catalogue
definition
brief introduction
The proposal of Schrodinger equation
Introduction of Schrodinger
Brief introduction of Schrodinger equation
Mathematical expression of single particle Schrodinger equation
Solution of Schrodinger Equation —— Properties of Wave Function
Hilbert space and Schrodinger equation
launch
Schrodinger equation
Schrodinger equation is also called Schrodinger wave equation. In quantum mechanics, the state of the system is not determined by the value of the mechanical quantity (for example, X), but by the function ψ (x, t) of the mechanical quantity, that is, the wave function (also called probability amplitude and state function), so the wave function becomes quantum. How the probability distribution of mechanical quantities changes with time can be solved by solving Schrodinger equation of wave function. This equation was put forward by Austrian physicist Schrodinger in 1926. It is one of the most basic equations in quantum mechanics, and its position in quantum mechanics is equivalent to that of Newton's equation in classical mechanics.
Schrodinger equation is the most basic equation and a basic hypothesis of quantum mechanics, and its correctness can only be tested by experiments.
Detailed entry: Erwin Schr?dinger
Irving Schrodinger (1887— 196 1 year) was born in August 1887 in Vienna, Austria. 1906- 19 10 studied in the physics department of Vienna university. 19 10 received his doctorate. After graduation, I worked in the Second Institute of Physics of Vienna University. During the First World War, he was recruited to a remote artillery fortress to study theoretical physics in his spare time. After the war, he still returned to the Second Institute of Physics. 1920 went to Jena University to assist Wayne in his work. 192 1 year, Schrodinger was hired as a professor of mathematical physics at the University of Zurich, Switzerland, where he worked for six years, during which he put forward the Schrodinger equation.
[Erwin Schr?dinger]
Erwin Schr?dinger
1927 Schrodinger succeeded Planck as a professor of theoretical physics at the University of Berlin. 1933 after Hitler came to power, Schrodinger was deeply indignant at the fascist behavior of the Nazi regime persecuting Einstein and other outstanding scientists, and moved to Oxford University to be a visiting professor at Madelen College. In the same year, he won the Nobel Prize in Physics with Dirac.
1936, he returned to Austria and became a professor of theoretical physics at Graz University. Less than two years later, Austria was annexed by the Nazis, and he fell into adversity again. 1939 10 went into exile in Dublin, Ireland, and became the director of the Institute of Advanced Studies in Dublin, engaged in theoretical physics research. During this period, he also conducted research on philosophy of science and biophysics, and made great achievements. Published a book "What is Life", trying to explain the stability of gene structure with quantum physics. From 65438 to 0956, Schrodinger returned to Austria and was hired as a professor of theoretical physics at Vienna University. The Austrian government gave him great honor and established a national prize named after Schrodinger, which was awarded by the Austrian Academy of Sciences.
Edit the Schrodinger equation in this paragraph.
2
?
-—— —— ψ(x，t)+V(x)ψ(x，t)=i？ ——ψ(x，t)=Hψ(x，t)
2
2m？ x？ x
Where h is Hamiltonian.
Steady-state Schrodinger equation;
2 ?
-—— ▽ ψ(r，t)+V(r)ψ(r，t)=i？ ——ψ(x，t)=Hψ(x，t)
2m？ x
Edit the mathematical expression of this single particle Schrodinger equation.
Mathematical form
This is a second-order linear partial differential equation, and ψ(x, y, z) is a function that requires solution, and it is a complex function of three variables (that is, the function value is not necessarily a real number, but may also be a complex number). The left-most inverted triangle of the formula is an operator, which means to find the sum of squares of partial derivatives of x, y and z coordinates of ψ(x, y and z) respectively.
physical significance
This is a stationary Schrodinger equation describing particles in a three-dimensional potential field. The so-called potential field is the field where particles have potential energy. For example, the electric field is the potential field of charged particles. The so-called steady state is to assume that the wave function does not change with time. Where e is 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. Schrodinger equation has a very good property, that is, time and space are separated from each other. Calculate the space part of the steady-state wave function and multiply it by the time part E (-T * I * E * 2π/h) to form a complete wave function.
Edit the solution of Schrodinger equation in this paragraph-the properties of wave function.
