Quantum mechanics is based on abstract thinking, a physical theory that studies microscopic objects that real physical processes cannot directly observe. How to visualize and visualize abstract and complex quantum theory is a direction of people's exploration and research. In recent years, with the rapid development of computer technology, the three-dimensional computer reconstruction technology has become more and more mature. The combination of computational physics methods and the powerful functions of computer drawing software has made the visual analysis of abstract quantum theory more vivid. The Schrödinger equation can be solved strictly, and the wave function can be written in an analytic form. Therefore, whether it is in the study of quantum mechanics or the analysis of atomic and molecular structure, revealing the nature of chemical bonds has an important role. In-depth research has been done157, but because the theory is relatively abstract, the visual description of the wave function of hydrogen atoms has also attracted people's attention. Zhang Jianhua et al. Used the Origin6.0 software to draw the hydrogen atom wave function graph 1891. Chen Zhongxian also used the MC method to make a simple discussion on the simulation of the hydrogen atom electron cloud. The electron cloud distribution under the state is visually analyzed, which intuitively reveals the probability distribution law of the hydrogen atom electron cloud, which provides ideas for the visual analysis of the abstract wave function of quantum mechanics.
1 Wave function of single-electron system The Schrodinger equation of electrons in the spherical polar coordinates of the nuclear potential field in the single-electron system is 111121: Fund project: Xi'an University of Technology Fund Project (108210309).
ma (lemigxaut.e (lu.cn Shi Wei (1957), male, Zhejiang Jinhua, professor, doctoral supervisor, research direction is the design and development of solid-state electronic devices. E set Xr0, outer = R (r) Y (e, Sentence, according to the separation variable method, the radial angular distribution function formula (2) and the spherical harmonic function formula (3) can be obtained: where the normalization can be obtained: according to the wave function solution of the stationary Schrodinger equation of the hydrogen atom, you can The probability distribution of electrons in hydrogen atoms at various points in space, but this physical model is very abstract and difficult to understand.
The following uses Matlab software to visually analyze and discuss the distribution of hydrogen atom electron cloud from three aspects: radial, angle and space.
2 The distribution of the hydrogen atom electron cloud will be (4) from e to 1, from-and from, and note that Y /, m (0, decay is normalized, you can get the radius r to r + dr The probability of finding electrons in the spherical shell is: use Matlab to draw the radial wave function Rn / (r) and the radial probability distribution function D (r) = r2R2 (r) curves a ', b', c in each state of the hydrogen atom , D 'and e' are the corresponding radial probability distribution functions. It can be seen that the electron cloud density of the hydrogen atom at the origin is the largest, and it decays exponentially as the radius r increases; the radial distribution function Rn, / (r ) There are nodes, that is, at some r value (r ping 0), Rn, / (r) = 0, and the number of nodes is related to the quantum number of the electronic state. For example, for R20 (r), there is 1 node (see a ) For R30 (r), there are 2 nodes (see b).
/ (R) is normalized, and the probability of electrons in the solid angle dfi = in the vicinity of (0 outside direction) can be obtained. Bookmark7. As can be seen, the probability of the electron density of the hydrogen atom angle under various states, where the color The deep indicates that the electron probability density wave function and its angular probability distribution function pattern shape are basically the same, and the degree is large. From the spatial probability distribution slice diagram, it can be seen more intuitively that the state of the hydrogen atom is different, and the angle distribution direction of the electron cloud is different from the shape of the hydrogen atom electron. The distribution of clouds. For example, b made a slice graph at n =. It can be seen that the probability density distribution of electrons in two dense spherical crowns and three-dimensional space is: a thin spherical belt ring with the dense spherical crown center at z On the axis, the spherical belt ring is on the Wn, /, m (r, e, two X / m (r, e, 2 (7) z = 0 plane, the radius phase of the spherical crown and the maximum density of the spherical belt ring using Matlab The software draws the space of hydrogen atoms in each state, etc. From b ', you can clearly see the probability distribution of electrons appearing in the spherical band. See the slice diagram. The shade of the color on the slice diagram indicates that the probability is relatively small.
3 Results discussion Through the visual analysis of the radial, angular and spatial distribution of the hydrogen atom electron cloud in different states, people have a more systematic and intuitive understanding of the distribution law of the hydrogen atom electron cloud.
For s-state electrons, IY and m2 are constants, so the electron cloud has nothing to do with the size of angle 9, and the shape of the electron cloud should be spherical.
However, it can be seen that the s-state electron cloud is not a hollow ball but a solid ball with the highest density of electron clouds at the center. If the space above a certain probability density value is an electron cloud sphere and this density value is called the boundary density, drawing the electron cloud such as a can intuitively see the distribution law of the electron cloud. For different states, that is, the main quantum number n takes different values, its electron cloud density distribution is different. As shown, the distribution is not necessarily monotonously decreasing, but may change after being reduced to zero (see figure lb '), and two peaks have been formed since then. If the appropriate boundary density is selected, concentric electron cloud shells will appear outside the solid electron cloud sphere, but the electron cloud density of these shells is much smaller than the center. Although these electron cloud densities are the largest at the origin, we can see from figures la 'and b' that the probability of electrons appearing at radius r is not at the origin. Because if the probability distribution with radius r as the parameter is considered, the spherical surface with radius r must be considered. Although the electron cloud density is closer to the origin, the area of ​​the spherical surface decreases with r-2.
For p electrons, although the relationship between the electron probability density and the angle is similar to a ', the distribution shape of the electron cloud is not the traditional spindle shape, because the radial probability distribution of electrons is also considered, and the total probability distribution is the two. The product of. c made a slice map of the p electron cloud with n = 1, / = l, m = 1 in x =, less = 0, z = 0, according to the depth of the color on the slice map, the electrons can be seen intuitively The size distribution of probability density.
In quantum mechanics, the state of electrons is described by the probability that electrons appear everywhere, and the concept of orbit is abandoned. But from the visual analysis of the hydrogen atom electron cloud, the traces of the orbit can also be clearly seen. Unlike the orbit theory, the orbit here is wider, not linear in the orbit theory, and much wider than the electron itself.
Through the visual analysis of the hydrogen atom electron cloud, we can visually understand the movement form of microscopic particles and the abstract wave function probability wave interpretation of quantum theory; at the same time, it provides a A feasible research idea and method.
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