Schrodinger equation

5.
2.
1 Schrodinger equation

In 1926, the Austrian physicist Schrodinger established a wave equation describing microscopic particles, which is a second-order partial differential equation, namely

In the formula, the wave function Ψ is the function of x, y, z; E is the total energy of the electron; V is the potential energy of the electron; m is the mass of the electron; h is the Planck constant; π is the circumference ratio
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The relationship between the potential energy V of the extranuclear electron in the Schrodinger equation and the atomic number Z, the original charge e, and the distance r between the electron and the nucleus is

In the formula, ε ₀ is the vacuum dielectric constant
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The r in the potential energy term of Schrodinger's equation is related to the three variables of x, y, and z at the same time, which brings great difficulties to the solution of the equation
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In order to solve the equation, people carry out coordinate transformation on the Schrodinger equation, transforming the three variables of rectangular coordinates (x, y, z) into three variables of spherical coordinates (r, θ, Φ), as shown in Figure 5-2

Variable separation

Ψ(r,θ,Φ)=R(r)·Θ(θ)·Φ(φ)

After the variables are separated, the partial differential equations of the three variables are decomposed into three ordinary differential equations with one variable each
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Among them, R(r) is only related to r, that is, only related to the distance between the electron and the nucleus, which is called the radial part of the wave function

Y(θ,Φ)=Θ(θ)·Φ(φ)

Y(θ, Φ) has nothing to do with r, only related to angles θ and φ, which is called the angle part of the wave function
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Figure 5-2 The relationship between rectangular coordinates and spherical coordinates

Solve the three ordinary differential equations of R(r), Θ(θ), and Φ(φ) respectively, and obtain the solutions of three univariate functions of r, θ and φ
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When solving ordinary differential equations to find Φ(φ), a parameter m must be introduced, and only when m takes some special value, Φ(φ) has a reasonable solution; when solving ordinary differential equations to find Θ(θ) , We need to introduce a parameter l, and only when l takes some special value, Θ(θ) can have a reasonable solution; when solving ordinary differential equations to find R(r), we need to introduce a parameter n, and only when n When taking some special values, R(r) has a reasonable solution

The solution of Schrodinger’s equation is a series of three-variable, three-parameter functions, namely

Each wave function corresponding to Ψn, l, m (r, θ, φ), has a specific energy E
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For H atoms and hydrogen-like ions with only one electron

In the formula, Z is the atomic number and n is the parameter (the principal quantum number mentioned later)
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Below are a few examples of wave function Ψ
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In the formula, a0=52.

9pm, which is the Bohr radius; the subscripts "1, 0, 0", "2, 1, 0", "3, 2, 2" are the values ​​of the parameters "n, l, m"

The wave function that describes the motion state of the electron obtained by solving the Schrodinger equation is called the atomic orbit
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But unlike Bohr's orbit, the wave function is an orbital function, which is the space area where electrons move outside the nucleus, not the trajectory
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