2. Report1 [hydrogen atom wave function]

The most beautiful experiment in the world

Calculation of the Hydrogen atom wave function for the ground state

 

3.1. Goal

Find the potential energy of a hydrogen atom and substitute it into the time-independent Schrödinger equation. The wave function can be obtained by solving the Schrödinger equation, and the squaring of the wave function becomes the probability density function. The potential energy and wave function are output as coordinates in a txt file and the probability density function is expressed as a graph.

 

3.2. Theoretical background

Computational physics

In computational physics, there are numerical analysis methods, computer simulation methods, and special-purpose machine development methods. The method used in the work is a Numerical Analysis method, and although this method knows the differential equation that is believed to govern the phenomenon to be analyzed, it mainly refers to the method commonly used when it is difficult to analyze the solution analytically. can do. This method has a significant effect, and it is well known which calculation method (Algorithm) is effective to use according to the form of various differential equations. In addition, fitting, interpolation, and extrapolation, which effectively complement experimental data, are also called numerical methods, and effective calculation methods for these are also known. If such a method is meaningful, it will be possible to overcome the problems of cost, effort, and time in experimental physics considerably. However, it can be said that there is a problem in how meaningful data supplementation in complex phenomena is. In this work, we did not use other data supplementary calculation methods (algorithm), but simply compare the numerically obtained values ​​with actual theoretical values ​​and output the results as graphs if they exist within the error range.

 

Hydrogen atom wave function

It is usually expressed in the form of a function ψ(x) of the spatial coordinate x, and is the basic equation of quantum mechanics. It is obtained by E. Schrödinger's wave equation under the necessary conditions. As a direct expression, it represents the wavelike aspect of the particle as a material wave, but physically, it means that the square of its absolute value ψ(x)2 is proportional to the probability of finding a particle at the place x, and the wave and It can be said to be a quantity that uniformly expresses particle nature. Solving the Schrodinger equation for a hydrogen atom gives four quantum numbers (n, l, ml, ms). These four quantum numbers determine the energy, angular momentum, distribution, and spin state of each electron. And the four quantum numbers were regarded as one quantum state, and the symbols for each quantum state were determined by the formulas 1s, 2s, 2p, 3s, 3p, and 3d. In the Bohr model, each electron orbits a given orbit, and the electron moves to another orbit only when it absorbs or emits light corresponding to the energy of the other orbit. However, according to the Schrodinger equation, electrons do not have a definite orbit, but rather a distributed distribution like a cloud. So we name it orbital because it is different from orbital.

 

 

Only the final expression reflected in the code is summarized here. Although I did not learn all the physical meanings of the expressions, I understood the connection of each method and reflected it in the code. All formulas appearing in the bibliography are in the appendix. The method used during the process was used differently depending on the required accuracy. Analytical solutions are more detailed by referring to university physics or modern physics books. The numerical method was coded by referring to the examples of the method in the references.