4.4.1. De Broglie's hypothesis
An important step in the creation of quantum mechanics was the discovery of the wave properties of microparticles. The idea of wave properties was originally put forward as a hypothesis by the French physicist Louis de Broglie.
In physics for many years the theory dominated, according to which light is an electromagnetic wave. However, after the work of Planck (thermal radiation), Einstein (photoelectric effect) and others, it became obvious that light has corpuscular properties.
To explain some physical phenomena, it is necessary to consider light as a stream of photon particles. The corpuscular properties of light do not reject, but complement its wave properties.
So, photon is an elementary particle of light with wave properties.
Formula for photon momentum
| . | (4.4.3) |
According to de Broglie, the motion of a particle, for example, an electron, is similar to a wave process with a wavelength λ defined by formula (4.4.3). These waves are called de Broglie waves. Therefore, particles (electrons, neutrons, protons, ions, atoms, molecules) can exhibit diffractive properties.
K. Davisson and L. Germer were the first to observe electron diffraction on a single crystal of nickel.
The question may arise: what happens to individual particles, how are the maxima and minima formed during the diffraction of individual particles?
Experiments on the diffraction of electron beams of very low intensity, that is, as if individual particles, showed that in this case the electron is not "smeared" in different directions, but behaves like a whole particle. However, the probability of electron deflection in separate directions as a result of interaction with the diffraction object is different. The electrons are most likely to hit the places that, according to the calculation, correspond to the diffraction maxima, their hit to the minima is less likely. Thus, wave properties are inherent not only to the collective of electrons, but also to each electron individually.
4.4.2. Wave function and its physical meaning
Since a wave process is associated with a microparticle, which corresponds to its motion, the state of particles in quantum mechanics is described by a wave function that depends on coordinates and time: .
If the force field acting on the particle is stationary, that is, does not depend on time, then the ψ-function can be represented as a product of two factors, one of which depends on time and the other on coordinates:
This implies the physical meaning of the wave function:
4.4.3. Uncertainty relation
One of the important provisions of quantum mechanics are the uncertainty relations proposed by W. Heisenberg.
Let the position and momentum of the particle be measured simultaneously, while the inaccuracies in the definitions of the abscissa and the projection of the momentum on the abscissa axis are Δx and Δр x , respectively.
In classical physics, there are no restrictions that prohibit simultaneously measuring both one and the other quantity with any degree of accuracy, that is, Δx→0 and Δр x→ 0.
In quantum mechanics, the situation is fundamentally different: Δx and Δр x , corresponding to the simultaneous determination of x and р x , are related by the dependence
Formulas (4.4.8), (4.4.9) are called uncertainty relations.
Let us explain them with one model experiment.
When studying the phenomenon of diffraction, attention was drawn to the fact that a decrease in the width of the slit during diffraction leads to an increase in the width of the central maximum. A similar phenomenon will also occur in the case of electron diffraction by a slit in a model experiment. Reducing the width of the slot means a decrease in Δ x (Fig. 4.4.1), this leads to a greater "smearing" of the electron beam, that is, to a greater uncertainty in the momentum and velocity of the particles.
Rice. 4.4.1. Explanation of the uncertainty relation.
The uncertainty relation can be represented as
| , | (4.4.10) |
where ΔE is the energy uncertainty of some state of the system; Δt is the period of time during which it exists. Relationship (4.4.10) means that the shorter the lifetime of any state of the system, the more uncertain its energy value. Energy levels E 1 , E 2 etc. have a certain width (Fig. 4.4.2)), depending on the time the system is in the state corresponding to this level.
Rice. 4.4.2. Energy levels E 1, E 2, etc. have some width.
The "blurring" of the levels leads to the uncertainty of the energy ΔE of the emitted photon and its frequency Δν during the transition of the system from one energy level to another:
,
where m is the mass of the particle; ; E and E n are its total and potential energies (potential energy is determined by the force field in which the particle is located, and for the stationary case does not depend on time)
If the particle moves only along a certain line, for example, along the OX axis (one-dimensional case), then the Schrödinger equation is substantially simplified and takes the form
 | (4.4.13) |
One of the simplest examples of the use of the Schrödinger equation is the solution of the problem of the motion of a particle in a one-dimensional potential well.
4.4.5. Application of the Schrödinger equation to the hydrogen atom. quantum numbers
Describing the states of atoms and molecules using the Schrödinger equation is a rather difficult task. It is most simply solved for one electron located in the field of the nucleus. Such systems correspond to the hydrogen atom and hydrogen-like ions (singly ionized helium atom, doubly ionized lithium atom, etc.). However, in this case, the solution of the problem is also complicated, so we restrict ourselves to a qualitative presentation of the problem.