Simple systems such as Schrodinger equation of electrons in hydrogen atoms can be solved, while complex systems must be solved approximately. Because for atoms with z electrons, the potential energy of their electrons will change due to the shielding effect, so they can only be solved approximately. Approximate solutions mainly include variational method and perturbation method.
Under the bound state boundary condition, not all solutions corresponding to the value of e are physically acceptable. Principal quantum number, angular quantum number and magnetic quantum numbers are all solutions of Schrodinger equation. To describe the electronic state completely, we must have four quantum numbers. The spin magnetic quantum number is not the solution of Schrodinger equation, but is accepted as an experimental fact.
Principal quantum number n
Quantum number related to energy. Atoms have discrete energy levels, and energy can only take a series of values, and each wave function corresponds to the corresponding energy. The discrete values of hydrogen atom and hydrogen-like atom are:
En =-1/n * 2× 2.18×10 * (18) j, the greater the energy, the farther away the electron layer is from the nucleus. Principal quantum number determines the distance between the region with the highest probability of electron appearance and the nucleus, and determines the energy of the electron. N= 1，2，3，； Commonly used k, l, m, n.
Angular quantum number l
Quantum number related to energy. Electrons have a definite angular momentum L in atoms, and its value is not arbitrary, and it can only take a series of discrete values, which is called angular momentum quantization. L=√l(l+ 1) (h/2π)，l=0， 1，2，(n- 1)。 The greater the L, the greater the angular momentum and energy, and the different shapes of electron clouds. L = 0, 1, 2, commonly expressed by S, P, D, F, G, which is simply the electron sublayer mentioned above. Angular quantum number determines the orbital shape, so it is also called the non-orbital quantum number. S is spherical, P is dumbbell-shaped, D is petal-shaped, and F orbit is more complicated.
Magnetic quantum number m
And quantum numbers without energy. The orbital angular momentum of electrons in atoms moving around the nucleus is quantized in the direction of external magnetic field, which is determined by quantum number m, called magnetic quantum number. For any selected external magnetic field direction Z, the component LZ of angular momentum L in this direction can only take a series of discrete values, which is called spatial quantization. LZ=m h/2π, m=0, 1, 2L. The magnetic quantum number determines the spatial extension direction of the atomic orbit, that is, the orientation of the atomic orbit in space. One direction (ball) is S orbit, three directions are P orbit, five directions are D orbit and seven directions are F orbit. L is the same, m is different, that is, the orbital energy of atoms with the same shape and different spatial orientations is the same. The phenomenon that different atomic orbits have the same energy is called energy degeneracy.
Atomic orbits with the same energy are called degenerate orbits, and their number is called degeneracy. For example, the P orbit has three degenerate orbits, and the degeneracy is 3. Degenerate orbits will produce energy differences under the action of external magnetic field, which is the reason why linear spectra split under magnetic field.
Spin magnetic quantum number m
The spin of particles also produces angular momentum, which depends on the spin quantum number. The angular momentum of electron spin is quantized, and its value is ls = √ s (s+ 1) (h/2π), s= 1/2, and s is the quantum number of spin. A component Lsz of spin angular momentum should take the following discrete values: lsz = ms (h/2π), ms = 66π.
Atomic spectrum, under the high-resolution spectrometer, each ray is composed of two very close spectral lines, and the spin of particles is proposed to explain this phenomenon. The spin of electrons indicates two different states of electrons, which have different spin angular momentum.
The spin of electrons is not the rotation of machinery itself, but its inherent properties and new degrees of freedom. Just as mass and charge are its intrinsic properties, the spin angular momentum of an electron is:? /2。
Edit this Hilbert space and Schrodinger equation.
General physical states correspond to vectors in Hilbert space, and physical quantities correspond to operators in Hilbert space. This form of Schrodinger equation is shown on the right.
[Schrodinger equation]
Schrodinger equation
H is Hamiltonian operator. This equation fully shows the correspondence between time and space in this form (time corresponds to energy, just like space corresponds to momentum, which will be discussed later). This description method of natural phenomena in which the operator (physical quantity) does not change with time but the state changes with time is called Schrodinger's landscape painting. Accordingly, Heisenberg painted landscapes.
The spatial coordinate operator X and its corresponding momentum operator P satisfy the following exchange relation: