Detection wave properties microparticles indicated that classical mechanics cannot give a correct description of the behavior of such particles. A theory covering all properties elementary particles, should take into account not only them corpuscular properties, but also wave ones. From the experiments discussed earlier, it follows that a beam of elementary particles has the properties of a plane wave propagating in the direction of the particle speed. In the case of propagation along the axis, this wave process can be described by the de Broglie wave equation (7.43.5):
(7.44.1)
where is the energy and is the momentum of the particle. When propagating in any direction:
(7.44.2)
Let's call the function a wave function and figure it out physical meaning by comparing the diffraction of light waves and microparticles.
According to wave representations On the nature of light, the intensity of the diffraction pattern is proportional to the square of the amplitude of the light wave. According to views photon theory, the intensity is determined by the number of photons entering the this point diffraction pattern. Consequently, the number of photons at a given point in the diffraction pattern is given by the square of the amplitude of the light wave, while for one photon the square of the amplitude determines the probability of the photon hitting a particular point.
The diffraction pattern observed for microparticles is also characterized by an unequal distribution of microparticle fluxes. The presence of maxima in the diffraction pattern from the point of view wave theory means that these directions correspond to the highest intensity of de Broglie waves. The intensity is greater where larger number particles. Thus, diffraction pattern for microparticles is a manifestation of a statistical pattern and we can say that knowledge of the type of de Broglie wave, i.e. Ψ -function allows one to judge the probability of one or another of the possible processes.
So, in quantum mechanics the state of microparticles is described in a fundamentally new way - using the wave function, which is the main carrier of information about their corpuscular and wave properties. The probability of finding a particle in an element with volume is
(7.44.3)
Magnitude
(7.44.4)
has the meaning of probability density, i.e. determines the probability of finding a particle in a unit volume in the vicinity given point. Thus, it is not the function itself that has a physical meaning, but the square of its module, which sets the intensity of de Broglie waves. The probability of finding a particle at a moment in time in a finite volume, according to the theorem of addition of probabilities, is equal to
(7.44.5)
Since a particle exists, it is sure to be found somewhere in space. Probability reliable event is equal to one, then
. (7.44.6)
Expression (7.44.6) is called the probability normalization condition. The wave function characterizing the probability of detecting the action of a microparticle in a volume element must be finite (the probability cannot be greater than one), unambiguous (the probability cannot be an ambiguous value) and continuous (the probability cannot change abruptly).
> Wave function
Read about wave function and probability theories of quantum mechanics: the essence of the Schrödinger equation, the state quantum particle, harmonic oscillator, circuit.
We are talking about the probability amplitude in quantum mechanics, which describes the quantum state of a particle and its behavior.
Learning Objective
- Combine the wave function and the probability density of identifying a particle.
Main points
- |ψ| 2 (x) corresponds to the probability density of identifying a particle in a specific 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 characterizing a change in state physical system. It was formulated in 1925 by Erwin Schrödinger.
- A harmonic oscillator is a system that, when displaced from its original position, is influenced 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 a particle and its behavior. Usually the value is complex number. The most common symbols for the wave function are ψ (x) or Ψ(x). Although ψ is a complex number, |ψ| 2 – real and corresponds to the probability density of finding a particle in a specific place and time.
The trajectories are shown here harmonic oscillator in classical (A-B) and quantum (C-H) mechanics. The quantum ball has a wave function displayed with the real part in blue and imaginary in red. TrajectoriesC-F - examples 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, such as 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 following restrictions:
- always final.
- always continuous and continuously differentiable.
- satisfies the appropriate normalization condition for the particle to exist 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 observations of a quantum system, we will not get finite and definite values.
WAVE FUNCTION, in QUANTUM MECHANICS, a function that allows you to find the probability that quantum system is in some state s at time t. Usually written: (s) or (s, t). The wave function is used in the SCHRÖDINGER equation... Scientific and technical encyclopedic dictionary
WAVE FUNCTION Modern encyclopedia
Wave function- WAVE FUNCTION, in quantum mechanics the main quantity (in general case complex), describing the state of the system and allowing one to find the probabilities and average values characterizing this system physical quantities. Wave module square... ... Illustrated Encyclopedic Dictionary
WAVE FUNCTION- (state vector) in quantum mechanics is the main quantity that describes the state of a system and allows one to find the probabilities and average values of physical quantities characterizing it. Wave function modulus squared equal to probability given... ... Big encyclopedic Dictionary
WAVE FUNCTION- in quantum mechanics (probability amplitude, state vector), a quantity that completely describes the state of a micro-object (electron, proton, atom, molecule) and any quantum in general. systems. Description of the state of a microobject using V. f. It has… … Physical encyclopedia
wave function- - [L.G. Sumenko. English-Russian dictionary on information technology. M.: State Enterprise TsNIIS, 2003.] Topics information Technology in general EN wave function … Technical Translator's Guide
wave function- (probability amplitude, state vector), in quantum mechanics the main quantity that describes the state of a system and allows one to find the probabilities and average values of physical quantities characterizing it. The squared modulus of the wave function is... ... encyclopedic Dictionary
wave function- banginė funkcija statusas T sritis fizika atitikmenys: engl. wave function vok. Wellenfunktion, f rus. wave function, f; wave function, f pranc. fonction d’onde, f … Fizikos terminų žodynas
wave function- banginė funkcija statusas T sritis chemija apibrėžtis Dydis, apibūdinantis mikrodalelių ar jų sistemų fizikinę būseną. atitikmenys: engl. wave function rus. wave function... Chemijos terminų aiškinamasis žodynas
WAVE FUNCTION - complex function, describing the state of quantum mechanics. system and allows you to find probabilities and cf. the meanings of the physical characteristics it characterizes. quantities Square modulus V. f. equal to probability this state, therefore V.f. called also amplitude... ... Natural science. encyclopedic Dictionary
Books
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The diffraction pattern observed for microparticles is characterized by an unequal distribution of microparticle fluxes in different directions - there are minima and maxima in other directions. The presence of maxima in the diffraction pattern means that de Broglie waves are distributed in these directions with the greatest intensity. And the intensity will be maximum if the maximum number of particles propagates in this direction. Those. The diffraction pattern for microparticles is a manifestation of a statistical (probabilistic) pattern in the distribution of particles: where the intensity of the de Broglie wave is maximum, there are more particles.
