## Introduction

The science that discusses and elucidates about very small particles’ scientific principles and behaviors and their interaction with atoms and subatomic particles is called Quantum Mechanics. Stedl (2005) states that “quantum mechanics is the study of matter and radiation at an atomic level”. Classical physics provides details about matter and energy on a certain scale that is familiar to human understanding. It also explains behaviour of astronomical bodies. Indeed, it is one of the important subject-matters of science and technology. Classical physics failed to explain the macro- and microworld that was successfully elucidated by Quantum Mechanics. It took place in the late 19th century. The major revolution in physics comes after the development of quantum mechanics. During the early decades of the 20th century, physicists exposed the limitations of standard physics. They developed the theory of Quantum Mechanics. The very word ‘Quantum’ refers to the minimum of material body occupied in a contact. The characteristics of light, to some extent, are like particles and waves. On the other hand, electrons and atoms, which are matter particles, behave like the same as well. Quantum mechanics elucidates that light comes in separate units. These separate units are photons, electromagnetic radiation, spectral intensities, and colours, including some inconsistent behaviour that is often observed at larger length scales. On the other hand, an atom of hydrogen is a particle of the chemical-component hydrogen. The atom that is electrically neutral includes a proton of positive charge and an electron of negative charge that are hurdles to the nucleus of that atom. Nearly 75% of the constituent weight (mass) of the world comprised of atomic hydrogen. ‘Atomic’ or ‘monatomic hydrogen’ is extremely rare to find in everyday life on this planet. Indeed, the characteristics of this atom are to unite with itself or other atoms to form gas, such as hydrogen (H2), or liquid, such as water (H2O). Such configurations include hydrogen atoms but do not include atomic hydrogen.

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## Max Planck’s black-body radiation is the first quantum theory

Thermal radiation is considered to be electromagnetic radiation that is emitted from the outer surface of a body due to that particular body’s temperature. A body will start emitting light at the red end of the spectrum if it is heated adequately. The Quantum Mechanics: The Hydrogen Atom (2008) states that, “these spectra can be used as analytical tools to assess the composition of matter” (p. 5). The change of color depends on further heating. The perfect emitter is a perfect absorber as well. When the body becomes cold, it looks like black because it emits no light and absorbs all the light that falls on it. Therefore, a perfect thermal emitter is considered a black body. The emitting radiation from this black body is known as black-body radiation. Scientists moderately characterized such radiation with experiments in the late 19th century. Quantum Mechanics and Atomic Theory state that “energy can be gained or lost only in integer multiples of hv., where n is an integer; h is Planck’s constant = 6.626 x 10-34J s; ? is the frequency of the electromagnetic radiation absorbed or emitted” (p. 7).

## Emission of spectra

### The photoelectric effect

If light with adequate frequency hits a metallic surface, it starts emitting electrons. Such an incident was observed by Heinrich Hertz in 1887. In 1902, Philipp Lenard exposed that the highest probable energy of an emitted electron depends on the frequency of the light but it does not depend on its strength. There will be no electron ejection if the frequency level is very low. The minimum frequency of light that causes electron ejection is defined as the threshold frequency. The threshold frequency varies depending on the different metals.

### Isotopes

The abundant isotope is hydrogen-1 which contains no neutrons. On the other hand, deuterium and tritium are hydrogen isotopes enclosed from one to more than one neutron.

## Quantum hypothetical analysis

In Quantum Mechanics, a hydrogen particle (atom) has particular importance. It is a two-body problem of physical coordination. In the closed form, it has yielded lots of easy methodical problems. After making several simplifying assumptions, Niels Bohr achieved the energy intensities and the hydrogen particle (atom) spectral frequencies. The Model provided by Bohr identifies energy level by the numeral quantum number n. It is recognized as the Principal Quantum Number. The energy level is provided by the following equation:

where me = electron mass,

c = speed of light, and

? = fine-structure constant (solution of the radial part of the Schrodinger equation (Quantum Mechanics and Atomic Theory, p. 8).

