§4 Wave function and its physical meaning. Wave function and its physical meaning Physical meaning of the electron wave function

Wave function
Wave function

Wave function (or state vector) is a complex function that describes the state of a quantum mechanical system. Her knowledge allows you to get the most complete information about the system, fundamentally attainable in the microworld. So with its help it is possible to calculate all measurable physical characteristics of the system, the probability of its stay in a certain place in space and 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).
The square of the modulus of the wave function of an individual particle | ψ (, t) | 2 \u003d 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 of volume dv \u003d dxdydz around the 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. So the average observed value of the physical quantity F for the system is given by the expression

,

where is the operator of this quantity and the integration is carried out over the entire region of the multidimensional space.
Instead of the coordinates of particles x, y, z, as independent variables of the wave function, their momenta p x, p y, p z or other sets of physical quantities can be chosen. This choice depends on the representation (coordinate, pulse, 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 motion 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 set 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 function of orbital motion ψ and the internal function φ:

since usually the internal characteristics of a particle and its degrees of freedom, describing the orbital motion, do not depend on each other.
As an example, we restrict 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 1/2. A particle with such a spin can be in one of two states - with the spin projection on the z axis equal to +1/2 (spin up) and with the 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 \u003d χ +1/2 ψ will describe the motion of a particle with spin 1/2 directed upward along the trajectory determined by the function ψ, and the wave function -1/2 \u003d χ -1/2 ψ will describe the motion along the same trajectory of the same particle, but with a downward spin.
In conclusion, we note that in quantum mechanics such states are possible that cannot be described using the wave function. Such states are called mixed and they 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.

Experimental confirmation of de Broglie's idea of \u200b\u200bthe universality of wave-particle dualism, the limited application of classical mechanics to micro-objects, dictated by the uncertainty relation, as well as the contradiction of a number of experiments with those used at the beginning of the 20th century. theories led to a new stage in the development of quantum theory - the creation of quantum mechanics, which describes the laws of motion and interaction of microparticles, taking into account their wave properties.

In quantum mechanics, the state of microparticles is described using wave function, which is the main carrier of information about their corpuscular and wave properties... The probability of finding a particle in an element of volume dV equals

dW\u003d │Ψ│ 2 dV. (33.6)

The quantity │Ψ│ 2 \u003d dW / dV- makes sense of the probability density, i.e. determines the probability of finding a particle in a unit volume in the vicinity of a point with coordinates x, at, z... Thus, it is not the Ψ-function itself that has physical meaning, but the square of its modulus | Ψ | 2, which sets the intensity of the de Broglie waves.

The probability of finding a particle at time t in a finite volume V, equals

W \u003d \u003d │Ψ │ 2 dV. (33.7)

Because │ Ψ │ 2 dV is defined as a probability, then it is necessary to normalize the wave function Ψ so that the probability of a reliable event turns into unity, if the volume V accept the infinite volume of all space. This means that under this condition, the particle must be somewhere in space. Therefore, the normalization condition for the probabilities

│ Ψ │ 2 dV=1, (33.8)

where this integral (8) is calculated over the entire infinite space, i.e., over the coordinates x, at, z from - ∞ to ∞. Function Ψ must be finite, unambiguous , and continuous.

Schrödinger equation

The equation of motion in quantum mechanics, which describes the motion of microparticles in different fields of force, must be an equation from which the wave properties of particles would follow. It should be an equation for the wave function Ψ ( x, at, z, t), since the quantity │ Ψ │ 2 determines the probability of the particle being in the volume at the moment of time.

The main equation was formulated by E. Schrödinger: the equation is not derived, but postulated.

Schrödinger equation looks like:

- ΔΨ + U(x,y, z, t)Ψ \u003d iħ, (33.9)

where ħ \u003d h /(2π ), t-particle mass, Δ-Laplace operator , i- imaginary unit, U(x,y,z,t) is the potential function of a particle in a force field in which it moves, Ψ ( x,y, z, t) is the desired wave function of the particle.

