Schr ö dinger equation, also known as Schrodinger wave equation, is a basic equation in quantum mechanics proposed by Austrian physicist Schr ö dinger, and also a basic assumption of quantum mechanics.
It willMatter waveConcept andwave equation Second order established by combiningpartial differential equation , which can describe the movement of microscopic particles. Each microscopic system has a corresponding Schrodinger equation, which can be obtained by solving the equationwave functionAnd the corresponding energy, so as to understand the properties of the microscopic system.In quantum mechanics, particles appear in the form of probability, with uncertainty, and failure can be ignored at the macro scale.
Schrodinger equation isquantum mechanicsIn 1926, the Austrian theoretical physicist Schr ö dinger proposed the basic equation of.It describes the state of microscopic particles changing with time.The state of the micro system is described by the wave function, and the Schrodinger equation is the wave functiondifferential equation 。If the initial conditions and boundary conditions are given, the equation can be solvedwave function。
Equation definition
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Schrodinger equation inquantum mechanicsIn, the state of the system cannot be determined by the value of mechanical quantity (such as x), but is a function of mechanical quantityΨ（x,t）, i.ewave function(also known asProbability amplitude，State function）Therefore, the wave function has become the main object of quantum mechanics research.How the probability distribution of the value of the mechanical quantity and how this distribution changes with time can be solved by solving the Schrodinger equation of the wave function.This equation was put forward by Austrian physicist Schr ö dinger in 1926. It is one of the most basic equations in quantum mechanics. Its position in quantum mechanics is equal to that of Newton's equation in classical mechanics. Superstring theory attempts to unify the two theories.
Schrodinger equation is the most basic equation and a basic assumption of quantum mechanics. Its correctness can only be determined by experiment.
Schrodinger equation
In quantum mechanics, solving the particle problem often boils down to solving the Schrodinger equation or the stationary Schrodinger equation.Schrodinger equation is widely used in atomic physics, nuclear physics and solid physics. The results of solving a series of problems of atoms, molecules, nuclei, solids and so on are in good agreement with the reality.
Schrodinger equation is only applicable to non relativistic particles with low speed, and it does not include the description of particle spin.When the relativistic effect is involved, the Schrodinger equation is replaced by the relativistic quantum mechanical equation, which naturally includes the spin of particles.
Schrodinger's fundamental equation of quantum mechanics.It was founded in 1926.It is a non relativistic wave equation.It reflects the law describing the state of microscopic particles changing with time. Its position in quantum mechanics is equivalent to that of Newton's law for classical mechanics, and it is one of the basic assumptions of quantum mechanics.Let the wave function describing the state of microscopic particles beΨ（r，t）, the microscopic particles with mass m in the potential fieldV（r，t）Schrodinger equation of motion in.Given the initial conditions, boundary conditions and the single value, finite and continuous conditions that the wave function satisfies, the wave function can be solvedΨ（r，t）。From this, the distribution probability of particles and the average value (expected value) of any possible experiment can be calculated.When the potential function V is independent of time t, the particle has a certain energy, and its state is called a stationary state.The wave function in stationary state can be written in the formulaΨ（r）It is called the stationary wave function and satisfies the stationary Schrodinger equation, which is mathematically called the eigenequation, whereEIs the eigenvalue, which is the steady state energy,Ψ（r）It is also called the eigenfunction belonging to the eigenvalue E.
Schrodinger equation is the basic equation of quantum mechanics, which reveals the basic laws of the movement of matter in the microscopic physical world. As Newton's law plays a role in classical mechanics, it is a powerful tool to deal with all non relativistic problems in atomic physicsatom, molecular, solid physicsnuclear physics, chemistry and other fields.[1]
Background and development
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In 1900, Max Planck made the hypothesis of quantifying the energy of electromagnetic radiation in the study of blackbody radiation, so he found the Planck relationship that relates energy and frequency.In 1905, Albert Einstein gave a new interpretation of this relationship from his research on photoelectric effect: frequency isνOf photons have energyhν；Where, factorhyesPlanck constant。This idea became laterWave particle dualityOne of the early signposts of the concept.Since in the special theory of relativity, the correlation between energy and momentum is similar to that between frequency and wave number, it can be speculated that the momentum of photons is inversely proportional to the wavelength and directly proportional to the wave number, and the equation is used to express this relationship.
Louis de Broglie believes that not only photons obey this relationship, but all particles obey it.He further proposed in 1924De Broglie hypothesisIt shows that every microscopic particle has wave and particle properties, which is calledWave particle duality。The electron is no exception to this nature.The electron is aMatter wave, called "electronic wave".The energy and momentum of the electron determine the frequency and wave number of the matter wave accompanying it.In atoms, bound electrons formstanding wave；This means that his rotation frequency can only show some discrete values.These quantized orbits correspond to discreteenergy level。From these ideas, de Broglie copiedBohr modelThe energy level of.