First of all, the potential energy should be substituted into the Schrödinger equation (4.4.12), which for two interacting point charges - e (electron) and Ze (nucleus), - located at a distance r in vacuum, is expressed as follows:
This expression is a solution of the Schrödinger equation and completely coincides with the corresponding formula of the Bohr theory (4.2.30)
Figure 4.4.3 shows the levels of possible values of the total energy of the hydrogen atom (E 1 , E 2 , E 3 , etc.) and a plot of the potential energy E n versus the distance r between the electron and the nucleus. As the principal quantum number n increases, r increases (see 4.2.26), and the total (4.4.15) and potential energies tend to zero. The kinetic energy also tends to zero. The shaded area (E>0) corresponds to the state of a free electron.
Rice. 4.4.3. The levels of possible values of the total energy of the hydrogen atom are shown
and a plot of potential energy versus distance r between the electron and the nucleus.
Second quantum number - orbital l, which for a given n can take the values 0, 1, 2, ...., n-1. This number characterizes the orbital angular momentum L i of the electron relative to the nucleus:
Fourth quantum number - spin m s. It can take only two values (±1/2) and characterizes the possible values of the electron spin projection:
| .(4.4.18) |
The state of an electron in an atom with given n and l is denoted as follows: 1s, 2s, 2p, 3s, etc. Here, the number indicates the value of the main quantum number, and the letter - the orbital quantum number: the symbols s, p, d, f, correspond to the values l=0, 1, 2. 3, etc.
An equation that takes 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, which can
paint: 
;; U is the 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 you 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) Density of probabilities of finding a particle in a 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 passage 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
because
depends only on time 
divide we get:
The left side of the equality depends only on time, the right one only on the coordinates, this equality is valid only if both sides = const, such a constant is the total energy of the particle E.
Consider the right side of this equality: 
, transform: 
is the equation for the steady state.
Consider the left side of the Schrödinger equation: ;;
we divide the variables , integrate the resulting equation:
using mathematical transformations:
In this case, the probability of finding a particle can be determined:
Or after transformations:
– this probability does not depend on time, this equation characterizing microparticles is called – the stationary state of the particle.
It is usually required that the wave function be defined and continuous (an infinite number of times differentiable) throughout space, and also that it be single-valued. One type of ambiguity of wave functions is admissible - the 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 of space is one. In the general case, integration should be performed over all variables on which the wave function in a given representation depends.
26 A particle in a one-dimensional rectangular potential well of infinite depth. Energy quantization. Bohr's correspondence principle.
Let us 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 an electron to leave a metal, work must be done, which corresponds to the potential energy in the Schrödinger equation.
Under this condition, the particle does not penetrate beyond the "well", i.e.
y(0)= y(l)=0 Within the well (0
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=1,2,3…
An analysis of this equation shows that the energy in a potential well cannot be a discrete quantity.
the state with min energy is called the ground state, all the rest are excited.
Consider 
because Since the potential well is one-dimensional, we can write down that, we substitute into the expression in place and we get. Since a one-dimensional potential well with a flat bottom, then 
Let's graphically depict
It can be seen from the figure 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 necessary for understanding the structural organization of matter, i.e. the existence of stable, repeating in their properties, molecules, atoms and smaller structural units that make up both matter and radiation.
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 body, a molecule, an atom or an atomic nucleus - can do this only at certain values of 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 in the description of microscopic objects, such as atoms and elementary particles. On the other hand, experiments show that a variety of 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 that exhibit quantum behavior, for example, a 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 the premise for Bohr's correspondence principle, which states that classical physics should emerge as an approximation to quantum physics as systems get larger.
The conditions under which quantum and classical mechanics coincide are called the classical limit. Bohr proposed a rough criterion for the classical limit: the transition occurs when the quantum numbers describing the system are large, meaning either the system is excited 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 of the action
corpuscular-wave dualism in quantum physics describes the state of a particle using the wave function ($\psi (\overrightarrow(r),t)$- psi-function).
Definition 1
wave function is a function that is used in quantum mechanics. It describes the state of a system that has dimensions in space. It is a state vector.
This function is complex and formally has wave properties. The movement of any particle of the microworld is determined by probabilistic laws. The probability distribution is revealed when making a large number of observations (measurements) or a large number of particles. The obtained distribution is similar to the distribution of wave intensity. That is, in places with maximum intensity, the maximum number of particles is noted.
The set of wave function arguments determines its representation. Thus, the coordinate representation is possible: $\psi(\overrightarrow(r),t)$, momentum representation: $\psi"(\overrightarrow(p),t)$, etc.
In quantum physics, the goal is not to accurately predict an event, but to estimate the probability of an event. Knowing the magnitude of the probability, find the average values of physical quantities. The wave function allows you to find similar probabilities.
