Dual Nature Lights and substances. De Broil equation.

The coexistence of two serious scientific theories, each of which explained some of the properties of the world, but could not explain others. However, these two theories fully complemented each other.

Shine At the same time has the properties of continuous electromagnetic waves and discrete photons.

The relationship between the corpuscular and wave properties of the light finds a simple interpretation with a statistical approach to the spread of light.

The interaction of photons with a substance (for example, when the light passes through the diffraction grid) leads to the redistribution of photons in the space and the occurrence of the diffraction pattern on the screen. It is obvious that the illumination at different points of the screen is directly proportional to the likelihood of photons entering these points of the screen. But, on the other hand, from the wave representations it can be seen that the illumination is proportional to the intensity of light J, and the one, in turn, is proportional to the square of amplitude A 2. Hence the conclusion: the square of the amplitude of the light wave at some point is the measure of the probability of photons from entering this point.

De Broil equation.

The physical meaning of the relationship of de Broglya: one of physical characteristics Any particles - its speed. The wave is described in length or frequency. The ratio that binds the pulse of the quantum particle p with a wavelength λ, which describes it: λ \u003d H / P where H is a constant plank. In the words, the wave and corpuscular properties of the quantum particle are standardly interrelated.

14) Probabilistic interpretation of the waves de Broil. If you consider an electron particle, so that the electron remains in its orbit, it must have the same speed (or rather, the impulse) at any distance from the kernel. If the electron wave is considered, so that it fit into the orbit of a given radius, it is necessary that the circumference of this orbit is equal to an integer number of the length of its wave. The main physical meaning of the relation of de Broglyl is that we can always determine the allowed pulses or the wavelengths of electrons in orbits. However, the relation de Broglie shows, for most orbits with a specific radius or wave, or a corpuscular description will show that the electron cannot be located at this distance from the kernel.

De Brogly waves are not E.M. or mechanical waves, but are probability waves. The wave module characterizes the probability of finding a particle in space.

The ratio of the uncertainty of Heisenberg.

Δx * Δp x\u003e h / 2

where Δx is the uncertainty (measurement error) of the spatial coordinate of the microparticle, Δp is the uncertainty of the particle pulse on the x axis, and H is a constant plank, equals approximately 6.626 x 10 -34 J · s.

The smaller the uncertainty with respect to one variable (for example, Δx), the more uncertain becomes another variable (ΔV) in fact, if we manage to absolutely define one of the measured values, the uncertainty of the other value will be equal to infinity. Those. If we were able to absolutely accomplish the coordinates of the quantum particle, we would not have the slightest idea of \u200b\u200bits speed.

Schrödinger equation and its meaning.

Schrödinger applied to the concept of probability waves a classical differential equation of the wave function. The Schrödinger equation describes the distribution of the wave of probability of finding a particle in set point space. The peaks of this wave (point of maximum probability) are shown in which space is most likely a particle will be. The aforementioned wave function The probability distribution, denoted by the Greek letter ψ ("PSI), is a solution to the following differential equation (nothing terrible, if it is not clear to you; the main thing is to faith that this equation indicates that the probability behaves like a wave):

where X is the coordinate, H is a constant planet, and M, E and U - respectively, the mass, total energy and potential particle energy.

The picture of the quantum events that the Schrödinger equation gives us is that electrons and others elementary particles behave like waves on the surface of the ocean. Over time, the wave peak (corresponding to the place in which the electron will most likely be) is shifted in space in accordance with the equation describing this wave. That is, what we have traditionally considered a particle, in the quantum world behaves in many ways like a wave.

The statistical interpretation of the de Brogly wave and the ratio of the uncertainties of Heisenberg led to the conclusion that the movement equation in quantum mechanics describing the movement of microparticles in various power fields should be an equation from which the wave properties of particles observed on the experiment. The main equation should be the equation for the wave function Y ( h., y, z, t), Since it is she, or, more precisely, the value | Y | 2, determines the likelihood of a particle stay at the time of time. T. in volume D. V. , i.e. in the area with coordinates h. and x + DX, and y + dy, z and z + dz.. Ta as the desired equation should take into account the wave properties of particles, then it must be wave equation Like an equation describing electromagnetic waves.

