation must be a wave equation; experiments on the diffraction of microparticles, confirming their wave nature, must derive their explanation from it.

The Schrödinger equation has the following form:

. (3.3)

Where m– particle mass, i– imaginary unit,
– Laplace operator,
,U– particle potential energy operator.

The form of the Ψ-function is determined by the function U, i.e. the nature of the forces acting on the particle. If the force field is stationary, then the solution to the equation has the form:

, (3.4)

Where E is the total energy of the particle, it remains constant in each state, E=const.

Equation (3.4) is called the Schrödinger equation for stationary states. It can also be written in the form:

.

This equation is applicable to non-relativistic systems provided that the probability distribution does not change over time, i.e. when functions ψ look like standing waves.

The Schrödinger equation can be obtained as follows.

Let's consider the one-dimensional case - a freely moving particle along the axis X. It corresponds to a plane de Broglie wave:

,

But
, That's why
. Let us differentiate this expression by t:

.

Let us now find the second derivative of the psi function with respect to the coordinate

,

In non-relativistic classical mechanics, energy and momentum are related by the relation:
Where E- kinetic energy. The particle moves freely, its potential energy U= 0, and full E=E k. That's why

,

is the Schrödinger equation for a free particle.

If a particle moves in a force field, then E– all energy (both kinetic and potential), therefore:

,

then we get
, or
,

and finally

This is the Schrödinger equation.

The above reasoning is not a derivation of the Schrödinger equation, but an example of how this equation can be established. The Schrödinger equation itself is postulated.

In expression

the left side denotes the Hamiltonian operator – the Hamiltonian is the sum of the operators
And U. The Hamiltonian is an energy operator. We will talk in detail about operators of physical quantities later. (The operator expresses some action under the function ψ , which is under the operator sign). Taking into account the above we have:

.

It does not have a physical meaning ψ -function, and the square of its modulus, which determines the probability density of finding a particle in a given location in space. Quantum mechanics makes statistical sense. It does not allow one to determine the location of a particle in space or the trajectory along which the particle moves. The psi function only gives the probability with which a particle can be detected at a given point in space. In this regard, the psi function must satisfy the following conditions:

It must be unambiguous, continuous and finite, because determines the state of the particle;

It must have a continuous and finite derivative;

Function I ψ I 2 must be integrable, i.e. integral

must be finite because it determines the probability of detecting a particle.

Integral

,

This is a normalization condition. It means that the probability that a particle is located at any point in space is equal to one.

To describe the particle-wave properties of an electron in quantum mechanics, a wave function is used, which is denoted by the Greek letter psi (T). The main properties of the wave function are:

• at any point in space with coordinates x, y, z it has a certain sign and amplitude: BHd:, at, G);
• squared modulus of the wave function | CHH, y,z)| 2 is equal to the probability of finding a particle in a unit volume, i.e. probability density.

The probability density of detecting an electron at various distances from the nucleus of an atom is depicted in several ways. It is often characterized by the number of points per unit volume (Fig. 9.1, A). A dotted probability density image resembles a cloud. Speaking about the electron cloud, it should be borne in mind that an electron is a particle that simultaneously exhibits both corpuscular and wave

Rice. 9.1.

properties. The probability range for detecting an electron does not have clear boundaries. However, it is possible to select a space where the probability of its detection is high or even maximum.

In Fig. 9.1, A The dashed line indicates a spherical surface within which the probability of detecting an electron is 90%. In Fig. Figure 9.1b shows a contour image of the electron density in a hydrogen atom. The contour closest to the nucleus covers a region of space in which the probability of detecting an electron is 10%, the probability of detecting an electron inside the second contour from the nucleus is 20%, inside the third - 30%, etc. In Fig. 9.1, the electron cloud is depicted as a spherical surface, within which the probability of detecting an electron is 90%.

Finally, in Fig. 9.1, d and b, shows the probability of detecting an electron Is at different distances in two ways G from the kernel: at the top is a “cut” of this probability passing through the kernel, and at the bottom is the function itself 4lr 2 |U| 2.

Schrödingsr's equation. This fundamental equation of quantum mechanics was formulated by the Austrian physicist E. Schrödinger in 1926. It relates the total energy of a particle E, equal to the sum of potential and kinetic energies, potential energy?„, particle mass T and wave function 4*. For one particle, for example an electron with mass that is, it looks like this:

From a mathematical point of view, this is an equation with three unknowns: Y, E And?". Solve it, i.e. These unknowns can be found by solving it together with two other equations (three equations are required to find three unknowns). The equations for potential energy and boundary conditions are used as such equations.

The potential energy equation does not contain the wave function V. It describes the interaction of charged particles according to Coulomb’s law. When one electron interacts with a nucleus having a +z charge, the potential energy is equal to

Where g = Y* 2 + y 2+ z 2 .

This is the case of the so-called one-electron atom. In more complex systems, when there are many charged particles, the potential energy equation consists of the sum of the same Coulomb terms.

