In quantum mechanics, the **principal quantum number** (symbolized * n*) is one of four quantum numbers assigned to each electron in an atom to describe that electron's state. Its values are natural numbers (from 1) making it a discrete variable.

Apart from the principal quantum number, the other quantum numbers for bound electrons are the azimuthal quantum number *ℓ*, the magnetic quantum number *m _{l}*, and the spin quantum number

As *n* increases, the electron is also at a higher energy and is, therefore, less tightly bound to the nucleus. For higher *n* the electron is farther from the nucleus, on average. For each value of *n* there are *n* accepted *ℓ* (azimuthal) values ranging from 0 to *n* − 1 inclusively, hence higher-*n* electron states are more numerous. Accounting for two states of spin, each *n*-shell can accommodate up to 2*n*^{2} electrons.

In a simplistic one-electron model described below, the total energy of an electron is a negative inverse quadratic function of the principal quantum number *n*, leading to degenerate energy levels for each *n* > 1.^{ [1] } In more complex systems—those having forces other than the nucleus–electron Coulomb force—these levels split. For multielectron atoms this splitting results in "subshells" parametrized by *ℓ*. Description of energy levels based on *n* alone gradually becomes inadequate for atomic numbers starting from 5 (boron) and fails completely on potassium (*Z* = 19) and afterwards.

The principal quantum number was first created for use in the semiclassical Bohr model of the atom, distinguishing between different energy levels. With the development of modern quantum mechanics, the simple Bohr model was replaced with a more complex theory of atomic orbitals. However, the modern theory still requires the principal quantum number.

There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers *n*, *ℓ*, *m*, and *s* specify the complete and unique quantum state of a single electron in an atom, called its wave function or orbital. Two electrons belonging to the same atom cannot have the same values for all four quantum numbers, due to the Pauli exclusion principle. The Schrödinger wave equation reduces to the three equations that when solved lead to the first three quantum numbers. Therefore, the equations for the first three quantum numbers are all interrelated. The principal quantum number arose in the solution of the radial part of the wave equation as shown below.

The Schrödinger wave equation describes energy eigenstates with corresponding real numbers *E _{n}* and a definite total energy, the value of

The parameter *n* can take only positive integer values. The concept of energy levels and notation were taken from the earlier Bohr model of the atom. Schrödinger's equation developed the idea from a flat two-dimensional Bohr atom to the three-dimensional wavefunction model.

In the Bohr model, the allowed orbits were derived from quantized (discrete) values of orbital angular momentum, *L* according to the equation

where *n* = 1, 2, 3, … and is called the principal quantum number, and *h* is Planck's constant. This formula is not correct in quantum mechanics as the angular momentum magnitude is described by the azimuthal quantum number, but the energy levels are accurate and classically they correspond to the sum of potential and kinetic energy of the electron.

The principal quantum number *n* represents the relative overall energy of each orbital. The energy level of each orbital increases as its distance from the nucleus increases. The sets of orbitals with the same *n* value are often referred to as an electron shell.

The minimum energy exchanged during any wave–matter interaction is the product of the wave frequency multiplied by Planck's constant. This causes the wave to display particle-like packets of energy called quanta. The difference between energy levels that have different *n* determine the emission spectrum of the element.

In the notation of the periodic table, the main shells of electrons are labeled:

*K*(*n*= 1),*L*(*n*= 2),*M*(*n*= 3), etc.

based on the principal quantum number.

The principal quantum number is related to the radial quantum number, *n*_{r}, by:

where *ℓ* is the azimuthal quantum number and *n*_{r} is equal to the number of nodes in the radial wavefunction.

The definite total energy for a particle motion in a common Coulomb field and with a discrete spectrum, is given by:

- ,

where is the Bohr radius.

This discrete energy spectrum resulted from the solution of the quantum mechanical problem on the electron motion in the Coulomb field, coincides with the spectrum that was obtained with the help application of the Bohr–Sommerfeld quantization rules to the classical equations. The radial quantum number determines the number of nodes of the radial wave function .^{ [2] }

In chemistry, values *n* = 1, 2, 3, 4, 5, 6, 7 are used in relation to the electron shell theory, with expected inclusion of *n* = 8 (and possibly 9) for yet-undiscovered period 8 elements. In atomic physics, higher *n* sometimes occur for description of excited states. Observations of the interstellar medium reveal atomic hydrogen spectral lines involving *n* on order of hundreds; values up to 766^{ [3] } were detected.

In atomic theory and quantum mechanics, an **atomic orbital** is a mathematical function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term *atomic orbital* may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital.

In atomic physics, the **Bohr model** or **Rutherford–Bohr model**, presented by Niels Bohr and Ernest Rutherford in 1913, is a system consisting of a small, dense nucleus surrounded by orbiting electrons—similar to the structure of the Solar System, but with attraction provided by electrostatic forces in place of gravity. After the cubical model (1902), the plum pudding model (1904), the Saturnian model (1904), and the Rutherford model (1911) came the *Rutherford–Bohr model* or just *Bohr model* for short (1913). The improvement over the 1911 Rutherford model mainly concerned the new quantum physical interpretation.

A **hydrogen atom** is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively charged proton and a single negatively charged electron bound to the nucleus by the Coulomb force. **Atomic hydrogen** constitutes about 75% of the baryonic mass of the universe.