De Broglie waves in quantum mechanics are considered like waves probabilities, those. probability of detecting a particle in different points in space varies according to wave law(i.e. e - iωt). But for some points in space this probability will be negative (i.e. the particle does not fall into this region). M. Born (German physicist) suggested that according to the wave law, it is not the probability itself that changes, and the probability amplitude, which is also called the wave function or -function (psi - function).
The wave function is a function of coordinates and time.
The square of the modulus of the psi function determines the probability that the particle will be detected within the volumedV - it is not the psi-function itself that has a physical meaning, but the square of its modulus.
Ψ * - function complex conjugate to Ψ
(z = a +ib, z * =a- ib, z * - complex conjugate)
If the particle is in a finite volume V, then the possibility of detecting it in this volume is equal to 1, (reliable event)
R=
1 
In quantum mechanics it is accepted that Ψ and AΨ, where A = const, describe the same state of the particle. Hence,
Normalization condition
integral over , means that it is calculated over an infinite volume (space).
- the function must be
1) final (since R cannot be more than 1),
2) unambiguous (it is impossible to detect a particle under constant conditions with a probability of, say, 0.01 and 0.9, since the probability must be unambiguous).
continuous (follows from the continuity of space. There is always a probability of detecting a particle at different points in space, but for different points it will be different)
The wave function satisfies principle superpositions: if the system can be in different states described by wave functions 1 , 2 ... n , then it can be in state described by linear combinations of these functions:
With n (n=1,2...) - any numbers.
Using the wave function, the average values of any physical quantity of a particle are calculated
§5 Schrödinger equation
The Schrödinger equation, like other basic equations of physics (Newton's, Maxwell's equations), is not derived, but postulated. It should be considered as the initial basic assumption, the validity of which is proven by the fact that all the consequences arising from it are in exact agreement with experimental data.
(1)
Schrödinger time equation.
Nabla - Laplace operator 
Potential function particles in a force field,
Ψ(y,z,t) - the required function
If the force field in which the particle moves is stationary (i.e. does not change over time), then the function U does not depend on time and has the meaning of potential energy. In this case, the solution to the Schrödinger equation (i.e. Ψ is a function) can be represented as a product of two factors - one depends only on coordinates, the other only on time:
(2)
E is the total energy of the particle, constant in the case of a stationary field.
Substituting (2) (1):
(3)
Schrödinger equation for stationary states.
There are infinitely many solutions. By imposing boundary conditions, solutions that have a physical meaning are selected.
Border conditions:
the wave functions must be regular, i.e.
1) final;
2) unambiguous;
3) continuous.
Solutions that satisfy the Schrödinger equation are called own functions, and the corresponding energy values are eigenvalues energy. The set of eigenvalues is called spectrum quantities. If E n takes discrete values, then the spectrum - discrete, if continuous - solid or continuous.
Based on the idea that an electron has wave properties. Schrödinger in 1925 suggested that the state of an electron moving in an atom should be described by the standing equation known in physics electromagnetic wave. Substituting into this equation instead of the wavelength its value from the de Broglie equation, he obtained a new equation relating the energy of the electron with spatial coordinates and the so-called wave function, corresponding in this equation to the amplitude of the three-dimensional wave process.
Especially important to characterize the state of the electron has a wave function. Like the amplitude of any wave process, it can take both positive and negative values. However, the value is always positive. At the same time, she has remarkable property: how more value in a given region of space, the higher the probability that the electron will manifest its action here, that is, that its existence will be detected in some physical process.
It would be more accurate the following statement: the probability of finding an electron in a certain small volume is expressed by the product . Thus, the value itself expresses the probability density of finding an electron in the corresponding region of space.
Rice. 5. Electron cloud of the hydrogen atom.
To understand the physical meaning of the squared wave function, consider Fig. 5, which depicts a certain volume near the nucleus of a hydrogen atom. The density of points in Fig. 5 is proportional to the value in the corresponding place: than larger value, the denser the points are. If an electron had the properties of a material point, then Fig. 5 could be obtained by repeatedly observing the hydrogen atom and each time noting the location of the electron: the density of points in the figure would be greater, the more often the electron is detected in the corresponding region of space, or, in other words, the more more likely detecting it in this area.
We know, however, that the idea of an electron as material point does not correspond to its true physical nature. Therefore Fig. It is more correct to consider 5 as a schematic representation of an electron “smeared” throughout the entire volume of an atom in the form of a so-called electron cloud: the denser the points are located in one place or another, the greater the density of the electron cloud. In other words, the density of the electron cloud is proportional to the square of the wave function.
The idea of the state of an electron as a kind of cloud electric charge turns out to be very convenient, well conveys the main features of the behavior of the electron in atoms and molecules and will be often used in the subsequent presentation. At the same time, however, it should be borne in mind that the electron cloud does not have definite, sharply defined boundaries: even at long distance from the nucleus there is some, albeit very small, probability of finding an electron. Therefore, by electron cloud we will conventionally understand the region of space near the nucleus of an atom in which the predominant part (for example, ) of the charge and mass of the electron is concentrated. More precise definition this area of space is given on page 75.