Equation (32.9) is general Schrödinger equation... It is also called the time-dependent Schrödinger equation. For many physical phenomena occurring in the microworld, equation (33.9) can be simplified by eliminating the dependence of Ψ on time, in other words, to find the Schrödinger equation for stationary states - states with fixed values \u200b\u200bof energy. This is possible if the force field in which the particle moves is stationary, i.e., the function U(x,y,z,t) does not explicitly depend on time and has the meaning of potential energy.

∆ Ψ + ( E-U) Ψ \u003d 0. (33.10)

Equation (33.10) is called the Schrödinger equation for stationary states.

This equation includes the total energy as a parameter E particles. The solution to the equation does not take place for any values \u200b\u200bof the parameter E, but only with a certain set typical for a given task. These energy values \u200b\u200bare called eigenvalues. Eigenvalues E can form both continuous and discrete series.

The equation taking into account the wave and corpuscular properties of a particle was obtained by Schrödinger in 1926.

Schrödinger compared the motion of a particle to a complex function of coordinates and time, which is called a function, this function is a solution to the Schrödinger equation:

Where Laplace that you can

paint: ;; U-potential energy of the particle; Where is a function of coordinates and time.

In quantum physics, it is impossible to accurately predict any events, but we can only talk about the probability of a given event, the probability of events determines.

1) The probability of finding a microparticle in the volume dV at time T:

Related functions.

2) The probability density of finding a particle per unit volume:

3) The wave function must satisfy the condition:

where 3 integrals are calculated over the entire volume where the particle can be located.

This condition means that the probing of a particle is a reliable event with a probability of 1

25 Schrödinger equation for stationary states. Conditions imposed on the wave function. Wave function normalization.

For some practical problems, the potential energy of a particle does not depend on time. In this case, the wave function can be represented as the product

since depends only on time, then we divide it into:

The left side of the equality depends only on time, the right side only on the coordinates, this equality is true only if both sides \u003d const, such a constant is the total energy of the particle E.

Consider the right side of this equality: , we transform: - the equation for the stationary state.

Consider the left side of the Schrödinger equation: ;;

divide the variables, integrate the resulting equation:

using mathematical transformations:

In this case, the probability of finding a particle can be determined:

Or after the transformations:

- this probability does not depend on time, this equation, which characterizes microparticles, is called the stationary state of a particle.

Usually it is required that the wave function be defined and continuous (infinite number of times differentiable) in the entire space, and also that it be single-valued. One kind of ambiguity of wave functions is acceptable - ambiguity of the sign "+ /".

The wave function, 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 all space is equal to unity. In the general case, integration should be carried out over all variables on which the wave function in this representation depends.

26 A particle in a one-dimensional rectangular potential well of infinite depth. Energy quantization. Bohr's principle of correspondence.

Consider the motion of a microparticle along the x axis in a potential field.

Such a potential field corresponds to an infinitely deep potential well with a flat bottom. An example of motion in a potential well is the motion of an electron in a metal. But for the electron to escape from the metal, it is necessary to do work, which corresponds to the potential energy in the Schrödinger equation.

Under this condition, the particle does not penetrate beyond the "well";

y (0) \u003d y (l) \u003d 0 Within the well (0 will reduce to the equation

or this equation is a differential equation and, according to mathematics, its solution is where it can be determined from the boundary conditions.

n-principal quantum number n \u003d 1,2,3 ...

Analysis of this equation shows that energy in a potential well cannot be a discrete quantity.

the state with min energy is called the ground state, all others are excited.

Consider since the potential well is one-dimensional, then we can write that, in the place we substitute in the expression and we get. Since a one-dimensional potential well with a flat bottom, then

Let's graphically depict

The figure shows that the probability of a microparticle being in different places of the segment is not the same, with an increase in n, the probability of finding a particle increases

Energy quantization is one of the key principles required to understand the structural organization of matter, i.e. the existence of stable, repeating in their properties, molecules, atoms and smaller structural units, of which both matter and radiation are composed.