Basic equation nonrelativistic quantum mechanics Formulated in 1926 by E. Schrödinger. The Schrödinger equation, as well as all the main equations of physics (for example, Newton's equations in classical mechanics and Maxwell equations for the electromagnetic field), is not displayed, but is postulated. The correctness of this equation is confirmed by agreement with the experiments of the results obtained with its help, which, in turn, gives it the nature of the law of nature. Schrödinger equation has a view

where ћ =h./ (2p), t-mass of a particle, d - Laplace operator i. - Imaginary unit, U (x, y, z, t) - Potential particle function in the power field in which it moves, y (x, y, z, t) - the desired wave function of the particle.

Equation (217.1) is valid for any particle (with spin equal to 0) moving with a small (compared with the speed of light) speed, i.e. at speed v.<<с. Оно дополняется условиями, накладываемыми на волновую функцию: 1) волновая функция должна быть конечной, однозначной и непрерывной производные must be continuous; 3) Function | Y | 2 must be integrated; This condition in the simplest cases is reduced to the fault condition for probabilities (216.3).

To come to the Schrödinger equation, consider a freely moving particle, which, according to the idea of \u200b\u200bDe Brogly, is compared with a flat wave. For simplicity, consider a one-dimensional case. Equation of a flat wave propagating along the axis x, Has appearance , or in a comprehensive record . Consequently, the plane wave de Broglie has the view

(217.2)

(taken into account that w \u003d E / ћ, k \u003d p / ћ). In quantum mechanics, the exponent is taken with a minus sign, but since physical meaning has only | Y | 2, then this (see (217.2)) is insignificant. Then

Using the relationship between energy E.and impulse p (E \u003d P 2 /(2m)) and substituting expressions (217.3), we obtain a differential equation

which coincides with equation (217.1) for the case U \u003d.0 (we considered a free particle). If the particle moves in the power field characterized by potential energy U, That complete energy E. It consists of kinetic and potential energies. Conducting similar arguments and using the relationship between E.and R (for this case P. 2 /(2m.)=E -U.), We will span to the differential equation that coincides with (217.1).

The above arguments should not be perceived as the output of the Schrödinger equation. They only explain how to come to this equation. Proof of the correctness of the Schrödinger equation is consent with the experience of the conclusions to which it leads.

Equation (217.1) is the general Schrödinger equation. It is also called schrödinger equation depending on time. For many physical phenomena occurring in the micrometer, equation (217.1) can be simplified, eliminating the dependence of Y on time, in other words, to find the Schrödinger equation for stationary states - states with fixed energy values. This is possible if the power field in which the particle moves is stationary, i.e. function U \u003d U (x, y, z) It does not depend on time and makes sense of potential energy. In this case, the solution of the Schrödinger equation can be represented as a product of two functions, one of which is the function of only coordinates, the other is only time, and the dependence on time is expressed by a multiplier, so

where E - The total energy of the particles, constant in the case of a stationary field. Substituting (217.4) in (217.1), we get

where, after dividing on a general multiplier and corresponding transformations, we will come to the equation defining the function y:

(217.5)

Equation (217.5) is called Schrödinger Equation for Stationary states. In this equation, the parameter includes full energy E. Particles. In the theory of differential equations, it is proved that such equations have countless decisions, of which, by the imposition of boundary conditions, decisions are taken with physical meaning. For the Schrödinger equation, conditions are the conditions for the regularity of wave functions: the wave functions must be finite, unambiguous and continuous together with their first derivatives. Thus, the real physical meaning has only such solutions that are expressed by regular functions. y.. But regular solutions take place not at any values \u200b\u200bof the parameter E, But only with their defined set characteristic of this task. These energy values \u200b\u200bare called Own. Solutions that match Own energy values \u200b\u200bcalled own functions. Own meanings E. Can form both continuous and discrete row. In the first case talk about continuous, or solid, spectré, in the second - about discrete spectrum.

Schrödinger equation is generally. Schrödinger Equation for Stationary States

Statistical interpretation of de Brogly waves (see § 216) and the ratio of the uncertainty of Heisenberg (see 5 215) led to the conclusion that the equation of movement in quantum mechanics describing the movement of microparticles in various power fields should be the equation from which the observed on Experience the wave properties of particles. The main equation should be a equation for the wave function ψ (x, y, z, t), as it is it, or, more precisely, the value | ψ | 2, determines the probability of residence particles at time t in the volume of DV, i.e. in the region with coordinates x and x + dx, y and + dy, z and z + dz. Since the desired equation should take into account the wave properties of particles, it should be a wave equation, like an equation describing electromagnetic waves.