During 1925-1926, the Schrodinger equation was proved to be the same with the frequencies and underlying energy values of Bohr’s results. Schrodinger equation provides solutions for the hydrogen atom and is analytical with simple articulation for the energy intensity of hydrogen. Schrodinger equation’s solution goes further since it yields two other quantum numbers. Schrodinger equation provides the figure of the wave function of electrons for several probable quantum-mechanical conditions. Application of the Schrodinger equation to the problematical atoms as well as molecules is more helpful in solving problems and easier. The equation that was established by Schrodinger is applicable to the evaluation of non-relativistic Quantum Mechanics. Thus, the solution it achieves for the atom of hydrogen is not appropriate. For the improvement of the relativistic quantum hypothesis, the Dirac equation has helped a lot. According to

Schrodinger’s equation, “scientists can find the wave function, which solves a particular problem in quantum mechanics. Unfortunately, it is usually impossible to find an exact solution to the equation, so certain assumptions are used in order to obtain an approximate answer for the particular problem” (Quantum Mechanics and Atomic Theory).

## Solution of Schrodinger equation: an overview of results

The nucleus produces Coulomb potential. The solution of the Schrodinger equation uses the fact for the hydrogen element (atom). The potential that is produced by the nucleus as Coulomb’s theory suggests is isotropic. The result that comes for the energy eigenfunctions is not isotropic. It depends on the angular directions that later follow the isotropy of fundamental potential. It is defined as the angular momentum operator. Around the nucleus, it is preserved in the orbital movement of electrons. Consequently, the energy eigenstates are classified with two angular momenta. These are *?* and *m. *Where, *?* = 0, 1, 2, ... and *m* = ?*?*, ..., +*? *since both of them are numerals. The magnitude of angular momentum is determined by the angular energy quantum number *?*. On the other hand, the outcrop of the bony momentum of the *z*-axis that is arbitrarily chosen is determined by the magnetic quantum number *m.* The main quantum numeral *n* = 1, 2, 3, ...... The angular energy of the greatest value of the quantum number is restricted by the principal quantum number. The number can reach up to *n* – 1. Thus, *?* = 0, 1, ....., *n* ? 1.

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## Mathematical summary of eigenstates of a hydrogen atom

An equation was established by Paul Dirac in 1928 that was well-matched with special relativity. The equation has prepared the wave function for a 4-element Dirac spinor. It made ‘up’ and ‘down’ spin components as well. It has equally positive as well as negative energy. The result of this solution was more perfect in comparison to the Schrodinger solution.

### Energy levels

Picture of Hydrogen Spectrum

Electron level calculations

A report on Electron Transitions suggests that “the basic hydrogen energy level structure is in agreement with the Bohr model. Common pictures are those of a shell structure with each main shell associated with a value of the principal quantum number n”.

The following equation is used to calculate the energy level of a hydrogen atom

,

where

*?* = the fine-structure constant and

*j* = the ‘total angular momentum’ quantum number.

*j* is equal to |*?* ± 1?2| and depends on the direction of the electron spin.

A small correction is represented through the formula that was obtained by Bohr and Schrodinger. The factor in the latter equations’ square bracket is nearly one. This additional term comes due to the relativistic effects. Indeed, the value of this factor was nothing while it was first obtained by A. Sommerfeld in 1916. He obtained it on the basis of a relativistic description of the old theory provided by Bohr. Nevertheless, Sommerfeld used unusual information for the quantum numerals.

The value of the following equation is known as the Rydberg constant.

The value comes from the Bohr model as given bellow:

where, *m*e = electron mass,

*e* = elementary charge,

*h =* Planck constant, and

*?*0 = vacuum permittivity.

The constant, as the Rydberg energy unit, is frequently used in atomic physics:

*RM* is the constant of Rydberg *RM* for the hydrogen particle (atom), *R* is defined by

where *M = *atomic nucleus mass.

The quantity is about 1/1836 (for hydrogen-1) and the ratios are nearly 1/5497 and 1/3670 for tritium and deuterium, respectively.

### Wave function

The standardized location of the wave functions of spherical coordinates is provided below:

where

,

= Bohr radius

= a generalized Laguerre polynomial of level *n* ? *?* ? 1, and

= a spherical vocal function of *?* (quality) and *m *(order).

, Laguerre polynomial appears in the wave function of hydrogen is

.

The quantum integers may get the subsequent values:

Moreover, these wave functions are regularized:

where

represents the wave function in Dirac information, and stands for the Kronecker delta function.

3D illustration of the eigenstate

## Features going beyond the Schrodinger solution

Schrodinger equation neglects some significant effects. Such effects are dependable for tiny but quantifiable divergences of the existent spectral lines.

I. The average speed of hydrogen electrons is 1/137th in comparison to the pace of light. Thus there is an entire hypothetical clarification that should be performed so that a relativistic behavior of the difficulty can be solved. Due to smaller wavelengths, there is a contraction occurred by the orbitals containing advanced-pace electrons because the wavelength is measured by the momentum.

