Stationary solutions of the Schrödinger equation.
Appendix A
Finding a solution to the Schrödinger equation for a free electron in the form of a wave packet .
Let us write the Schrödinger equation for a free electron
After transformations, the Schrödinger equation takes the form
(A.2)
We solve this equation with the initial condition
(A.3)
Here - wave function electron at the initial moment of time. We are looking for a solution to equation (A.2) in the form of the Fourier integral
(A.4)
We substitute (A.4) into (A.2) and get
Solution (A.4) can now be written in the following form
(A.6)
We use initial condition(A.3), and from (A.6) we obtain the expansion of the initial wave function of the electron into the Fourier integral.
(A.7)
To expression (A.7) we apply inverse conversion Fourier
(A.8)
Let us summarize the transformations made. So, if the wave function of the electron at the initial moment of time is known, then after integration (A.8) we find the coefficients. Then, after substituting these coefficients into (A.6) and integrating, we obtain the wave function of the electron at an arbitrary moment in time at any point in space.
For some distributions, integration can be performed explicitly and an analytical expression for solving the Schrödinger equation can be obtained. As the initial wave function, we take the Gaussian distribution modulated by a plane monochromatic wave.
Here is the average electron momentum. Choosing the initial wave function in this form will allow us to obtain a solution to the Schrödinger equation in the form of a wave packet.
Let us consider in detail the properties of the initial wave function (A.9).
Firstly, the wave function is normalized to unity.
(A.10)
Normalization (A.10) is easily proven using the following table integral.
(A.11)
Secondly, if the wave function is normalized to unity, then the squared modulus of the wave function is the probability density of finding an electron at a given point in space.
Here the quantity will be called the amplitude of the wave packet at the initial moment of time. The physical meaning of the packet amplitude is the maximum value of the probability distribution. Figure 1 shows a graph of the probability density distribution.
Probability density distribution at the initial time.
Let us note some features of the graph in Fig. 1.
1. Coordinate is a point on an axis x, in which the probability distribution has a maximum value. Therefore, we can say that with the greatest probability it is possible to detect an electron near the point.
2. The value will determine the deviation from the point at which the distribution value decreases by e times the maximum value.
(A.13)
In this case, the quantity is called the width of the wave packet at the initial moment of time, and the quantity is called the half-width of the packet.
3. Calculate the probability of finding an electron in the interval 
.
(A.14)
Thus, the probability of detecting an electron in a region with a center and half-width is 0.843. This probability is close to unity, so usually the region with half-width is spoken of as the region where the electron is located at the initial moment of time.
Third, the initial wave function is not an eigenfunction of the momentum operator. Therefore, an electron in a state with a wave function does not have a specific momentum; we can only talk about the average momentum of the electron. Let's calculate the average electron momentum.
Therefore, the value in formula (A.9) is the average value of the electron momentum. Formula (A.15) is easily proven if you use the table integral (A.11).
Thus, the properties of the initial wave function have been analyzed. Now let's substitute the function into the Fourier integral (A.8) and find the coefficients.
In the integral (A.16) we make the following change of the integration variable.
(A.17)
As a result, integral (A.16) takes next view.
(A.18)
As a result, we obtain the following expression for the coefficients.
(A.18)
Substituting the coefficients into formula (A.6), we obtain the following integral expression for the wave function.
In the integral (A.19) we make the following change of the integration variable.
(A.20)
As a result, integral (A.19) takes the following form.
We finally obtain the formula for the wave packet.
(A.22)
It is easy to see that for the initial moment of time, formula (A.22) turns into formula (A.9) for the initial wave function. Let us find the probability density for function (A.22).
We substitute the wave packet (A.22) into formula (A.23), and as a result we obtain the following expression.
(A.24)
Here the center of the wave packet, or the maximum of the probability density distribution, moves with a speed equal to the following value.
The half-width of the wave packet increases with time and is determined by the following formula.
(A.26)
The amplitude of the wave packet decreases with time and is determined by the following formula.
(A.27)
Thus, the probability distribution for a wave packet can be written in the following form.
(A.28)
In Fig.2. shows the probability distribution at three consecutive points in time.
Probability distribution at three consecutive points in time.
Appendix B
General information on solving the Schrödinger equation .
Introduction.
The motion of a quantum particle is generally described by the Schrödinger equation:
Here i is the imaginary unit, h =1.0546´10 -34 (J×s) is Planck’s constant. Operator Ĥ is called the Hamilton operator. The form of the Hamilton operator depends on the type of interaction of the electron with external fields.