The main equation of nonrelativistic quantum mechanics was formulated in 1926 by E. Schrödinger. The Schrödinger equation, as well as all the main equations of physics (for example, Newton's equations in classical mechanics and Maxwell equations for the electromagnetic field), is not displayed, but is postulated. The correctness of this equation is confirmed by agreement with the experiments of the results obtained with its help, which, in turn, gives it the nature of the law of nature. Schrödinger equation has a view

where H \u003d h / (2π), the M-mass of the particle, δ-operator of Laplace ( ),

i - imaginary unit, u (x, y, z, t) - The potential function of the particle in the power field in which it moves, ψ (x, y, z, t ) - The desired wave particle function.

Equation (217.1) is valid for any particle (with spin equal to 0; see § 225) moving with a small (compared with the speed of light) speed, i.e. at the speed υ<<с. Оно дополняется условиями, накладываемыми на волновую функцию: 1) волновая функция должна быть конечной, однозначной и непрерывной (см. § 216); 2) производные

must be continuous; 3) Function | ψ | 2 must be integrated; This condition in the simplest cases is reduced to the fault condition for probabilities (216.3).

To come to the Schrödinger equation, consider a freely moving particle, which, according to the idea of \u200b\u200bDe Brogly, is compared with a flat wave. For simplicity, consider a one-dimensional case. The equation of a flat wave propagating along the x axis has the form (see § 154)

Or in a comprehensive record . Consequently, the plane wave de Broglie has the view

(217.2)

(It is taken into account that ω \u003d e / h, k \u003d p / h). In quantum mechanics, the exponent is taken with a minus sign, but since physical meaning has only | ψ | 2, then this (see (217.2)) is insignificant. Then

,

; (217.3)

Using the relationship between Energy E and Pulse P (E \u003d P 2 / (2M)) and substituting the expressions (217.3), we obtain a differential equation

which coincides with equation (217.1) for the case of U \u003d 0 (MI considered a free particle).

If the particle moves in a power field, characterized by the potential energy U, then the total energy E consists of kinetic and potential energies. Conducting similar arguments using the relationship between EI P (for a given case p 2 / (2m) \u003d E -U), spinning to the differential equation that coincides with (217.1).

The above arguments should not be perceived as the output of the Schrödinger equation. They only explain how to come to this equation. Proof of the correctness of the Schrödinger equation is consent with the experience of the conclusions to which it leads.

Equation (217.1) is the common Schrödinger equation. It is also called the Schrödinger equation, depending on time. For many physical phenomena occurring in the micrometer, equation (217.1) can be simplified, eliminating the dependence ψ on other words, find the Schrödinger equation for stationary state - a state with fixed energy values. This is possible if the power field in which the particle moves is stationary, i.e. the function u \u003d U (x, y, z ) It does not depend on time and makes sense of potential energy. In this case, the solution of the Schrödinger equation can be represented as a product of two functions, one of which is the function of only coordinates, the other is only time, and the dependence on time is expressed by a multiplier

,

where E. - The total energy of the particles, constant in the case of a stationary field. Substituting (217.4) in (217.1), we get

where, after dividing on the general factor E - I (E / H) T T and the corresponding transformations, we will come to the equation defining the function ψ:

(217.5)

Equation (217.5) is called the evident of Schrödinger for stationary states.

In this equation, the total energy E particle includes the parameter. In the theory of differential equations, it is proved that such equations have countless decisions, of which, by the imposition of boundary conditions, decisions are taken with physical meaning. For the Schrödinger equation, conditions are the conditions for the regularity of wave functions: the wave functions must be finite, unambiguous and continuous together with their first derivatives. Thus, the actual physical meaning has only such solutions that are expressed by regular functions ψ . But regular solutions take place not at any values \u200b\u200bof the parameter E, but only with their defined set characteristic of this task. These energy values \u200b\u200bare called. Decisions that are convened for the common energy values \u200b\u200bare called common features. Own values \u200b\u200bE can form both continuous and discrete row. In the first case, they speak a continuous, or solid, spectrum, in the second - about the discrete spectrum.

Schrödinger's equation is named after the Austrian physics of Erwin Schrödinger (E. Schrödinger). This is the main theoretical instrument of quantum mechanics. In quantum mechanics, the Schrödinger equation plays the same role as the movement equation (Newton's second law) in the mechanics of classical. Schrödinger equation is written for the so-called y. - Functions (psi - functions). In the general case, the PSI function is the function of coordinates and time: y. = y. (x, Y, Z, T). If the microparticle is in a stationary state, then the PSI is not dependent on time: y.= y. (x, Y, Z).