II. Though there is an unavailability of the magnetic area in the external part, the electromagnetic area contains a magnetic element. The rotation of the electron co-operates with the magnetic area.

These features are integrated with the ‘relativistic Dirac equation’. For instance, like the hydrogen atom, the Dirac equation can be resolved systematically in the particular problem of a two-element method.

## Conclusion

The study proves that Quantum Mechanics discusses the subject of matter and radiation. Formerly, classical physics failed to explain some experimental results. Modern physics left behind most of the obstacles of quantum physics and obtained the most upgraded explanation of the such subject matter. A hydrogen atom is one of the most complicated subject matters of physics that is explained by Quantum Mechanics. Several equations have been improved to get an accurate result in this context. Classical physics has some imperfections in this regard, which are being improved by quantum physics. In this context, the more complicated problems are being solved and novel explanations are offered by modern physics.

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### News & Events

April 29th: Our BEC machine now produces stable pure BEC with more than 100 000 atoms using a hybrid trap that combines a magnetic quadrupole and a crossed dipole trap.

October 2th: Our thesis proposal on a microscope of quantum fluctuations.

June 18th: CNRS Institut de physique - Actualités scientifique.

May 28th: Our article on the reduction of local velocity dispersion is now published: Phys. Rev. A 89, 053626 (2014).

May 9th: Our article on the Boltzmann equation is now published (Phys. Rev. Lett. 112, 180602 (2014)).

March 27th: Our article on reverse engineering applied to the Boltzmann equation has been accepted for publication in Phys. Rev. Lett.

September: Our article on a new kind of tunnel barrier relying on spatial gaps is now published in EPL.

July 25th: Our review article on Shortcut to adiabaticity is now published in Advances in Atomic, Molecular, and Optical Physics.

January 22th: our paper entitled «Matter-wave scattering on an amplitude-modulated optical lattice» is now published in Physical Review A.

Conference announcement: A conference on New Magnetic Field Frontiers at Les Houches 6th-10th May 2013.

July 19th, Our paper entitled "Optically guided beam splitter for propagating matter waves " has been published and selected as an Editor’s suggestion of PRL and is accompanied by a Physics Synopsis; This article has also been selected in Nature physics in research highlights (Link) and by Nature Photonics in the News and Views section (Link).

June 18th, 2012: Our paper entitled "Optically guided beam splitter for propagating matter waves " has been accepted for publication in Physical Review Letters.

Conference announcement: A conference on Shortcut to adiabaticity will be held in Bilbao in July 2012 (conference website).

December 16th, 2011, Our paper entitled "Exploring classically chaotic potentials with a matter wave quantum probe" has been published G. L Gattobigio et al. Phys. Rev. Lett. 107, 254104 (2011). A figure of this article has been selected for PRL’s cover, 16 December 2011, Volume 107, Issue 25. Cet article a fait l’objet d’un communiqué de presse régional (Lien).

November 29th, 2011, Our paper entitled "Realization of a distributed Bragg reflector for propagating guided matter waves" has been published, by C.M. Fabre et al. Phys. Rev. Lett. 107, 230401 (2011). Our article has been selected by Nature's physics research highlights. Ce travail a également fait l’objet d’un actualité scientifique de l’Institut de Physique du CNRS (Lien).

November 8th, 2011: Our paper entitled "Exploring classically chaotic potentials with a matter wave quantum probe" has been accepted for publication in Physical Review Letters.

June 30th, 2011: The paper about our Zeeman slower with permanent magnets in a Halbach configuration is now published: Rev. Sci. Instrum. 82, 063115 (2011).

November 15th, 2010: We have an MOT with several 10^10 atoms in the new setup, loaded in about one second by a novel type of Zeeman slower using permanent magnets in a Halbach configuration. See here for a video of the MOT, and here for many more details about the slower. The preprint arXiv:1101.3243 (submitted to Rev. Sci. Instr.) describes the slower in great detail.

October 22th, 2010: The paper Cold-atom dynamics in crossed-laser-beam waveguides, Phys. Rev. A 82, 043420 (2010), is now published.

September 20th, 2010: The paper Shortcut to Adiabatic Passage in Two- and Three-Level Atoms, Phys. Rev. Lett. 105, 123003 (2010), is now published.

August 30th, 2010: Our paper Interaction of a propagating guided matter wave with a localized potential, New J. Phys. 12, 085013 (2010), is now published.