# Algebraic solution of hydrogenic atom

• kowalski
In summary, the hydrogen atom was solved algebraically by Wolfgang Pauli in 1925, before the works of Heisenberg and Schroedinger. Kurt Gottfried's book on QM (2nd edition, 2003) provides more information and references on the issue. Prior to Pauli's solution, there were models such as Bohr and Bohr-Sommerfeld, but they did not produce the same results as QM. Biedenharn's "Theory of angular momentum" also offers a detailed account of the algebraic solution for the H-atom. More information can also be found in the Laplace–Runge–Lenz vector Wikipedia page.
kowalski
I've read that the hydrogen atom was solved by means of algebraic methods (similar to creation and annihilation operator for the harmonic oscillator) even before the works of Heisenberg and Schroedinger.

Thanks, Kowalski.

It was actually solved algebraically in 1925 by Wolfgang Pauli. See the pages 235 up to 244 of Kurt Gottfried's book on QM (2nd edition, 2003).

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There was the Bohr and Bohr-Sommerfeld models. But there was no model that got the QM result prior to Schrödinger, I'm almost certain.

bigubau said:
It was actually solved algebraically in 1925 by Wolfgang Pauli. See the pages 235 up to 244 of Kurt Gottfried's book on QM (2nd edition, 2003).

Thank you very much, bigubau!. I will look for this book by Gottfried in the library as soon as possible. Probably he citates the original (Pauli paper) reference. Thanks!.

alxm said:
There was the Bohr and Bohr-Sommerfeld models. But there was no model that got the QM result prior to Schrödinger, I'm almost certain.

Bohr was an ad-hoc model (with rules for producing desired known results) ; and Sommerfeld-Ishiwara applied more general ideas (extremizing the Action integral associated to the problem) based on analytical mechanics.(Not algebraic but analytical calculus--calculus of variations).

I think the good answer is bigubau's, about Pauli (algebraic) solution --1925.

A better account on the algebraic solution for the H-atom can be found in Biedenharn's "Theory of angular momentum", Vol 1, Chapter 7, dection 4. A detailed bibliography can be found at the end of the section.

bigubau said:
A better account on the algebraic solution for the H-atom can be found in Biedenharn's "Theory of angular momentum", Vol 1, Chapter 7, dection 4. A detailed bibliography can be found at the end of the section.

Thank you bigubau. I will look for this reference. K.

Avodyne said:

OK, thank you Avodyne, very interesting wikipedia item!. K.

## 1. What is the hydrogenic atom?

The hydrogenic atom is a type of atom that has only one electron and is similar to the hydrogen atom. It is considered a simplified model for other atoms, as it only contains a nucleus (with a single proton) and an electron.

## 2. How is the hydrogenic atom solved algebraically?

The hydrogenic atom can be solved algebraically by using the Schrodinger equation, which is a mathematical equation that describes the behavior of quantum particles, such as electrons. By solving this equation for the hydrogenic atom, we can determine the energy levels and wavefunctions of the electron.

## 3. What is the significance of the algebraic solution of the hydrogenic atom?

The algebraic solution of the hydrogenic atom allows us to understand and predict the behavior of electrons in other atoms and molecules. It also provides a foundation for the study of quantum mechanics and helps us understand the properties of matter at a microscopic level.

## 4. What are the limitations of the algebraic solution of the hydrogenic atom?

The algebraic solution of the hydrogenic atom only applies to atoms with one electron, so it cannot be used to solve more complex systems. It also does not take into account the effects of relativity, which become significant for heavier elements.

## 5. How does the algebraic solution of the hydrogenic atom relate to the periodic table?

The algebraic solution of the hydrogenic atom provides the basis for understanding the organization of the periodic table. The energy levels and wavefunctions of electrons in different atoms can be mapped onto the periodic table, allowing us to predict the properties of elements based on their position in the table.

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