If we do not take into account the spin properties of the electron, for example, if we do not consider the motion of the electron in a magnetic field, then the Hamilton operator can be represented in the form.
(B.2)
Here is the operator kinetic energy:
, (B.3)
Where m=9.1094´10 -31 (kg) – electron mass. Potential energy describes the interaction of an electron with an external electric field.
In this laboratory work we will consider the one-dimensional motion of an electron along the axis x. The Schrödinger equation in this case takes the following form:
. (B.4)
Equation (B.4) is, from a mathematical point of view, a partial differential equation for the unknown wave function Y=Y(x,t). It is known that such an equation has a definite solution if the appropriate initial and boundary conditions are given. The initial and boundary conditions are selected based on the specific physical problem.
Let, for example, an electron move from left to right with some average momentum p 0 . In addition, at the initial time t=0, the electron is localized in a certain region of space x m -d< x < x m +d. Здесь x m – центр области локализации электрона, а d – эффективная полуширина этой области.
In this case, the initial condition will look like this:
. (B.5)
Here Y 0 (x) is the wave function at the initial time. The wave function is a complex function, so it is convenient to graphically represent not the wave function itself, but the probability density.
The probability density of finding an electron in a given place at a given time is expressed through the wave function as follows:
Note that the probabilities must be normalized to unity. From here we obtain the normalization condition for the wave function:
. (B.7)
Probability density distribution at the initial time
, (B.8)
can be depicted graphically. In Fig.3. the possible location of the electron at the initial moment of time is shown.
The location of the electron at the moment t=0.
From this figure it is clear that with the greatest probability the electron is located at point x m. Letter A we will denote the amplitude (maximum value) of the probability distribution. This figure also shows how the width 2d or half-width d of the distribution is determined. If the distribution has an exponential or Gaussian character, then the width of the distribution is determined at a level in e times less than the maximum value.
In Fig.3. the vector of the average electron momentum is shown. This means that the electron is moving from right to left, and the probability distribution will also move from right to left. In Fig.2. shows the probability distribution at three consecutive points in time. In Fig.2. it can be seen that the maximum of the distribution x m (t) moves from left to right.
In Fig.2. one can notice that the movement of an electron from right to left is accompanied by a deformation of the probability density distribution. Amplitude A(t) decreases, and the half-width d(t) increases. All of the above details of the electron's motion can be obtained by solving the Schrödinger equation (B4) with the initial condition (B.5).
Summary . Depending on the formulation of the physical problem, the form of the Schrödinger equation may change. When researching certain physical phenomena described by the Schrödinger equation, the necessary initial and boundary conditions are selected to find a solution to the Schrödinger equation.
Stationary solutions of the Schrödinger equation.
If an electron moves in a time-constant external field, then its potential energy will not depend on time. In this case, one of possible solutions Schrödinger equation (B.4) is a time-separable solution t and along the x coordinate.
We use a technique known in mathematics for solving differential equations. We look for a solution to equation (B.4) in the form:
. (B.9)
We substitute (B.9) into equation (B.4) and obtain the following relations:
. (B.10)
Here E– a constant, which in quantum mechanics is given the meaning of the total energy of an electron. Relations (B.10) are equivalent to the following two differential equations:
. (B.11)
The first equation in system (B.11) has the following common decision:
Here C is an arbitrary constant. We substitute (B.12) into expression (B.9) and obtain a solution to the Schrödinger equation (B.4) in the form:
, (B.13)
where is the function y(x) satisfies the equation.
(B.14)
Constant C contained in the function y(x).
The solution to the Schrödinger equation (B.4) in the form of expression (B.13) is called stationary solution of the Schrödinger equation. Equation (B.14) is called stationary Schrödinger equation. Function y(x) is called wave function, independent of time.
The state of the electron, which is described by the wave function (B.13), is called stationary state. Quantum mechanics states that in a stationary state an electron has certain energy E.
The results obtained can be generalized to the Schrödinger equation (B.1) for three-dimensional electron motion. If the Hamilton operator Ĥ does not depend explicitly on time, then one of the possible solutions to the Schrödinger equation (B.1) is a stationary solution of the following form:
, (B.15)
where the wave function satisfies the stationary Schrödinger equation.