The principle of energy quantization states that any system of interacting particles capable of forming a stable state - be it a piece of a solid, a molecule, an atom or an atomic nucleus - can do this only at certain values \u200b\u200bof energy.

In quantum mechanics, the correspondence principle is the statement that the behavior of a quantum mechanical system tends to classical physics in the limit of large quantum numbers. This principle was introduced by Niels Bohr in 1923.

The rules of quantum mechanics are very successfully applied to describe microscopic objects such as atoms and elementary particles. On the other hand, experiments show that various macroscopic systems (spring, capacitor, etc.) can be described quite accurately in accordance with classical theories, using classical mechanics and classical electrodynamics (although there are macroscopic systems demonstrating quantum behavior, for example, superfluid liquid helium or superconductors). However, it is quite reasonable to believe that the ultimate laws of physics should be independent of the size of the physical objects being described. This is a prerequisite for Bohr's correspondence principle, which states that classical physics should emerge as an approximation to quantum physics as systems grow large.

The conditions under which quantum and classical mechanics coincide are called the classical limit. Bohr proposed a crude criterion for the classical limit: a transition occurs when the quantum numbers describing the system are large, meaning either the excitation of the system to large quantum numbers, or that the system is described by a large set of quantum numbers, or both. A more modern formulation says that the classical approximation is valid for large values \u200b\u200bof the action

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

| ψ (t)⟩ \u003d ∫ Ψ (x, t) | x⟩ d x (\\ displaystyle \\ left | \\ psi (t) \\ right \\ rangle \u003d \\ int \\ Psi (x, t) \\ left | x \\ right \\ rangle dx)

where | x⟩ \u003d | x 1, x 2,…, x n⟩ (\\ displaystyle \\ left | x \\ right \\ rangle \u003d \\ left | x_ (1), x_ (2), \\ ldots, x_ (n) \\ right \\ rangle) is the coordinate basis vector, and Ψ (x, t) \u003d ⟨x | ψ (t)⟩ (\\ displaystyle \\ Psi (x, t) \u003d \\ 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, it 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 unity. In the general case, the integration should be performed over all the variables on which the wave function in this representation depends.

Superposition principle of quantum states

The principle of superposition is valid for wave functions, which states that if the 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 the state described by the wave function

Ψ Σ \u003d c 1 Ψ 1 + c 2 Ψ 2 (\\ displaystyle \\ Psi _ (\\ Sigma) \u003d 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 \u003d c 1 Ψ 1 + c 2 Ψ 2 +… + c N Ψ N \u003d ∑ n \u003d 1 N cn Ψ n (\\ displaystyle \\ Psi _ (\\ Sigma) \u003d c_ (1) \\ Psi _ (1) + c_ (2) \\ Psi _ (2) + \\ ldots + (c) _ (N) (\\ Psi) _ (N) \u003d \\ sum _ (n \u003d 1) ^ (N) (c) _ (n) ( \\ Psi) _ (n)).

In this state, the squared modulus of the coefficient c n (\\ displaystyle (c) _ (n)) determines the probability that the measurement will detect the system in the state described by the wave function Ψ n (\\ displaystyle (\\ Psi) _ (n)).

Therefore, for the normalized wave functions ∑ n \u003d 1 N | c n | 2 \u003d 1 (\\ displaystyle \\ sum _ (n \u003d 1) ^ (N) \\ left | c_ (n) \\ right | ^ (2) \u003d 1).

Regularity conditions for 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 regularity conditions of the wave function.

Wave function in various representations uses states in different representations - will match the expression of the same vector in different coordinate systems. The rest of the operations with wave functions will also have analogs 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 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.

Any microparticle is a special kind of formation that combines 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 observed the field of an electron. At the same time, the wave can be divided into parts and then perceive each part separately.

The difference between a microparticle in quantum mechanics and an ordinary microparticle is that it does not have simultaneously definite values \u200b\u200bof coordinates and momentum, therefore the concept of a trajectory for a microparticle loses its meaning.

The distribution of the probability 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 in the volume element dVproportional to
and the volume element dV:

dP =
dV.