In the simplest case of one-dimensional movement of microparticles (for example, only along the axis x. ) The Schrödinger equation has the form:

where y (x) - PSI - function depending only from one coordinate x. ; m. – particle mass; - Permanent Planck (\u003d h / 2π.); E. - Full particle energy, U. - potential energy. In classical physics, the magnitude (E -U. ) it would be equal to the kinetic particle energy. In quantum mechanics due to ratios of uncertainty The concept of kinetic energy is deprived of meaning. Note that potential energy U.- This is characteristic external power fieldin which the particle is moving. This value is quite definite. It is also a function of coordinates, in this case U. = U. (x, y, z).

In a three-dimensional case when y. = y. (x, y, z),instead of the first term in the Schrödinger equation, the sum of three partial derivatives from Psi-function in three coordinates should be recorded.

What is the Schrödinger equation? As already noted, this is the main equation of quantum mechanics. If it is written and decide (which is generally not a simple task) for a particular microparticle, then we will obtain the value of the Psi-function at any point of the space in which the particle moves. What does it give? Psi-function module square characterizes probabilityparticle detection in a particular area of \u200b\u200bspace. Take some point in space with coordinates x. , y. , z. (Fig. 6). What is the probability of detecting a particle at this point? Answer: This probability is zero! (The point does not have the sizes, to get to the point of the particle simply can not physically). So the question is incorrect. We will put it differently: what is the probability of detecting a particle in a small area of \u200b\u200bspace dV \u003d DX DY DZ With the center in the selected point? Answer:

where dP. - elementary probability to detect a particle in the elementary volume dV . Equation (22) is valid for a valid psi-function (it can also be complex, in this case to equation (22) it is necessary to substitute the square of the Psi-function module). If the space area has a finite volume V. That probability P. detect a particle in this volume is the integration of expression (22) by volume V. :

Recall that probabilistic description of the movement of microparticles - The main idea of \u200b\u200bquantum mechanics. Thus, with the help of the Schrödinger equation, the main task of quantum mechanics is solved: a description of the movement of the object under study, in this case a quantum-mechanical particle.

We note a number of important circumstances. As can be seen from formula (21), the Schredinger equation is a second-order differential equation. Consequently, in the process of its decision there will be two arbitrary constants. How to find them? For this use the so-called border conditions: From the specific content of the physical problem, the value of the Psi-function should be known at the boundaries of the microparticle movement area. In addition, the so-called the condition is normalizationTo which the Psi Function must satisfy:

The meaning of this condition is simple: the probability of detecting a particle at least somewhere inside the region of its movement is a reliable event, the likelihood of which is equal to one.

It is the boundary conditions that fill the solution of the Schrödinger equation with a physical meaning. Without these conditions, the solution of the equation is a purely mathematical task, devoid of physical meaning. In the following section, on a specific example, the application of boundary conditions and the normalization conditions in solving the Schrödinger equation are considered.

Psi-function

Wave function (status function, psi-function, amplitude probability) - comprehensive functionused in quantum mechanics for probabilistic description State quantum mechanical system. In a broad sense - the same as status vector.

The option of the "probability amplitude" name is associated with statistical interpretation Wave Function: The probability density of finding a particle at a given point of space is currently equal to the square of the absolute value of the wave function of this state.

Physical meaning of the square of the wave function module

The wave function depends on the coordinates (or generalized coordinates) of the system and, in the general case, on time, and is formed so that square her module Represented density probability (For discrete spectra - just probability) detect the system in the position described by the coordinates at the time of time:

Then in a given quantum state of the system described by the wave function, it is possible to calculate the likelihood that the particle will be detected in any area of \u200b\u200bthe space of the final volume: .

The set of coordinates that act as arguments function, represents full set of physical quantitiesthat can be measured in the system. In quantum mechanics it is possible to choose several complete sets of values, so the wave function of the same state can be recorded from different arguments. The full set of values \u200b\u200bis selected to record the wave function. view of the wave function. So, possible coordinate representation, pulse Presentation, B. quantum field theory Used secondary quantization and presentation of filling numbers or presentation of Fok. and etc.

If the wave function, for example, an electron in the atom, is specified in the coordinate representation, the square of the wave function module is a probability density to detect an electron at a particular point of space. If the same wave function is specified in a pulse view, the square of its module is the probability density to detect one or another impulsefrom.