In 1925, Zurich, Switzerland held a physics academic seminar every two weeks.Once, the sponsorPeter Debye Schrodinger was invited to talk about De Broglie's doctoral thesis on wave particle duality.During that period, Schrodinger was studying gas theoryBose Einstein statisticsIn the discussion, I contacted De Broglie's doctoral thesis and had a deep understanding of this aspect.In the seminar, he explained the wave particle duality thoroughly, and everyone listened with great interest.Debye pointed out that since particles have wave properties, there should be a kind ofwave equation 。His opinion gave Schrodinger great inspiration and encouragement, and he began to look for the wave equation.The simplest and basic way to test this equation is to use this equation to describe the physical behavior of bound electrons in the hydrogen atom, and it must be able to replicateBohr modelIn addition, this equation must be able to explain the fine structure given by Sommerfeld model.
Soon, Schrodinger deduced a relativistic wave equation through the relativistic theory in De Broglie's paper. He applied this equation tohydrogen atom, calculate the wave function of the bound electron.Because Schrodinger didn't convert thespinTaking into account, the fine structure formula derived from this equation does not conform to the Sommerfeld model.He had to modify the equation, remove the relativistic part, and use the remaining non relativistic equation to calculate the spectral line of the hydrogen atom.The work of analyzing this differential equation is quite difficult. With the help of his good friend, mathematician Herman Weir, he copied the answer exactly the same as that of Bohr model.Therefore, he decided not to publish the relativistic part for the time being, but only to write the non relativistic wave equation and hydrogen atomic spectrum analysis results as a paper.In 1926, he officially published this paper.
This paper quickly shocked the quantum academia.Planck said, "He has read the whole paper, just like a child who has been puzzled by a riddle for a long time and yearns to know the answer. Now he finally heard the answer".Einstein praised that the inspiration of this work came from a real genius like a spring.Einstein felt that Schrodinger had made a decisive contribution.Because the wave mechanics created by Schrodinger involves the familiar wave concepts and mathematics, rather thanMatrix mechanicsAbstract and unfamiliarMatrix algebraQuantum scholars are happy to start learning and applying wave dynamics.Discoverer of spinGeorge Eugene Uhlenbeck Surprised, "Schrodinger equation has brought us great relief!"Wolfgang PauliI think this paper is one of the most important works.
Schrodinger equation given by Schrodinger can correctly describe the quantum behavior of wave function.At that time, physicists did not know how to interpret the wave function. Schrodinger tried tocharge densityIt is not successful to interpret the absolute value square of the wave function.In 1926, Born put forward the concept of probability amplitude, which successfully explained the physical meaning of wave function.But Schr ö dinger and Einstein both disagree with this statistical or probabilistic method and its associated discontinuitywave function collapse 。Einstein claimed that quantum mechanics is aDeterminismStatistical approximation of.In the last year of Schrodinger's life, he wrote a letter to Born, in which he clearly stated that he did not acceptCopenhagen Interpretation 。[2]
About the author
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Erwin Schrodinger (Erwin Schrodinger, 1887-1961) Born on August 12, 1887AustriacapitalVienna。From 1906 to 1910, he studied physics at the University of Vienna.He received his doctor's degree in 1910.After graduation, he was engaged in experimental physics at the Second Institute of Physics, University of Vienna.During the First World War, he was drafted to serve in a remote artillery fortress and used his spare time to studyTheoretical physics。
Erwin Schrodinger
After the war, he returned to the Second Institute of Physics.In 1920, he went to Jena University to help Wayne work.In 1921, Schrodinger was employed as a professor of mathematical physics at Zurich University in Switzerland, where he worked for six years. Schrodinger equation was proposed during this period.In 1927, Schrodinger succeeded Planck as professor of theoretical physics at Berlin University.After Hitler came to power in 1933, Schr ö dinger was deeply indignant at the Nazi regime's fascist behavior of persecuting Einstein and other outstanding scientists, and moved to Oxford, where he served as a member of the Magdalen Academyvisiting professor。In the same year, he anddirac And jointly won the Nobel Prize in Physics.
In 1936, he returned to Austria as a professor of theoretical physics at Graz University.Less than two years later, after Austria was annexed by the Nazis, he fell into adversity again.In October 1939, he exiled to Dublin, the capital of Ireland, and became the director of the Dublin Institute of Advanced StudiesTheoretical physicsResearch.During this period, he also conducted research on scientific philosophy and biophysics, which made great achievements.He published what is life, and tried to useQuantum physicsclarifygenetic structure Stability.In 1956, Schr ö dinger returned to Austria and was employed as a professor of theoretical physics at the University of Vienna. The Austrian government gave him great honor and set up a national prize named after Schr ö dinger, which was awarded by the Austrian Academy of Sciences.
