Schrodinger equationIt is the most basic equation of quantum mechanics and a basic assumption of quantum mechanics. Its correctness can only be tested by experiments.Schrodinger equation is the basic equation of quantum mechanics, which reveals thebasic lawJust as Newton's law plays a role in classical mechanics, it is a powerful tool in atomic physics to deal with all non relativistic problemsnuclear physics, chemistry and other fields.
Schrodinger was born in Vienna, the capital of Austria, on August 12, 1887.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.the First World WarDuring this period, he was drafted to serve in a remote artillery fortress, and used his spare time to study theoretical physics.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. He moved to Oxford and worked as a visiting professor at the Magdalen College.In the same year, he and Dirac jointly obtainedThe nobel prize in physics。
In 1936, he returned to Austria as a professor of theoretical physics at Graz University.In less than two years,AustriaAfter being 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 Studies, engaged in theoretical physics research.During this period, he also conducted research on scientific philosophy and biophysics, which made great achievements.He published What Is Life, trying to illustrate the stability of genetic structure with quantum physics.In 1956, Schrodinger 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 a national prize named after SchrodingerAustrian Academy of SciencesGrant.
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Schrodinger equationSchrdinger equation is a basic equation in quantum mechanics proposed by Austrian physicist Schr ö dinger. It is also a basic assumption of quantum mechanics. Its correctness can only be tested by experiments.
In quantum mechanics, a basic problem is that the Hamiltonian hat is not a function of time.At this time,
It can be decomposed into the product of a function only related to space and a function only related to time, that is
。If you take it into the Schrodinger equation, you will get
。and
Then the following equation is satisfied:
application
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In quantum mechanics, solving particle problems often boils down to solving Schrodinger equation orStationary 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 taken into account, the Schrodinger equation is replaced by the relativistic quantum mechanical equation, which naturally includes the spin of particles.
The basic equation of quantum mechanics proposed by Schrodinger was established in 1926.It is aNonrelativisticWave equation of.It reflects the law describing the state of microscopic particles changing with time, and its position in quantum mechanics is equivalent toNewton's lawAs for classical mechanics, it is one of the basic assumptions of quantum mechanics.Let the wave function describing the state of microscopic particles be
, the microscopic particles with mass m in the potential field
The Schrodinger equation of motion in is.Given the initial conditions, boundary conditions and the single value, finite and continuous conditions that the wave function satisfies, the wave function can be solved
。From this, the distribution probability of particles and the average value (expected value) of any possible experiment can be calculated.When the potential function U is independent of time t, the particle has a certain energy, and the state of the particle is called a stationary state.Stationary stateThe wave function of time can be written as
be calledStationary wave function, satisfies the stationary Schrodinger equation, which is mathematically called the eigenequation, where E is the eigenvalue and is the stationary energy,
It is also called the eigenfunction belonging to the eigenvalue E.[1]
Solution method
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Finite difference method
In view of the problem that most of the Hamiltonian operators of quantum systems in quantum mechanics are relatively complex, and the Schrodinger equation can not get strict solutions or analytical solutions, the paper proposes to use theFinite difference methodTo solve the eigenproblem of Schroeder's norm equation in computational quantum mechanics.The ordinary radial Schrodinger equation and time-dependent Schrodinger equation are analyzed by the finite difference method, and the discrete equations of the finite difference method of the two Schrodinger equations are givenLinear harmonic oscillatorAs an example, computer programming calculation is carried out.The results show that this method has a broad application prospect in the study of quantum mechanics problems.[2]
Factorization
Using De La Pena's expression of the ladder operator method, we canFactorizationTo standardize the steps in order to be widely used in the teaching of elementary quantum mechanics and quantum chemistry.We apply this method to free particles, three-dimensional isotropic harmonic oscillator and two-dimensional hydrogen atom in spherical coordinates, and obtain satisfactory results.[1]
