MATHEMATICAL PHYSICS. THE SCHRÖDINGER EQUATION FOR THE WAVE FUNCTION Ψ(X, T)

Summary. The article describes Schrödinger equation, which describes the dynamics of a quantum system through a wave function. It is discovered that the use of the Schrödinger equation allows studying quantum mechanics at the molecular and atomic levels, which has found application in many fields, such as quantum chemistry, quantum physics

Mathematical physics is the general name for mathematical methods of studying and solving differential equations that arise, in particular, in physics. Theory of mathematical models of physical phenomena; occupies a special position in both mathematics and physics, being at the junction of these sciences. Mathematical physics is closely related to physics in the part that concerns the construction of a mathematical model, and at the same time, mathematical physics is a branch of mathematics, since the methods of researching models are mathematical. The concept of methods of mathematical physics includes those mathematical methods that are used to build and study mathematical models that describe large classes of physical phenomena.
One of the key tools of mathematical physics is the Schrödinger equation, which describes the dynamics of a quantum system through a wave function.
The wave function describes the state of the quantum system and allows predicting the results of experiments with the measurement of physical quantities. The Schrödinger equation allows you to find this wave function and determine various characteristics of a quantum system, such as energy levels, the distribution of electrons in molecules, the interaction of light with matter, and others.
 History The methods of mathematical physics as the theory of mathematical models of physics began at the end of the 17th century. To be intensively developed in the works of Isaac Newton on the creation of the foundations of classical mechanics, СЕКЦІЯ XXI. ФІЗИКО-МАТЕМАТИЧНІ НАУКИ universal gravitation, and the theory of light. The further development (18th -1st half of the 19th century) of the methods of mathematical physics and their successful application to the study of mathematical models of a huge volume of various physical phenomena is associated with the names of Joseph Louis Lagrange, Leonard Euler, Pierre Simon Laplace, Joseph Fourier, Karl Gauss, Bernhard Riemann, Mykhailo Ostrogradsky and other scientists. A great contribution to the development of mathematical physics methods was made by Erwin Schrödinger, Richard Feynman, Hadamard Marcel and Jean Leplace. From the second half of the XIX century. the methods of mathematical physics have been successfully used to study mathematical models of physical phenomena associated with various physical fields and wave functions in electrodynamics, acoustics, theory of elasticity, hydrodynamics and aerodynamics, and other areas of research of physical phenomena in solid environments.
Mathematical models of this class of phenomena are most often described using differential equations with partial derivatives, which have received the name mathematical physics equations. In addition to the differential equations of mathematical physics, when describing mathematical models of physics, integral equations and integro-differential equations, variational and theoretical-probability methods, potential theory, methods of the theory of functions of a complex variable and a number of other sections of mathematics are used. In connection with the rapid development of computational mathematics, direct numerical methods that use computers, and primarily finite-difference methods for solving boundary value problems, are of particular importance for the study of mathematical models of physics, which allowed the methods of mathematical physics to effectively solve to solve new problems of gas dynamics, transport theory, plasma physics, including inverse problems of these directions of physical research.
 "The Schrödinger equation for the wave function ψ(x, t)" Erwin Rudolf Josef Alexander Schrödinger or Erwin Schrödinger -austrian theoretical physicist, one of the creators of quantum mechanics. Winner of the Nobel Prize in Physics (1933). A member of a number of world academies of sciences.
He was educated at home until the age of 11, and then entered the prestigious Academic Gymnasium. After grammar school, Erwin entered the University of Vienna, where he chose to study mathematics and physics.
Erwin became interested in theoretical problems of physics after meeting Friedrich Gasenerl, the successor of Ludwig Boltzmann at the Department of Theoretical Physics. It was from Gasenerl that the future scientist learned about current scientific problems and difficulties that arise in classical physics during attempts to solve them.
During his studies at the university, Schrödinger thoroughly mastered the mathematical methods of physics, but his dissertation work was experimental. It was devoted to the study of the influence of air humidity on the electrical properties of a number of insulating materials. After defending his thesis and successfully passing the oral exams, Schrödinger earned a PhD.
Schrödinger has a number of fundamental results in the field of quantum theory that formed the basis of wave mechanics: he formulated wave equations (stationary and time-dependent Schrödinger equation), proved the identity of the formalism he developed and matrix mechanics, developed quantum mechanical 288 SECTION XXI. PHYSICS AND MATHEMATICS perturbation theory, obtained solutions to many specific tasks. Schrödinger offered an original interpretation of the physical meaning of the wave function; in subsequent years repeatedly criticized the generally accepted the Copenhagen interpretation of quantum mechanics ("Schrödinger's cat" paradox).