Specific introduction
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Mathematical form
One-dimensional Schrodinger equation
Three dimensional Schrodinger equation
Stationary Schrodinger equation
Mathematical expression of single particle Schrodinger equation
This is a second-order linear partial differential equation,ψ(x, y, z) is the function to be solved, which is a complex function of the three variables of x, y, z (that is, the function value is not necessarily a real number, but may also be an imaginary number).The inverted triangle on the left of the formula is the Laplacian operator, which means thatψ(x, y, z).
Physical meaning
This is a stationary Schrodinger equation describing a particle in a three-dimensional potential field.The so-called potential field is the field in which particles will have potential energy, such aselectric fieldIs the potential field of a charged particle;so-calledStationary stateThat is, assuming that the wave function does not change with time.Among them,EIs the energy of the particle itself;U(x, y, z) is a function describing the potential field, assuming that it does not change with time.The Schrodinger equation has a good property that time and space are separated from each otherStationary wave functionThe space part of is multiplied by the time part
Then it becomes a complete wave function.
The Solution of Schrodinger Equation -- Properties of Wave Function
Simple systems, such as the Schrodinger equation of the electron in the hydrogen atom, can only be solved, and complex systems must be approximately solved.Because for those with Z electronsatomThe interaction potential energy of its electrons will change due to the shielding effect, so it can only be solved approximately.The methods of approximate solution mainly include variational method andPerturbation method。
The main quantum number n is the quantum number related to energy.atomWith separationenergy level, energy can only take a series of values, and each wave function corresponds to the corresponding energy.Hydrogen atom andHydrogen like atomDiscrete values for are:
，nThe higher the energy, the farther the electron layer is from the nucleus.The main quantum number determines the distance between the region with the greatest probability of electron appearance and the nucleus, and determines the energy of the electron.N=1，2，3，……；It is usually represented by K, L, M, N, etc.
Angular quantum numberlA quantum number related to energy.The electron has definiteangular momentumL, its value is not arbitrary, and can only take a series of discrete values, called angular momentumquantization。
。lThe larger the angular momentum is, the higher the energy is, and the shape of the electron cloud is different.l=0, 1, 2,... are usually expressed by s, p, d, f, g, which is simply the previous statementElectronic sublayer。The angular quantum number determines the orbital shape, so it is also called the orbital shape quantum number.sIt is spherical,pDumbbell type,dIs a petal,fThe orbit is more complex.
The magnetic quantum number m is independent of the electron energy.The orbital angular momentum of the electron moving around the nucleus in an atom, whose component in the direction of the external magnetic field isquantizationAnd quantum numbermdecision,mIt is called magnetic quantum number.For any selected external magnetic field direction Z, angular momentumLComponent L in this directionzOnly a series of discrete values can be taken, which is called spatial quantization.
。The magnetic quantum number determines the space extension direction of the atomic orbit, that is, the orientation of the atomic orbit in space,sTrack in one direction (ball),pTrack in 3 directions,d5 tracks,f7 tracks.lSame,mDifferent atomic orbitals with the same shape and different spatial orientations have the same energy.The phenomenon that different atomic orbits have the same energy is calledEnergy degeneracy。
The atomic orbitals with the same energy are calledDegenerate orbitThe number is called degeneracy.aspThere are three degenerate orbits,DegeneracyIs 3.Degenerate orbits will produce energy difference under the action of external magnetic field, which is the reason why the linear spectrum splits under the magnetic field.
The spin of a particle also produces angular momentum, which depends on the spin magnetic quantum number（ms）。electron spinAngular momentum isquantizationWhose value is
，sbySpin quantum number, a component of spin angular momentumLszThe following discrete values shall be taken:
。
Atomic spectrum. Under the high-resolution spectrometer, each ray is composed of two very close spectral lines. To explain this phenomenon, the spin of particles is proposed.The spin of the electron represents two different states of the electron, which have different spin angular momentum.
The spin of an electron is not the self rotation of a machine, it is its own intrinsic attribute, and it is also a new degree of freedom. For example, mass and charge are its internal attributes. The spin angular momentum of an electron is:ħ/2。[3]
Correspondence
Hilbert Space and Schrodinger Equation
Generally, in physics, the physical state corresponds to the vector on the Hilbert space, and the physical quantity corresponds to the operator on the Hilbert space.The Schrodinger equation in this form is
HIs the Hamiltonian operator.This equation fully shows the correspondence between time and space in this form (time corresponds to energy, just as space corresponds to momentum).This operator（physical quantity）The description method of natural phenomena that does not change with time but changes with time is called Schrodinger's painting, which corresponds to Heisenberg's painting.
The space coordinate operator x and its corresponding momentum operator p meet the following exchange relationship:
The so-called Schrodinger representation is that the space operator is directly regarded as x, while the momentum operator is the following differential operator containing differential:
