Solution of the hydrogen atom

In summary, the conversation discusses the possibility of solving the hydrogen atom problem using creation and annihilation operators, as is the case with the harmonic oscillator. The book <Principles of Quantum Mechanics - As Applied to Chemistry and Chemical Physics> by D. Fitts is recommended for further reference. However, it should be noted that the solution provided in the book only covers the discrete spectrum and it is unknown if a similar analysis has been performed for the continuous spectrum. The use of stepping operators in solving the hydrogen atom is considered a miracle due to the fact that the Hamiltonian can be factored into a product of linear factors, which can be attributed to the algebraic properties of Hermite polynomials and Laguerre functions or to additional
  • #1
crawf_777
8
0
Just a quick question regarding the solution of the hydrogen atom: is it possible to solve the hydrogen atom problem via creation and annihilation operators as is the case with the harmonic oscillator?

Any help here greatly appreciated!

Crawf.
 
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  • #2
Yes, of course. See the book of D. Fitts: <Principles of Quantum Mechanics - As Applied to Chemistry and Chemical Physics>, from page 161 onwards.
 
  • #3
Thanks very much for the ref! I'll have a look at said book!
 
  • #4
The solution provided in the book I mentioned covers only the discrete spectrum. I don't know if a similar analysis has been performed for the continuous spectrum as well.
 
  • #5
If you think the answer is 'yes of course' let's see you do it for the finite square well!

It's automatic only in the following sense: you have a bunch of states and can always define stepping operators that take you from one state to the next. But as a practical matter, the fact that the hydrogen atom can be solved by the use of stepping operators is far from automatic: it's a miracle!

The Hamiltonian is a quadratic expression in x and p, and the miracle is that this expression can be factored into a product of linear factors. You can explain this as do to the remarkable algebraic properties of Hermite polynomials and Laguerre functions. Or you can ascribe it to 'extra' symmetries, such as the Runge-Lenz vector. But there are very few systems that can be handled this way.
 
  • #6
It's called "supersymmetric method" if I recall correctly.
 
  • #7
Thanks for your useful input Bill_K. (and TTB).
 
  • #8
Visualize charge density

I want to use Matlab to visualize charge density aroud hydrogen nuclear atom. I have had wave function already. Does anyone know how to do it?
Thanks in advance
 

1. What is the solution of the hydrogen atom?

The solution of the hydrogen atom is a mathematical representation of the energy levels and wave functions of the electron in a hydrogen atom. It describes the probability of finding the electron at a certain distance from the nucleus.

2. How was the solution of the hydrogen atom discovered?

The solution of the hydrogen atom was first proposed by physicist Niels Bohr in 1913. He combined classical mechanics with the newly discovered quantum theory to develop the Bohr model of the atom, which provided a solution for the energy levels of the hydrogen atom.

3. What is the significance of the solution of the hydrogen atom?

The solution of the hydrogen atom is significant because it was the first successful attempt to explain the atomic structure and the behavior of electrons in an atom. It also laid the foundation for the development of quantum mechanics and led to a better understanding of the nature of matter.

4. What are the components of the solution of the hydrogen atom?

The solution of the hydrogen atom consists of the energy levels, wave functions, and quantum numbers. The energy levels represent the specific energies that the electron can have, while the wave functions describe the distribution of electron density in space. The quantum numbers specify the energy level and orbital of the electron.

5. How does the solution of the hydrogen atom relate to the periodic table?

The solution of the hydrogen atom provides a foundation for understanding the electronic structure of all elements, which is reflected in the periodic table. The number of electrons in an atom determines its chemical properties, and the solution of the hydrogen atom helps to explain the arrangement of electrons in different elements on the periodic table.

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