Let's make a drawing

In our problem, the function u (x) has a special, discontinuous look: it is zero between the walls, and at the edges of the pit (on the walls) addresses infinity:

We write the Schrödinger equation for stationary particle states at points located between the walls:

or, considering the formula (1.1)

To equation (1.3), it is necessary to add boundary conditions on the walls of the pit. We take into account that the wave function is associated with the probability of finding particles. In addition, under the conditions of the task outside the particle walls, the particle cannot be detected. Then the wave function on the walls and beyond the limits should be in zero, and the boundary conditions of the task take a simple form:

Now proceed to solve equation (1.3). In particular, it is possible to take into account that his decision is the de Broglie waves. But one wave of de-broille as a solution, it is clearly not applicable to our task, as it knows that it describes the free particle, "running" in one direction. We have a particle runs "there and here" between the walls. In such a case, on the basis of the principle of superposition, the desired decision can be attempted to represent in the form of two waves of de Broglie, traveling to each other with the pulses of P and -P, that is, in the form:

Permanent and can be found from one of the boundary conditions and normalization conditions. The latter says that if you add all the probabilities, it is to find the probability of detection of an electron between the walls in general in (any place), then the unit will be obtained (the probability of a reliable event is 1), i.e.:

According to the first boundary condition, we have:

Thus, we obtain the solution of our task:

As known, . Therefore, the solution found can be rewritten in the form:

Constant A is determined from the normalization condition. But here it is not of particular interest. The second boundary condition remained unused. What result does it allow to get? With regard to the solution found (1.5), it leads to an equation:

It sees that in our problem, the Pulse P can take not any values, but only values

By the way, n cannot be zero, since the wave function then everywhere on the interval (0 ... L) was zero! This means that the particle between the walls can not be alone! She must move. In similar conditions there are electrons conduction in metal. The resulting output applies to them: electrons in metal cannot be fixed.

The smallest possible pulse of the moving electron is equal

We indicated that the email pulse when reflected from the walls changes the sign. Therefore, to the question, what is the pulse of the electron when it is locked between the walls, it is impossible to answer it: then whether + p, or -p. The impulse is uncertain. Its degree of uncertainty is obviously determined by this: \u003d p - (- p) \u003d 2p. The uncertainty of the coordinate is equal to L; If you try to "catch" an electron, then it will be detected within the walls between the walls, but where exactly is unknown. Since the smallest value p is equal, we get:

We confirmed the ratio of Heisenberg in the conditions of our task, that is, subject to the existence of the smallest value p. If I keep in mind an arbitrary possible value of the impulse, the ratio of uncertainty gets the following form:

This means that the initial postulate of Heisenberg-Bow on uncertainty and establishes only the lower border of uncertainties possible during measurements. If the system has been endowed with minimal uncertainty, then over time they can grow.

However, formula (1.6) indicates to another extremely interesting conclusion: it turns out that the system's impulse in quantum mechanics is not always able to change continuously (as it always takes place in classical mechanics). The spectrum of the particle pulse in our example is discrete, the pulse of the particles between the walls can vary only by jumps (quanta). The magnitude of the jump in the considered problem is constant and equal.

In fig. 2. The spectrum of possible values \u200b\u200bof the particle pulse is clearly depicted. Thus, the discreteness of changes in mechanical values, completely alien to the classical mechanics, in quantum mechanics flows from its mathematical apparatus. When the question is why the impulse changes with jumps, it is impossible to find visual. These are the laws of quantum mechanics; Our conclusion flows out of them logically - this is all an explanation.

We now turn to the particle energy. Energy is associated with a pulse of formula (1). If the spectrum of the pulse is discrete, then automatically it turns out that the spectrum of the energy of the particle between the walls is discrete. And it is elementary. If possible values \u200b\u200baccording to formula (1.6) substitute in formula (1.1), we obtain:

where n \u003d 1, 2, ..., and is called a quantum number.

Thus, we got energy levels.

fig. 3.

Fig. 3 depicts the location of the energy levels corresponding to the conditions of our task. It is clear that for another task, the location of the energy levels will be different. If the particle is charged (for example, it is an electron), then, being not at the lowest energy level, it will be able to spontaneously emit light (as a photon). At the same time, it will turn to a lower energy level in accordance with the condition:

Wave functions for each stationary state in our problem are sinusoids, the zero values \u200b\u200bof which are necessarily falling on the walls. Two such wave functions for n \u003d 1,2 are shown in Fig. one.