(B.16)
Note that equations (B.14) and (B.16) in quantum mechanics also have this name. These equations are equations for native functions And eigenvalues Hamilton operator. In other words, by solving equation (B.16) we find the energies E(eigenvalues of the Hamilton operator) and the corresponding wave functions (eigenfunctions of the Hamilton operator).
Summary . Stationary solutions of the Schrödinger equation are a certain class of solutions from a huge set of other solutions of the Schrödinger equation. Stationary solutions exist if the Hamiltonian operator does not depend explicitly on time. In a stationary state, an electron has a certain energy. To find possible values energy, it is necessary to solve the stationary Schrödinger equation.
Wave packet.
It is easy to see that stationary solutions of the Schrödinger equation do not describe the motion of a localized electron, as shown in Fig. 1 and Fig. 2. Indeed, if we take the stationary solution (B.13) and find the probability distribution, we will obtain a function independent of time.
(B.17)
This is not surprising; the stationary solution (B.13) is one of the possible solutions to the partial differential equation (B.4).
But what’s interesting is that due to the linearity of the Schrödinger equation (B.4) with respect to the wave function Y(x,t), for solutions of this equation the principle of superposition is satisfied. For stationary states, this principle states the following. Any linear combination of stationary solutions (with different energies E) of the Schrödinger equation (B.4) is also a solution of the Schrödinger equation (B.4).
To give a mathematical expression for the superposition principle, we need to say a few words about the energy spectrum of the electron. If the solution to the stationary Schrödinger equation (B.14) has a discrete spectrum, this means that equation (B.14) can be written as follows:
(B.18)
where the index n runs through, generally speaking, an infinite series of values n=0,1,2,¼. In this case, the solution to the Schrödinger equation (B.4) can be represented as a sum of stationary solutions.
(B.19)
In quantum mechanics it is proven that the eigenfunctions y n(x) of a discrete spectrum can be made an orthonormal system of functions. This means that the following normalization condition is satisfied.
(B.20)
Here d n m is the Kronecker symbol.
y n (x) is orthonormal, then the coefficients C n in sum (B.19) have a simple physical meaning. Square modulus of coefficient C n is equal to the probability that an electron in a state with wave function (B.19) has energy E n.
The most important thing in this statement is that an electron in a state with wave function (B.19) does not have a specific energy. When measuring energy, this electron can have any energy from the set with probability (B.21).
Therefore, they say that an electron can have one or another energy with a probability determined by formula (B.21).
An electron that is in a stationary state and has a certain energy will be called monochromatic electron. An electron that is not in a stationary state and therefore does not have a certain energy will be called non-monochromatic electron.
If the solution to the stationary Schrödinger equation (B.14) has a continuous spectrum, this means that equation (B.14) can be written as follows:
, (B.22)
where is the energy E takes values on some continuous interval [ E min, E max]. In this case, the solution to the Schrödinger equation (B.4) can be represented as an integral of stationary solutions.
(B.23)
Eigenfunctions of the continuous spectrum y In quantum mechanics, E (x) is usually normalized to the d-function:
, (B.24)
The definition of the d-function is contained in the following integral relations:
To visualize the behavior of the d-function, the following description of this function is given:
So, if the system of functions y E (x) is normalized to the d-function, then the square of the modulus of the coefficient C(E) in the integral (B.23) equal to density the probability that an electron in a state with wave function (B.19) has energy E.
The wave function Y(x,t) presented as a sum (B.19) or as an integral (B.23) of stationary solutions of the Schrödinger equation is called wave packet.
Thus, the state of a non-monochromatic electron is described by a wave packet. One can also say this: the states of a monochromatic electron with their weight factors contribute to the state of a non-monochromatic electron.
In Fig.1. and Fig.2. The electron wave packets are depicted at different times.
Summary . The state of a non-monochromatic electron is described by a wave packet. A non-monochromatic electron does not have a specific energy. A wave packet can be represented as a sum or integral of wave functions of stationary states with their own energies. The probability that a non-monochromatic electron has one or another energy from this set of energies is determined by the contribution of the corresponding stationary states to the wave packet.
Free movement. General solution of the Schrödinger equation.
Depending on the field with which the electron interacts, the solution to the stationary Schrödinger equation (B.14) may have different type. This lab examines free movement. Therefore, in equation (B.14) we set the potential energy to zero. As a result, we get the following equation:
, (B.26)
the general solution to this equation has the following form:
. (B.27)
Here C 1 and C 2 are two arbitrary constants, k has the meaning of a wave number.
