Quantum numbers are of basic physical importance in quantum physics. In the early 1900s energy in atomic systems was found to exist as quanta or discrete, indivisible units. This concept along with the particle-wave duality led to the development of quantum physics and hence the Schrödinger equation. Unlike classical mechanics, in which energy is continuous and electrons follow fixed orbits described by Bohr's planetary model, quantum mechanics describes electrons as effectively taking up entire predefined spaces around the nucleus and energy as a quantized phenomenon. Quantum numbers are the four numbers used to describe not only the distribution of electrons in atoms and molecular systems but also the allowable values of certain physical quantities of an electron's behavior. A wave function for an electron describes the probability of finding that electron at various points in space. The first three quantum numbers describe the physical dimensions of space in which that electron can be found. Quantum numbers are used to describe atomic orbitals although they are also used to describe hybrid orbitals, orbitals composed of individual atomic orbitals, in molecular systems. The Schrödinger equation cannot be solved exactly for an atom containing more than one electron although methods of successive approximations have been used to obtain approximate solutions for systems containing more than one electron. The solutions to the Schrödinger equation reveals the values of the allowed or permitted energies which can be assumed by every electron in an atom or molecule and are described by the set of quantum numbers for that electron.
The energies of bound electrons are quantized and depend on angular momentum and orientation as well as spin of the electron and are derived from the mathematical solution to the Schrödinger or Dirac equations. The Schrödinger equation links the energy and position of electrons in atoms. The locations of the electrons, described by these numbers, are not circular orbits but rather fields of electron density. These fields are described by the four quantum numbers: n, l, m, and s. Although no two electrons in the same atom or molecule may have an identical set of quantum numbers each quantum number can have values that depend upon the values of the other quantum numbers describing the same electron.
The principle quantum number, n, determines the size of the orbital. It denotes the major shell that contains the electron. When the value of the principle quantum number increases the orbital size increases as well as its distance from the nucleus. It can be any positive integer (1, 2, 3, 4, etc.) with the higher number indicating a larger size. Different values correspond to different energy levels. As n grows larger, the relative spacing of the energy levels become closer together approaching the ionization limit for the atom or molecule.
The angular momentum quantum number, l, determines the geometric shape of the orbital and its value can range from zero to n-1. This quantum number is sometimes referred to as the subshell quantum number. So for any orbital there can be many shapes associated with a particular size depending upon what the principle quantum number is. The first five shapes associated with the orbital quantum numbers are referred to as s, p, d, and f. A pure s orbital is spherical. A pure p orbital is dumb-bell shaped with two parts separated by a nodal plane where the probabilit ity of finding the electron is zero. The pure d orbital has four lobes with nodal planes at the intersection of the four lobes. f orbitals have complicated shapes and are too complicated to describe here. Although these are the respective shapes of the orbitals in atomic systems, molecules combine atomic orbitals to make molecule orbitals and so the shapes are hybrid, more complex combinations of the individual atomic orbitals.
The third quantum number is the magnetic quantum number, m. It determines the orientation of an orbital with respect to an applied magnetic field and can range in value from -l to l. The magnetic quantum number denotes the energy levels available for occupation within a subshell. Because of the range of possible values of magnetic quantum numbers associated with the l = 2 orbital there are three possible orientations of the p orbital. There are five possible orientations for d orbitals because of the range of possible magnetic quantum numbers associated with this orbital and there are seven possible orientations for f orbitals described by the magnetic quantum numbers. The different values of the magnetic quantum number mean little differences in energies of the electrons.
There is a fourth quantum number that describes the spin of the electron in an orbital. This quantum number is called the spin quantum number, s, and can have a value of either 1/2 or -1/2 since an electron can orient in two ways in an applied magnetic field. It is an intrinsic quantum number that is unrelated to the s-shaped orbital. Because there are only two values associated with this quantum number each orbital can contain only two electrons and each electron must have a spin that is opposite to the spin of the other electron in that orbital. The electrons in these electron pairs have essentially the same energies.
Quantum numbers describe the distribution of electrons in atoms and molecules. The Pauli principle states that no two electrons may be in the same quantum state, hence no two electrons can have identical sets of quantum numbers. Hund's rule tells us that electrons will enter empty orbitals of equal energy when they are available. Quantum numbers, derived from quantum theory, are of basic physical importance in calculating energy as well as describing the physical space occupied by electrons in molecular systems.
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