So the probability of the presence of a microparticle in the volume dV at time t can be defined as:
where $\psi^*$ is the complex conjugate function to the function $\psi.$ The probability density (probability per unit volume) is:
Probability is a quantity that can be observed in an experiment. At the same time, the wave function is not available for observation, since it is complex (in classical physics, the parameters that characterize the state of the particle are available for observation).
Normalization condition for $\psi$-functions
The wave function is defined up to an arbitrary constant factor. This fact does not affect the state of the particle, which the $\psi$-function describes. However, the wave function is chosen in such a way that it satisfies the normalization condition:
where the integral is taken over the entire space or over a region in which the wave function is not equal to zero. The normalization condition (2) means that in the entire region where $\psi\ne 0$ the particle is reliably present. A wave function that obeys the normalization condition is called normalized. If $(\left|\psi\right|)^2=0$, then this condition means that there are certainly no particles in the region under study.
Normalization of the form (2) is possible for a discrete spectrum of eigenvalues.
The normalization condition may not be feasible. So, if $\psi$ is a de Broglie plane wave function and the probability of finding a particle is the same for all points in space. These cases are considered as an ideal model in which the particle is present in a large but limited region of space.
Wave function superposition principle
This principle is one of the main postulates of quantum theory. Its meaning is as follows: if for some system the states described by the wave functions $\psi_1\ (\rm u)\ $ $\psi_2$ are possible, then for this system there is a state:
where $C_(1\ )and\ C_2$ are constant coefficients. The principle of superposition is confirmed empirically.
We can talk about the addition of any number of quantum states:
where $(\left|C_n\right|)^2$ is the probability that the system is found in the state described by the wave function $\psi_n.$
Stationary states
In quantum theory, stationary states (states in which all observable physical parameters do not change in time) play a special role. (The wave function itself is fundamentally unobservable). In the stationary state, the $\psi$-function has the form:
where $\omega =\frac(E)(\hbar )$, $\psi\left(\overrightarrow(r)\right)$ does not depend on time, $E$ is the energy of the particle. In the form (3) of the wave function, the probability density ($P$) is a time constant:
From the physical properties of stationary states follow the mathematical requirements for the wave function $\psi\left(\overrightarrow(r)\right)\to \ (\psi(x,y,z))$.
Mathematical requirements for the wave function for stationary states
$\psi\left(\overrightarrow(r)\right)$ - the function must be at all points:
- continuous,
- unambiguous,
- finite.
If the potential energy has a discontinuity surface, then on such surfaces the function $\psi\left(\overrightarrow(r)\right)$ and its first derivative must remain continuous. In a region of space where the potential energy becomes infinite, $\psi\left(\overrightarrow(r)\right)$ must be equal to zero. The continuity of the function $\psi\left(\overrightarrow(r)\right)$ requires that $\psi\left(\overrightarrow(r)\right)=0$ on any boundary of this region. The continuity condition is imposed on the partial derivatives of the wave function ($\frac(\partial \psi)(\partial x),\ \frac(\partial \psi)(\partial y),\frac(\partial \psi)(\ partial z)$).
Example 1
Exercise: For some particle, a wave function of the form is given: $\psi=\frac(A)(r)e^(-(r)/(a))$, where $r$ is the distance from the particle to the center of force (Fig. 1 ), $a=const$. Apply the normalization condition, find the normalization factor A.
Picture 1.
Solution:
We write the normalization condition for our case in the form:
\[\int((\left|\psi\right|)^2dV=\int(\psi\psi^*dV=1\left(1.1\right)))\]
where $dV=4\pi r^2dr$ (see Fig.1 It is clear from the conditions that the problem has spherical symmetry). From the conditions of the problem we have:
\[\psi=\frac(A)(r)e^(-(r)/(a))\to \psi^*=\frac(A)(r)e^(-(r)/(a ))\left(1.2\right).\]
Let us substitute $dV$ and wave functions (1.2) into the normalization condition:
\[\int\limits^(\infty )_0(\frac(A^2)(r^2)e^(-(2r)/(a))4\pi r^2dr=1\left(1.3\ right).)\]
Let's integrate on the left side:
\[\int\limits^(\infty )_0(\frac(A^2)(r^2)e^(-(2r)/(a))4\pi r^2dr=2\pi A^2a =1\left(1.4\right).)\]
From formula (1.4) we express the desired coefficient:
Answer:$A=\sqrt(\frac(1)(2\pi a)).$
Example 2
Exercise: What is the most probable distance ($r_B$) of an electron from the nucleus if the wave function that describes the ground state of an electron in a hydrogen atom can be defined as: $\psi=Ae^(-(r)/(a))$, where $ r$ is the distance from the electron to the nucleus, $a$ is the first Bohr radius?