Now, using expression (B.23), we write down the general solution of the Schrödinger equation for free motion. We substitute function (B.27) into integral (B.23). At the same time, we take into account that the limits of integration over energy E for free movement are selected from zero to infinity. As a result, we get the following expression:
In this integral it is convenient to move from integration over energy E to integration over wave number k. We will assume that the wave number can take both positive and negative values. For convenience, we introduce the frequency w associated with the energy E, the following relation:
Transforming the integral (B.28), we obtain the following expression for the wave packet:
. (B.30)
Integral (B.30) gives the general solution to the Schrödinger equation (B.4) for free motion. Odds C(k) are found from the initial conditions.
Let's take the initial condition (B.5) and substitute the solution (B.30) there. As a result, we get the following expression:
(B.31)
Integral (B.31) is nothing more than the expansion of the initial wave function into the Fourier integral. Using the inverse Fourier transform, we find the coefficients C(k).
. (B.32)
Summary . By free motion of an electron we mean motion in the absence of an external field in an infinite region of space. If the wave function of the electron at the initial moment of time Y 0 (x) is known, then using formulas (B.32) and (B.30) one can find the general solution of the Schrödinger equation Y(x,t) for the free movement of the electron.
In development of de Broglie's idea about the wave properties of matter, E. Schrödinger received his famous equation in 1926. Schrödinger associated the movement of a microparticle with a complex function of coordinates and time, which he called the wave function and denoted by the Greek letter “psi” (). We will call it the psi function.
The psi function characterizes the state of the microparticle. The form of the function is obtained from the solution of the Schrödinger equation, which looks like this:
Here is the mass of the particle, i is the imaginary unit, is the Laplace operator, the result of the action of which on a certain function is the sum of the second partial derivatives with respect to the coordinates:
The letter U in equation (21.1) denotes the function of coordinates and time, the gradient of which, taken with the opposite sign, determines the force acting on the particle. In the case when the function U does not depend explicitly on time, it has the meaning of the potential energy of the particle.
From equation (21.1) it follows that the form of the psi function is determined by the function U, i.e., ultimately, by the nature of the forces acting on the particle.
The Schrödinger equation is the fundamental equation of non-relativistic quantum mechanics. It cannot be derived from other relations. It should be considered as an initial basic assumption, the validity of which is proven by the fact that all the consequences flowing from it are in the most accurate agreement with experimental facts.
Schrödinger established his equation based on an optical-mechanical analogy. This analogy lies in the similarity of the equations that describe the path of light rays with the equations that determine the trajectories of particles in analytical mechanics. In optics, the path of rays satisfies Fermat’s principle (see § 115 of the 2nd volume); in mechanics, the type of trajectory satisfies the so-called principle of least action.
If the force field in which the particle moves is stationary, then the function V does not explicitly depend on time and, as already noted, has the meaning of potential energy. In this case, the solution to the Schrödinger equation splits into two factors, one of which depends only on the coordinates, the other - only on time:
Here E is the total energy of the particle, which in the case of a stationary field remains constant. To verify the validity of expression (21.3), let us substitute it into equation (21.1). As a result, we obtain the relation
Reducing by a common factor we arrive at differential equation, defining the function
Equation (21.4) is called the Schrödinger equation for stationary states. In what follows we will deal only with this equation and for brevity we will simply call it the Schrödinger equation. Equation (21.4) is often written in the form
Let us explain how one can arrive at the Schrödinger equation. For simplicity, we restrict ourselves to the one-dimensional case. Let's consider a freely moving particle.
According to de Broglie's idea, it needs to be associated with a plane wave
(in quantum mechanics it is customary to take the exponent with a minus sign). Replacing in accordance with (18.1) and (18.2) through E and , we arrive at the expression
Differentiating this expression once with respect to t, and a second time twice with respect to x, we obtain
In non-relativistic classical mechanics, the energy E and the momentum of a free particle are related by the relation
Substituting expressions (21.7) for E and into this relation and then reducing by , we obtain the equation
which coincides with equation (21.1), if in the latter we put
In the case of a particle moving in a force field characterized by potential energy U, energy E and momentum are related by the relation
Extending expressions (21.7) for E to this case, we obtain
Multiplying this ratio by and moving the term to the left, we arrive at the equation
coinciding with equation (21.1).