The boundary condition equation is the expression

It means that the electron wave function tends to zero at large distances from the atomic nucleus.

Solving the Schrödinger equation allows one to find the electron wave function? = (x, y, z) as a function of coordinates. This distribution is called an orbital.

Orbital - it is a wave function defined in space.

A system of equations, including the Schrödinger equations, potential energy and boundary conditions, has not one, but many solutions. Each of the solutions simultaneously includes 4 x = (x, y, G) And E, i.e. describes the electron cloud and its corresponding total energy. Each of the solutions is determined quantum numbers.

The physical meaning of quantum numbers can be understood by considering the oscillations of a string, which result in the formation of a standing wave (Fig. 9.2).

Standing wave length X and string length b related by the equation

The length of a standing wave can only have strictly defined values ​​corresponding to the number P, which only accepts non-negative integer values ​​1,2,3, etc. As is obvious from Fig. 9.2, the number of maxima of the oscillation amplitude, i.e. the shape of a standing wave is uniquely determined by the value P.

Since an electron wave in an atom is a more complex process than a standing wave of a string, the values ​​of the electron wave function are determined not by one, but by four

Rice. 9.2.

four numbers, which are called quantum numbers and are designated by letters P, /, T And s. This set of quantum numbers P, /, T simultaneously correspond to a certain wave function Ch"lDl, and the total energy E„j. Quantum number T at E are not indicated, since in the absence of an external field the electron energy from T does not depend. Quantum number s does not affect any 4 *n xt, not at all E n j.

• , ~ elxv dlxv 62*p
• The symbols --, --- mean the second partial derivatives of the fir1 arcs of the 8z2 H"-function. These are derivatives of the first derivatives. Does the meaning of the first derivative coincide with the tangent of the slope of the function H" from the argument x, y or z on the graphs? = j(x), T =/2(y), H" =/:!(z).

Wave function
Wave function

Wave function (or state vector) is a complex function that describes the state of a quantum mechanical system. Knowing it allows you to obtain the most complete information about the system, which is fundamentally achievable in the microcosm. So, with its help, you can calculate all the measurable physical characteristics of the system, the probability of its being in a certain place in space and its evolution in time. The wave function can be found by solving the Schrödinger wave equation.
The wave function ψ (x, y, z, t) ≡ ψ (x,t) of a point structureless particle is a complex function of the coordinates of this particle and time. The simplest example of such a function is the wave function of a free particle with momentum and total energy E (plane wave)

.

The wave function of the system A of particles contains the coordinates of all particles: ψ ( 1 , 2 ,..., A ,t).
Squared modulus of the wave function of an individual particle | ψ (,t)| 2 = ψ *(,t) ψ (,t) gives the probability of detecting a particle at time t at a point in space described by coordinates, namely, | ψ (,t)| 2 dv ≡ | ψ (x, y, z, t)| 2 dxdydz is the probability of finding a particle in a region of space with volume dv = dxdydz around point x, y, z. Similarly, the probability of finding at time t a system A of particles with coordinates 1, 2,..., A in a volume element of a multidimensional space is given by | ψ ( 1 , 2 ,..., A ,t)| 2 dv 1 dv 2 ...dv A .
The wave function completely determines all the physical characteristics of a quantum system. Thus, the average observed value of the physical quantity F of the system is given by the expression

,

where is the operator of this quantity and integration is carried out over the entire region of multidimensional space.
Instead of particle coordinates x, y, z, their momenta p x , p y , p z or other sets of physical quantities can be chosen as independent variables of the wave function. This choice depends on the representation (coordinate, impulse or other).
The wave function ψ (,t) of a particle does not take into account its internal characteristics and degrees of freedom, i.e., it describes its movement as a whole structureless (point) object along a certain trajectory (orbit) in space. These internal characteristics of a particle can be its spin, helicity, isospin (for strongly interacting particles), color (for quarks and gluons) and some others. The internal characteristics of a particle are specified by a special wave function of its internal state φ. In this case, the total wave function of the particle Ψ can be represented as the product of the orbital motion function ψ and the internal function φ:

since usually the internal characteristics of a particle and its degrees of freedom, which describe orbital motion, do not depend on each other.
As an example, we will limit ourselves to the case when the only internal characteristic taken into account by the function is the spin of the particle, and this spin is equal to 1/2. A particle with such a spin can be in one of two states - with a spin projection on the z axis equal to +1/2 (spin up), and with a spin projection on the z axis equal to -1/2 (spin down). This duality is described by a spin function taken in the form of a two-component spinor:

Then the wave function Ψ +1/2 = χ +1/2 ψ will describe the motion of a particle with spin 1/2 directed upward along a trajectory determined by the function ψ, and the wave function Ψ -1/2 = χ -1/2 ψ will describe the movement along the same trajectory of the same particle, but with the spin directed downward.
In conclusion, we note that in quantum mechanics states are possible that cannot be described using the wave function. Such states are called mixed and are described within the framework of a more complex approach using the concept of a density matrix. The states of a quantum system described by the wave function are called pure.