Solution:
We use the formula that determines the probability of the presence of a microparticle in the volume $dV$ at time $t$:
where $dV=4\pi r^2dr.\ $Consequently, we have:
In this case, $p=\frac(dP)(dr)$ can be written as:
To determine the most probable distance, we equate the derivative $\frac(dp)(dr)$ to zero:
\[(\left.\frac(dp)(dr)\right|)_(r=r_B)=8\pi rA^2e^(-(2r)/(a))+4\pi r^2A^ 2e^(-(2r)/(a))\left(-\frac(2)(a)\right)=8\pi rA^2e^(-(2r)/(a))\left(1- \frac(r)(a)\right)=0(2.4)\]
Since the solution $8\pi rA^2e^(-(2r_B)/(a))=0\ \ (\rm at)\ \ r_B\to \infty $ does not suit us, it is rejected:
Experimental confirmation of de Broglie's idea about the universality of wave-particle duality, 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 is equal to
dW= │Ψ│ 2 dV. (33.6)
The value │Ψ│ 2 = dW/dV- has the meaning of a probability density, i.e. determines the probability of finding a particle in a unit volume in the neighborhood 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 de Broglie waves.
Probability of finding a particle at time t in a finite volume V, is equal to
W==│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 certain event turns into unity, if the volume V take the infinite volume of the whole space. This means that under this condition, the particle must be somewhere in space. Therefore, the condition for the normalization of 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 ∞. The function Ψ must be finite, single-valued , and continuous.
Schrödinger equation
The equation of motion in quantum mechanics, which describes the motion of microparticles in various force fields, should be an equation from which the wave properties of particles would follow. It must be an equation for the wave function Ψ( X,at,z,t), since the value │ Ψ │ 2 determines the probability of the particle being in the volume at the moment of time.
The basic equation was formulated by E. Schrödinger: the equation is not derived, but postulated.
Schrödinger equation looks like:
- ΔΨ + U(x,y,z,t)Ψ = iħ, (33.9)
Where ħ=h/(2π ), T-particle mass, Δ-Laplace operator , i- imaginary unit, U(x,y,z,t) is the potential function of the particle in the force field in which it moves, Ψ( x,y,z,t) is the desired wave function of the particle.
Equation (32.9) is the 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 energy values. 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 = 0. (33.10)
Equation (33.10) is called Schrödinger equation for stationary states.
This equation includes the total energy as a parameter E particles. The solution of the equation does not take place for any values of the parameter E, but only for a certain set characteristic of the given problem. These energy values are called eigenvalues. Eigenvalues E can form both continuous and discrete series.
> Wave function
Read about wave function and the theory of probability of quantum mechanics: the essence of the Schrödinger equation, the state of a quantum particle, a harmonic oscillator, a scheme.
We are talking about the probability amplitude in quantum mechanics, which describes the quantum state of the particle and its behavior.
Learning task
- Combine the wave function and the particle detection probability density.
Key Points
- |ψ| 2 (x) corresponds to the probability density of detecting a particle in a particular place and moment.
- The laws of quantum mechanics characterize the evolution of the wave function. The Schrödinger equation explains its name.
- The wave function must satisfy many mathematical constraints for computation and physical interpretation.
Terms
- The Schrödinger equation is a partial differential that characterizes the change in the state of a physical system. It was formulated in 1925 by Erwin Schrödinger.
- A harmonic oscillator is a system that, when displaced from its original position, is affected by a force F proportional to the displacement x.
Within quantum mechanics, the wave function reflects the probability amplitude that characterizes the quantum state of the particle and its behavior. Usually the value is a complex number. The most common wave function symbols are ψ (x) or Ψ(x). Although ψ is a complex number, |ψ| 2 is real and corresponds to the probability density of finding a particle in a particular place and time.
Here the trajectories of the harmonic oscillator are displayed in the classical (A-B) and quantum (C-H) mechanics. In the quantum ball, the wave function is shown with the real part in blue and the imaginary part in red. TrajectoriesC-F are examples of standing waves. Each such frequency will be proportional to the possible energy level of the oscillator
The laws of quantum mechanics evolve over time. The wave function resembles others, like waves in water or a string. The fact is that the Schrödinger formula is a type of wave equation in mathematics. This leads to the duality of wave particles.
The wave function must comply with the restrictions:
- always final.
- always continuous and continuously differentiable.
- satisfies the corresponding normalization condition so that the particle exists with 100% certainty.
If the requirements are not satisfied, then the wave function cannot be interpreted as a probability amplitude. If we ignore these positions and use the wave function to determine the observations of a quantum system, we will not get finite and definite values.