The stated reasoning has no evidentiary force and cannot be considered as a derivation of the Schrödinger equation. Their purpose is to explain how this equation could be arrived at.
In quantum mechanics, the concept plays an important role. An operator is a rule by which one function (let's denote it) is associated with another function (let's denote it). Symbolically this is written as follows:
Here is a symbolic designation of the operator (with the same success one could take any other letter with a “cap” above it, for example, etc.). In formula (21.2), the role of Q is played by the function F, and the role of f is right part formulas.
Introduction
It is known that the course of quantum mechanics is one of the most difficult to understand. This is due not so much to the new and “unusual” mathematical apparatus, but primarily to the difficulty of understanding the revolutionary, from the standpoint of classical physics, ideas underlying quantum mechanics and the complexity of interpreting the results.
In the majority teaching aids in quantum mechanics, the presentation of the material is based, as a rule, on the analysis of solutions to the stationary Schrödinger equations. However, the stationary approach does not allow one to directly compare the results of solving a quantum mechanical problem with similar classical results. In addition, many processes studied in the course of quantum mechanics (such as the passage of a particle through a potential barrier, the decay of a quasi-stationary state, etc.) are in principle non-stationary in nature and, therefore, can be understood in full only on the basis of solutions to the non-stationary equation Schrödinger. Since the number of analytically solvable problems is small, the use of a computer in the process of studying quantum mechanics is especially relevant.
The Schrödinger equation and the physical meaning of its solutions
Schrödinger wave equation
One of the basic equations of quantum mechanics is the Schrödinger equation, which determines the change in states of quantum systems over time. It is written in the form
where H is the Hamiltonian operator of the system, coinciding with the energy operator if it does not depend on time. The type of operator is determined by the properties of the system. For the nonrelativistic motion of a mass particle in a potential field U(r), the operator is real and is represented by the sum of the operators of the kinetic and potential energy of the particle
If a particle moves in an electromagnetic field, then the Hamiltonian operator will be complex.
Although equation (1.1) is a first-order equation in time, due to the presence of an imaginary unit, it also has periodic solutions. Therefore, the Schrödinger equation (1.1) is often called the Schrödinger wave equation, and its solution is called the time-dependent wave function. Equation (1.1) at known form operator H allows you to determine the value of the wave function at any subsequent time, if this value is known at the initial time. Thus, the Schrödinger wave equation expresses the principle of causality in quantum mechanics.
The Schrödinger wave equation can be obtained based on the following formal considerations. In classical mechanics it is known that if energy is given as a function of coordinates and momentum
then the transition to the classical Hamilton-Jacobi equation for the action function S
can be obtained from (1.3) by the formal transformation
In the same way, equation (1.1) is obtained from (1.3) by passing from (1.3) to the operator equation by formal transformation
if (1.3) does not contain products of coordinates and momenta, or contains products of them that, after passing to operators (1.4), commute with each other. Equating after this transformation the results of the action on the function of the operators of the right and left sides of the resulting operator equality, we arrive at the wave equation (1.1). However, these formal transformations should not be taken as a derivation of the Schrödinger equation. The Schrödinger equation is a generalization of experimental data. It is not derived in quantum mechanics, just as Maxwell’s equations are not derived in electrodynamics, the principle of least action (or Newton’s equations) in classical mechanics.
It is easy to verify that equation (1.1) is satisfied for the wave function
describing the free movement of a particle with a certain value impulse. In the general case, the validity of equation (1.1) is proven by agreement with experience of all conclusions obtained using this equation.
Let us show that equation (1.1) implies the important equality
indicating that the normalization of the wave function persists over time. Let us multiply (1.1) on the left by the function *, a the equation complex conjugate to (1.1) by the function and subtract the second from the first resulting equation; then we find
Integrating this relation over all values of the variables and taking into account the self-adjointness of the operator, we obtain (1.5).
If we substitute into relation (1.6) the explicit expression of the Hamiltonian operator (1.2) for the motion of a particle in a potential field, then we arrive at the differential equation (continuity equation)
where is the probability density, and the vector
can be called the probability current density vector.
The complex wave function can always be represented as
where and are real functions of time and coordinates. Thus, the probability density
and the probability current density
From (1.9) it follows that j = 0 for all functions for which the function Φ does not depend on the coordinates. In particular, j= 0 for all real functions.