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

Wave function
is the main characteristic of the state of micro-objects (micro-particles). With its help, in quantum mechanics, the average values \u200b\u200bof physical quantities can be calculated that characterize a given object in a state described by a wave function
.

3.2. The uncertainty principle

In classical mechanics, the state of a particle is set by coordinates, momentum, energy, etc. These are dynamic variables. A microparticle cannot be described with such dynamic variables. The peculiarity of microparticles is that certain values \u200b\u200bare not obtained for all variables during measurements. For example, a particle cannot have exact coordinate values \u200b\u200bat the same time x and pulse components r x ... Uncertainty of values x and r x satisfies the ratio:

(3.1)

- the smaller the uncertainty of the coordinate Δ x, the greater the uncertainty of the momentum Δ r x , and vice versa.

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

The quantities Δ x and Δ r x are called canonically conjugate. The same canonically conjugate Δ at and Δ r at , etc.

The Heisenberg Uncertainty Principle states: the product of the uncertainties of the values \u200b\u200bof two conjugate variables cannot be in order of magnitude less than Planck's constant ħ.

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

Δ t ~ ħ/ Δ E.

Determine the value of the coordinate x a free-flying microparticle by placing a slit of width Δ xlocated perpendicular to the direction of motion of the particle. Before the particle passes through the slit, its momentum component r x has a precise meaning, r x \u003d 0 (the slit is perpendicular to the momentum vector), so the momentum uncertainty is zero, Δ r x \u003d 0, but the coordinate x particle is completely undefined (Figure 3.1).

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

Indeed, due to diffraction, there is some probability that the particle will move within the angle 2 φ where φ - the angle corresponding to the first diffraction minimum (we neglect the higher-order maxima, since their intensity is small in comparison with the intensity of the central maximum).

Thus, uncertainty appears:

Δ r x =rsin φ ,

but sin φ = λ / Δ x- this is the first minimum condition. Then

Δ r x ~ pλ /Δ x,

Δ xΔ r x ~ pλ = 2πħ ≥ ħ/ 2.

The uncertainty relation indicates the extent to which one can use the concepts of classical mechanics in relation to microparticles, in particular, with what degree of accuracy one can speak about the trajectory of microparticles.

The movement along the trajectory is characterized by certain values \u200b\u200bof the particle velocity and its coordinates at each moment of time. Substituting in the uncertainty relation instead of r x momentum expression
, we have:

–

the greater the particle mass, the less the uncertainty of its coordinates and velocity, the more accurately the notions 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 motion of the particle is inseparable from the motion along the trajectory.

The uncertainty relation is a fundamental proposition of quantum mechanics. It, for example, makes it possible to 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 ratio

Δ rΔ p ≥ ħ/ 2,

and meaning r= 0 is impossible.

The energy of an electron in an atom will be minimal at r \u003d 0 and r \u003d 0; therefore, to estimate the smallest possible energy, we put 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 the quantities included in this ratio, so the factor can be discarded. In this case, we have
, from here p \u003d ħ /r... The energy of an electron in a hydrogen atom

(3.2)

Find rat which the energy E is minimal. We differentiate (3.2) and equate the derivative to zero:

,

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

You might think that with a microscope it would be possible to determine the position of a particle and thereby subvert the principle of uncertainty. However, the microscope will allow determining the position of the particle at best with an accuracy to the wavelength of the light used, i.e. Δ x ≈ λ, but since Δ r \u003d 0, then Δ rΔ x \u003d 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 \u003dk/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 observing a particle with coordinate uncertainty Δ x ≈ λ the impulse uncertainty should be Δ p ≥h/λ .

Multiplying these uncertainties, we get:

–

the uncertainty principle is fulfilled.

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

Δ r x ≥ ħ/ Δ x.

Therefore, the effect of the device on a microparticle cannot be considered small and insignificant, the device changes the state of the micro-object - as a result of the measurement, certain classical characteristics of the particle (momentum, etc.) are specified only within the limits limited by the uncertainty relation.