The **quantum Hall effect** is a quantized version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance *R*_{xy} exhibits steps that take on the quantized values at certain level

A quantum mechanical system or particle that is bound—that is, confined spatially—can only take on certain discrete values of energy, called **energy levels**. This contrasts with classical particles, which can have any amount of energy. The term is commonly used for the energy levels of the electrons in atoms, ions, or molecules, which are bound by the electric field of the nucleus, but can also refer to energy levels of nuclei or vibrational or rotational energy levels in molecules. The energy spectrum of a system with such discrete energy levels is said to be quantized.

The **Schrödinger equation** is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.

The **Bohr radius** (*a*_{0}) is a physical constant, equal to the most probable distance between the nucleus and the electron in a hydrogen atom in its ground state. It is named after Niels Bohr, due to its role in the Bohr model of an atom. Its value is 5.29177210903(80)×10^{−11} m.

In computational physics and chemistry, the **Hartree–Fock** (**HF**) method is a method of approximation for the determination of the wave function and the energy of a quantum many-body system in a stationary state.

In chemistry and quantum physics, **quantum numbers** describe values of conserved quantities in the dynamics of a quantum system. Quantum numbers correspond to eigenvalues of operators that commute with the Hamiltonian—quantities that can be known with precision at the same time as the system's energy—and their corresponding eigenspaces. Together, a specification of all of the quantum numbers of a quantum system fully characterize a basis state of the system, and can in principle be measured together.

The **azimuthal quantum number** is a quantum number for an atomic orbital that determines its orbital angular momentum and describes the shape of the orbital. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron. It is also known as the **orbital angular momentum** quantum number, **orbital quantum number** or **second quantum number**, and is symbolized as **ℓ**.

The **magnetic quantum number** is one of four quantum numbers in atomic physics. The set is: principal quantum number, azimuthal quantum number, magnetic quantum number, and spin quantum number. Together, they describe the unique quantum state of an electron. The magnetic quantum number distinguishes the orbitals available within a subshell, and is used to calculate the azimuthal component of the orientation of orbital in space. Electrons in a particular subshell are defined by values of *ℓ*. The value of *m _{l}* can range from -

A **rotational transition** is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

In atomic physics, the **spin quantum number** is a quantum number that describes the intrinsic angular momentum of a given particle. The spin quantum number is designated by the letter s, and is the fourth of a set of quantum numbers, which completely describe the quantum state of an electron. The name comes from a physical spinning of the electron about an axis that was proposed by Uhlenbeck and Goudsmit. However this simplistic picture was quickly realized to be physically impossible, and replaced by a more abstract quantum-mechanical description.

The **old quantum theory** is a collection of results from the years 1900–1925 which predate modern quantum mechanics. The theory was never complete or self-consistent, but was rather a set of heuristic corrections to classical mechanics. The theory is now understood as the semi-classical approximation to modern quantum mechanics.

**Relativistic quantum chemistry** combines relativistic mechanics with quantum chemistry to calculate elemental properties and structure, especially for the heavier elements of the periodic table. A prominent example is the explanation of the color of gold: due to relativistic effects, it is not silvery like most other metals.

In atomic physics, the **electron magnetic moment**, or more specifically the **electron magnetic dipole moment**, is the magnetic moment of an electron caused by its intrinsic properties of spin and electric charge. The value of the electron magnetic moment is approximately −9.284764×10^{−24} J/T. The electron magnetic moment has been measured to an accuracy of 7.6 parts in 10^{13}.

**Quantum mechanics** is the study of very small things. It explains the behavior of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to two major revolutions in physics that created a shift in the original scientific paradigm: the *theory of relativity* and the development of *quantum mechanics*. This article describes how physicists discovered the limitations of classical physics and developed the main concepts of the quantum theory that replaced it in the early decades of the 20th century. It describes these concepts in roughly the order in which they were first discovered. For a more complete history of the subject, see *History of quantum mechanics*.

The emission spectrum of atomic hydrogen has been divided into a number of **spectral series**, with wavelengths given by the Rydberg formula. These observed spectral lines are due to the electron making transitions between two energy levels in an atom. The classification of the series by the Rydberg formula was important in the development of quantum mechanics. The spectral series are important in astronomical spectroscopy for detecting the presence of hydrogen and calculating red shifts.

The **theoretical and experimental justification for the Schrödinger equation** motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

A **hydrogen-like atom/ion** (usually called a "hydrogenic atom") is any atomic nucleus bound to one electron and thus is isoelectronic with hydrogen. These atoms or ions can carry the positive charge , where is the atomic number of the atom. Examples of hydrogen-like atoms/ions are hydrogen itself, He^{+}, Li^{2+}, Be^{3+} and B^{4+}. Because hydrogen-like atoms/ions are two-particle systems with an interaction depending only on the distance between the two particles, their (non-relativistic) Schrödinger equation can be solved in analytic form, as can the (relativistic) Dirac equation. The solutions are one-electron functions and are referred to as *hydrogen-like atomic orbitals*.

- ↑ Here we ignore spin. Accounting for
*s*,*every*orbital (determined by*n*and*ℓ*) is degenerate, assuming absence of external magnetic field. - ↑ Andrew, A. V. (2006). "2. Schrödinger equation".
*Atomic spectroscopy. Introduction of theory to Hyperfine Structure*. p. 274. ISBN 978-0-387-25573-6. - ↑ Tennyson, Jonathan (2005).
*Astronomical Spectroscopy*(PDF). London: Imperial College Press. p. 39. ISBN 1-86094-513-9.

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