Solutions of the Schrödinger equation (1.1) in the general case are represented by complex functions. Using complex functions is quite convenient, although not necessary. Instead of one complex function, the state of the system can be described by two real functions and, satisfying two related equations. For example, if the operator H is real, then by substituting the function into (1.1) and separating the real and imaginary parts, we obtain a system of two equations
in this case, the probability density and probability current density will take the form
Wave functions in impulse representation.
The Fourier transform of the wave function characterizes the distribution of momentum in a quantum state. It is required to derive an integral equation for the potential with the Fourier transform as the kernel.
Solution. There are two mutually inverse relationships between the functions and.
If relation (2.1) is used as a definition and an operation is applied to it, then taking into account the definition of a 3-dimensional -function,
as a result, as is easy to see, we get the inverse relation (2.2). Similar considerations are used below in deriving relation (2.8).
then for the Fourier transform of the potential we have
Assuming that the wave function satisfies the Schrödinger equation
Substituting expressions (2.1) and (2.3) here instead of and, respectively, we obtain
In the double integral, we move from integration over a variable to integration over a variable, and then we again denote this new variable by. The integral over vanishes for any value only in the case when the integrand itself is equal to zero, but then
This is the desired integral equation with the Fourier transform of the potential as the kernel. Of course, the integral equation (2.6) can be obtained only under the condition that the Fourier transform of the potential (2.4) exists; for this, for example, the potential must decrease over large distances at least as, where.
It should be noted that from the normalization condition
equality follows
This can be shown by substituting expression (2.1) for the function into (2.7):
If we first perform integration over here, we can easily obtain relation (2.8).
1. Introduction
Quantum theory was born in 1900, when Max Planck proposed a theoretical conclusion about the relationship between the temperature of a body and the radiation emitted by that body - a conclusion that for a long time eluded other scientists. Like his predecessors, Planck proposed that radiation was emitted by atomic oscillators, but he believed that the energy of the oscillators (and therefore the radiation they emit) existed in the form of small discrete portions, which Einstein called quanta. The energy of each quantum is proportional to the frequency of radiation. Although the formula derived by Planck aroused universal admiration, the assumptions he made remained incomprehensible, since they contradicted classical physics.
In 1905, Einstein used quantum theory to explain some aspects of the photoelectric effect—the emission of electrons by the surface of a metal exposed to ultraviolet light. Along the way, Einstein noted an apparent paradox: light, which for two centuries had been known to travel as continuous waves, could, under certain circumstances, also behave as a stream of particles.
About eight years later, Niels Bohr extended quantum theory to the atom and explained the frequencies of waves emitted by atoms excited in a flame or an electric charge. Ernest Rutherford showed that the mass of the atom is almost entirely concentrated in the central nucleus, which carries a positive electric charge and is surrounded at relatively large distances by electrons carrying a negative charge, as a result of which the atom as a whole is electrically neutral. Bohr suggested that electrons could only be in certain discrete orbits corresponding to different energy levels, and that the “jump” of an electron from one orbit to another, with lower energy, is accompanied by the emission of a photon, the energy of which is equal to the difference in the energies of the two orbits. Frequency, according to Planck's theory, is proportional to the energy of the photon. Thus, Bohr's model of the atom established a connection between the various spectral lines characteristic of the substance emitting radiation and the atomic structure. Despite its initial success, Bohr's model of the atom soon required modifications to resolve discrepancies between theory and experiment. In addition, quantum theory at that stage did not yet provide a systematic procedure for solving many quantum problems.
New essential feature quantum theory emerged in 1924, when de Broglie put forward a radical hypothesis about the wave nature of matter: if electromagnetic waves, such as light, sometimes behave like particles (as Einstein showed), then particles, such as the electron, under certain circumstances, can behave like waves. In de Broglie's formulation, the frequency corresponding to a particle is related to its energy, as in the case of a photon (particle of light), but de Broglie's proposed mathematical expression was an equivalent relationship between the wavelength, the mass of the particle, and its speed (momentum). The existence of electron waves was experimentally proven in 1927 by Clinton Davisson and Lester Germer in the United States and John Paget Thomson in England.
Impressed by Einstein's comments on de Broglie's ideas, Schrödinger attempted to apply the wave description of electrons to the construction of a coherent quantum theory, unrelated to Bohr's inadequate model of the atom. In a certain sense, he intended to bring quantum theory closer to classical physics, which had accumulated many examples of mathematical descriptions of waves. The first attempt, made by Schrödinger in 1925, ended in failure.
