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The total energy of hydrogen-like atoms is a key concept in Quantum Mechanics and Atomic Physics. Hydrogen-like atoms, also known as one-electron ions, are atoms that have only one electron in their outer shell, similar to a hydrogen atom. The total energy of these atoms is determined by several factors, including the reduced mass, electronic charge, atomic number, permittivity, Planck's constant, and the principal quantum number.

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Total Energy (E) = |

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The total energy of a hydrogen-like atom can be calculated using the following formula:

E = - μ Z^{2} e^{4} / (8 ε_{0}^{2} h^{2} n^{2})

Where:

- E is the total energy of the atom.
- μ is the reduced mass of the electron.
- Z is the atomic number, representing the number of protons in the nucleus.
- e is the electronic charge.
- ε
_{0}is the permittivity of free space. - h is Planck's constant.
- n is the principal quantum number, which determines the energy level of the electron.

This formula for the total energy of hydrogen-like atoms is a result of the work done by Niels Bohr in the early 20th century. Bohr's model of the atom provided an explanation for the stability of the atom and the emission and absorption lines of hydrogen.

The concept of total energy of hydrogen-like atoms finds its application in atomic physics, astrophysics, and spectroscopy. Spectroscopy, in particular, uses this concept to identify elements based on their spectral lines, which are dependent on the energy states of their atoms. This is also crucial for understanding the behavior of stars, as their color and light intensity relate to the energy states of their constituent atoms.

Niels Bohr is a key figure in this field. His postulates about the quantized energy states of the electron in a hydrogen atom laid the foundation for Quantum Mechanics. Another important physicist, Max Planck, contributed the concept of Planck's constant, a crucial element in this formula.

- The total energy of hydrogen-like atoms is always negative. This is because the electron is bound to the nucleus and energy must be added to free it.
- The Bohr model, while not completely accurate (as it doesn't consider the wave nature of electrons), provides a good approximation for one-electron ions.
- Quantum mechanics, which emerged partly from the study of total energy in atoms, has completely transformed our understanding of the physical world, including the development of technologies like lasers and semiconductors.

Understanding the total energy of hydrogen-like atoms is vital in the study of Quantum Mechanics and Atomic Physics. This knowledge serves as a fundamental building block in modern physics and continues to have far-reaching implications in both theoretical developments and practical applications, such as spectroscopy and the study of stellar behavior.

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