He is the author of many works in various branches of physics: statistical mechanics and thermodynamics, physics of dielectrics, color theory, electrodynamics, general theory of relativity and cosmology; he made several attempts to build a unified field theory.
So, "Schrödinger's equation for the wave function ψ(x, t)": The Schrödinger equation is one of the basic equations of quantum mechanics. It determines the change of quantum states with the influence of time.
It is given in the form: ℏ = , where is the Hamiltonian operator of the system, coincides with the energy operator, if there is no dependence on time. For the non-relativistic motion of a particle of mass in the potential field ( ), the operator ̂ is valid, and is the sum of the kinetic and potential energy of the particle's motion = − ℏ +̂( ).
Due to the complex problem, the equation has periodic solutions. Therefore, it is also called the Schrödinger wave equation, and the solutions are wave functions. The Schrödinger equation reflects the principle of causality in quantum mechanics, because it determines the state of the system by the wave function (t) at any subsequent moment in time if the solution at the initial moment is known.
 Conservation of normalization of the wave function over time where = * is the probability density (probability of finding a particle in a certain area of space), and is the density vector of the probability flow.  General principles of finding its solutions, which have a physical meaning The conditions imposed on the wave function have a general physical meaning. The wave function must be defined, continuous and unambiguous throughout spacious The continuity condition must be satisfied in any field, even if ( , , ) has discontinuity surfaces. Continuity is also required at the break surface and the derivative of the wave function (crosslinking conditions). Continuity is not required in the case infinite potential pits and barriers. In the region of space where the potential = ∞, the particle cannot enter, that is ( = ). Continuity requires that on the edges the pit =0; derivatives in this case may have a discontinuity.
 Free movement of particles The equation ℏ = with the operator = − ℏ corresponds to the wave function describing the free movement of a particle in a certain value of momentum and energy. The wave functions of this equation are plane waves with СЕКЦІЯ XXI. ФІЗИКО-МАТЕМАТИЧНІ НАУКИ frequency ℏ and wave vector k= ℏ . The corresponding wavelength is = ℏ is called the de Broglie wavelength of the particle.  One-dimensional motion and its properties.
If the potential depends only on , then the wave function is sought as a product of the functions of , (the free particle equation for two dimensions) and the function of (the one-dimensional Schrödinger equation We will show that the energy levels for the one-dimensional problem are not degenerate: Proved from the opposite. Let 1 and 2 be two different eigenwave functions corresponding to a certain energy level (one). They satisfy the equation Integrating gives ′ − ′ = .
From the condition of convergence at infinity = =0, then the constant is equal to 0, and therefore ′ − ′ = .
After solving the equation, we got =const • -essentially the same functions. Therefore, the levels are not degenerate.
We will assume that the function ( ) tends to the finite limits as → ±∞ (but should by no means be a monotonic function).
And we will assume that > . The discrete spectrum lies in such values energies for which the particle cannot go to infinity; energy for this must be less than both limits (+∞), i.e. must be negative: < ; let us now consider the region of (+ energy) values smaller than : < < .
In this field, the spectrum will be continuous, and the motion of the particle in the corresponding ones stationary states -infinite, and the fraction goes to the side = +∞. It is easy to see that all energy eigenvalues in this part of the spectrum are not degenerate either. For this, it suffices to note that for the above (for a discrete spectrum) proof is sufficient that the functions and vanish at least at one of the infinities (in this case, they vanish at = −∞.
Finally, for E > , the spectrum will be continuous, and the motion will be infinite in both parties In this part of the spectrum, all levels are doubly degenerate.
This follows from the fact that the corresponding wave functions are defined by a second-order equation, and both independent solutions of this equation properly satisfy the conditions at infinity (while, for example, in the previous case, one of the solutions turned to infinity at → −∞ and therefore should have been rejected).
The asymptotic form of the wave function at → +∞ is = + − ; and similarly for → −∞. The term with corresponds to the particle − moving to the right, and the term to the left. Since no restrictions on E (except E >0) are imposed, the energy spectrum of particles is continuous.
The use of the Schrödinger equation allows studying quantum mechanics at the molecular and atomic levels, which has found application in many fields, such as quantum chemistry, quantum physics, electronics, medicine, technology, and others. Therefore, the Schrödinger equation and the wave function are important concepts of mathematical physics and allow studying and understanding the properties of quantum systems.