The speeds of electrons in Schrödinger's theory II were close to the speed of light, which required the inclusion of Einstein's special theory of relativity and the significant increase in electron mass at very high speeds that it predicted.
One of the reasons for Schrödinger's failure was that he did not take into account the presence of a specific property of the electron, now known as spin (the rotation of the electron around its own axis like a top), about which little was known at that time.
Schrödinger made the next attempt in 1926. This time the electron velocities were chosen so small that there was no need to invoke the theory of relativity.
The second attempt resulted in the conclusion wave equation Schrödinger, who gives a mathematical description of matter in terms of the wave function. Schrödinger called his theory wave mechanics. The solutions of the wave equation were in agreement with experimental observations and had a profound influence on the subsequent development of quantum theory.
Not long before, Werner Heisenberg, Max Born, and Pascual Jordan published another version of quantum theory, called matrix mechanics, which described quantum phenomena using tables of observable quantities. These tables are mathematical sets ordered in a certain way, called matrices, on which, according to known rules, it is possible to perform various mathematical operations. Matrix mechanics also allowed for agreement with observed experimental data, but unlike wave mechanics, it did not contain any specific reference to spatial coordinates or time. Heisenberg especially insisted on the rejection of any simple visual representations or models in favor of only those properties that could be determined from experiment.
Schrödinger showed that wave mechanics and matrix mechanics are mathematically equivalent. Now known collectively as quantum mechanics, these two theories provided a long-awaited common framework for describing quantum phenomena. Many physicists preferred wave mechanics because its mathematics was more familiar to them and its concepts seemed more “physical”; operations on matrices are more cumbersome.
Function Ψ. Probability normalization.
The discovery of the wave properties of microparticles indicated that classical mechanics cannot give correct description behavior of such particles. There was a need to create a mechanics of microparticles that would also take into account their wave properties. The new mechanics created by Schrödinger, Heisenberg, Dirac and others was called wave or quantum mechanics.
Plane de Broglie wave
(1)is a very special wave formation corresponding to free uniform movement particles in a certain direction and with a certain momentum. But a particle, even in free space and especially in force fields, can also perform other movements described by more complex wave functions. In these cases Full description the state of a particle in quantum mechanics is given not by a plane de Broglie wave, but by some more complex complex function
, depending on coordinates and time. It's called the wave function. In the particular case of free motion of a particle, the wave function transforms into a plane de Broglie wave (1). The wave function itself is introduced as an auxiliary symbol and is not one of the directly observable quantities. But its knowledge makes it possible to statistically predict the values of quantities that are obtained experimentally and therefore have a real physical meaning.The wave function determines the relative probability of detecting a particle in different places in space. At this stage, when only probability relations are discussed, the wave function is fundamentally determined up to an arbitrary constant factor. If at all points in space the wave function is multiplied by the same constant (generally speaking, complex) number, different from zero, then a new wave function is obtained that describes exactly the same state. It makes no sense to say that Ψ is equal to zero at all points in space, because such a “wave function” never allows us to conclude about the relative probability of detecting a particle in different places in space. But the uncertainty in determining Ψ can be significantly narrowed if we move from relative probability to absolute probability. Let us dispose of the indefinite factor in the function Ψ so that the value |Ψ|2dV gives the absolute probability of detecting a particle in the space volume element dV. Then |Ψ|2 = Ψ*Ψ (Ψ* is the complex conjugate function of Ψ) will have the meaning of the probability density that should be expected when trying to detect a particle in space. In this case, Ψ will still be determined up to an arbitrary constant complex factor, the modulus of which, however, equal to one. With this definition, the normalization condition must be met:
(2)where the integral is taken over the entire infinite space. It means that the particle will be detected with certainty throughout space. If the integral of |Ψ|2 is taken over a certain volume V1, we calculate the probability of finding a particle in the space of volume V1.
Normalization (2) may be impossible if integral (2) diverges. This will be the case, for example, in the case of a plane de Broglie wave, when the probability of detecting a particle is the same at all points in space. But such cases should be considered as idealizations of a real situation in which the particle does not go to infinity, but is forced to remain in a limited region of space. Then normalization is not difficult.
