1. Summary [hydrogen atom wave function]

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 a 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 a place called x, and the wave and It can be said to be a quantity that uniformly expresses particle nature. For example, the wavefunction corresponding to each energy level of a hydrogen atom gives a density distribution with a maximum on the Bohr orbital. The wavefunction is usually given as a complex number, then the value ψψ of it and the conjugated complex number gives the density distribution of the particle. The wave function for n particles is usually multiplied by an appropriate constant so that the integral of ψψ over the entire space becomes n. This is called normalization, and the constant is called normalization constant. An operator corresponding to an arbitrary physical quantity M is integrated with a wave function ψ related to a state and its conjugated complex number ψ to give the temporal average value of M in that state.

 

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1.1. Code

#define size  	500
#define hbarc 	(197.33*1000.*1.e-5)
#define ee    	0.01440
#define prm   	(938.28*1000.)
#define elm   	(0.511003*1000.)
#define ymg   	1.e-18
#define xmes  	0.01
#define l1    	0

It is a value referring to the appendix of University Physics. The size can be adjusted arbitrarily, and if this value is changed, the execution time may become longer, so it will be adjusted later.

 

Use the bore radius. The Bohr radius is the smallest radius of an orbit when the principal quantum number n = 1.

 

For the potential energy function of a hydrogen atom due to the electrical interaction between electrons and protons, refer to Equation 1053p in Chapter 42 of University Physics.

 

The equation for the allowable energy level refers to the equation in University Physics (1050p). At this time, since the unit is eV, keV units are used for ecm and de for unit conversion when calculating later.

 

In order to intensify, an integration process must be performed. Other mathematics courses must also be conducted. First of all, the integration process is attempted to approximate the micro-intervals by using the histogram method, which is the method used in autonomous research. We plan to use the approximation process of the NAMD tutorial used in autonomous research (numerov method will be used to solve the one-dimensional time-independent Schrödinger equation numerically) and the equations learned in university physics. Calculations are done on a nanometer scale, respectively.

 

 

1.2. 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.

 

 

1.3. Significance

Studying physical phenomena is not an easy task. In particular, experiments and research that require a quantum mechanics perspective will be more so. Through this work, we will be able to know more deeply about the hydrogen atom model, and furthermore, we will be able to check the probability distribution prediction of the multi-electron atom model and the probability distribution when the Bohr radius is manipulated differently. This work will be a jumping off point for me to become a future computational physicist.