So, the direct physical meaning is associated not with the function Ψ itself, but with its module Ψ*Ψ. Why in quantum theory do they operate with wave functions Ψ, and not directly with experimentally observed quantities Ψ*Ψ? This is necessary to interpret the wave properties of matter - interference and diffraction. Here the situation is exactly the same as in any wave theory. It (at least in a linear approximation) accepts the validity of the principle of superposition of the wave fields themselves, and not their intensities, and thus achieves inclusion in the theory of the phenomena of wave interference and diffraction. Likewise, in quantum mechanics the principle of superposition of wave functions is accepted as one of the main postulates, which consists in the following.
№1 Stationary equation Schrödinger has the form . This equation is written for...
The stationary Schrödinger equation in the general case has the form
, where is the potential energy of the microparticle. For the one-dimensional case. In addition, the particle cannot be inside the potential box, but outside the box, because its walls are infinitely high. Therefore, this Schrödinger equation is written for a particle in a one-dimensional box with infinitely high walls.
Linear harmonic oscillator
ü Particles in a one-dimensional potential box with infinitely high walls
Particles in a three-dimensional potential box with infinitely high walls
Electron in a hydrogen atom
Establish correspondences between quantum mechanical problems and Schrödinger equations for them.
General form the stationary Schrödinger equation has the form:
Particle potential energy,
Laplace operator. For simultaneous case
The expression for the potential energy of a harmonic oscillator, that is, a particle performing one-dimensional motion under the action of a quasi-elastic force, has the form U=.
The value of the potential energy of an electron in a potential box with infinitely high walls is U = 0. An electron in a hydrogen-like atom has potential energy For a hydrogen atom Z = 1.
Thus, for an electron in a one-dimensional potential box, the Schrödinger equation has the form:
Using the wave function, which is a solution to the Schrödinger equation, we can determine...
Answer options: (Indicate at least two answer options)
Average values of physical quantities characterizing a particle
The probability that a particle is located in a certain region of space
Particle trajectory
Particle location
The value has the meaning of probability density (probability per unit volume), that is, it determines the probability of a particle being in the corresponding place in space. Then the probability W of detecting a particle in a certain region of space is equal to
Schrödinger equation (specific situations)
No. 1The eigenfunctions of an electron in a one-dimensional potential box with infinitely high walls have the form 
where is the width of the box, a quantum number that has the meaning of a number energy level. If the number of function nodes on the segment and , then equals...
Number of nodes, i.e. the number of points at which the wave function on a segment vanishes is related to the number of the energy level by the relation . Then 
, and by condition this ratio is equal to 1.5. Solving the resulting equation for , we find that
Nuclear reactions.
№1 IN nuclear reaction the letter represents the particle...
From the laws of conservation of mass number and charge number it follows that the charge of the particle is zero, and mass number equals 1. Therefore, the letter denotes a neutron.
ü Neutron
Positron
Electron
The graph shows on a semi-logarithmic scale the dependence of the change in the number of radioactive nuclei of an isotope on time. Constant radioactive decay in is equal to...(round the answer to whole numbers)
The number of radioactive nuclei changes over time according to the law - the initial number of nuclei, - the radioactive decay constant. Taking the logarithm of this expression, we get
ln 
.Hence, 
=0,07
Conservation laws in nuclear reactions.
The reaction cannot proceed due to a violation of the conservation law...
In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric, baryon and lepton). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. To choose the correct answer, you need to check which conservation law prohibits and which allows the given reaction of interconversion of elementary particles. According to the law of conservation of lepton charge in a closed system during any process, the difference between the number of leptons and antileptons is preserved. We agreed to calculate for leptons: . lepton charge and for antileptons: . lepton charge. For all other elementary particles, lepton charges are assumed to be zero. The reaction cannot proceed due to a violation of the law of conservation of lepton charge, because
ü Lepton charge
Baryon charge
Spin angular momentum
The reaction cannot proceed due to a violation of the conservation law...
In all fundamental interactions, conservation laws are satisfied: energy, momentum, angular momentum (spin) and all charges (electric Q, baryon B and lepton L). These conservation laws not only limit the consequences of various interactions, but also determine all the possibilities of these consequences. According to the law of conservation of baryon charge B, for all processes involving baryons and antibaryons, the total baryon charge is conserved. Baryons (n, p nucleons and hyperons) are assigned a baryon charge
B = -1, and for all other particles the baryon charge is B = 0. The reaction cannot proceed due to a violation of the law of baryon charge B, because (+1)+(+1)
Answer options: lepton charge, spin angular momentum, electric charge. Q=0, antiproton (