In
atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned wit ...
, the magnetic quantum number () is one of the four
quantum number
In quantum physics and chemistry, 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 ...
s (the other three being the
principal,
azimuthal
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematically, ...
, and
spin
Spin or spinning most often refers to:
* Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning
* Spin, the rotation of an object around a central axis
* Spin (propaganda), an intentionally ...
) which describe the unique
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of an
electron
The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family,
and are generally thought to be elementary particles because they have n ...
. 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 (such as s, p, d, or f) are defined by values of (0, 1, 2, or 3). The value of can range from to , including zero. Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each, with values of within the ranges 0, ±1, ±2, ±3 respectively. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the
periodic table
The periodic table, also known as the periodic table of the (chemical) elements, is a rows and columns arrangement of the chemical elements. It is widely used in chemistry, physics, and other sciences, and is generally seen as an icon of ch ...
.
Derivation
There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers
,
,
, and
specify the complete
quantum state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
of a single electron in an atom called its
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
or
orbital. The
Schrödinger equation
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 th ...
for the wavefunction of an atom with one electron is a
separable partial differential equation. (This is not the case for the neutral
helium atom or other atoms with mutually interacting electrons, which require more sophisticated methods for solution) This means that the wavefunction as expressed in
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:
:
The differential equation for
can be solved in the form
. Because values of the azimuth angle
differing by 2
(360 degrees in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s) represent the same position in space, and the overall magnitude of
does not grow with arbitrarily large
as it would for a real exponent, the coefficient
must be quantized to integer multiples of
, producing an
imaginary exponent:
. These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of
tend to decrease the magnitude of
, and values of
greater than the
azimuthal quantum number
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 that describe ...
do not permit any solution for
.
As a component of angular momentum
The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number
refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the
-direction or
quantization axis.
, the magnitude of the angular momentum in the
-direction, is given by the formula:
[
:.
This is a component of the atomic electron's total orbital angular momentum , whose magnitude is related to the azimuthal quantum number of its subshell by the equation:
:,
where is the ]reduced Planck constant
The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
. Note that this for and approximates for high . It is not possible to measure the angular momentum of the electron along all three axes simultaneously. These properties were first demonstrated in the Stern–Gerlach experiment
The Stern–Gerlach experiment demonstrated that the spatial orientation of angular momentum is quantized. Thus an atomic-scale system was shown to have intrinsically quantum properties. In the original experiment, silver atoms were sent throug ...
, by Otto Stern
:''Otto Stern was also the pen name of German women's rights activist Louise Otto-Peters (1819–1895)''.
Otto Stern (; 17 February 1888 – 17 August 1969) was a German-American physicist and Nobel laureate in physics. He was the second most ...
and Walther Gerlach.
The energy of any wave is its frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
multiplied by Planck's constant. The wave displays particle-like packets of energy called quanta. The formula for the quantum number of each quantum state uses Planck's reduced constant, which only allows particular or discrete or quantized energy levels.[
]
Effect in magnetic fields
The quantum number refers, loosely, to the direction of the angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
. The magnetic quantum number only affects the electron's energy if it is in a magnetic field because in the absence of one, all spherical harmonics corresponding to the different arbitrary values of are equivalent. The magnetic quantum number determines the energy shift of an atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a 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 ...
due to an external magnetic field (the Zeeman effect
The Zeeman effect (; ) is the effect of splitting of a spectral line into several components in the presence of a static magnetic field. It is named after the Dutch physicist Pieter Zeeman, who discovered it in 1896 and received a Nobel pr ...
) — hence the name ''magnetic'' quantum number. However, the actual magnetic dipole moment
In electromagnetism, the magnetic moment is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include loops of electric current (such as electromagnet ...
of an electron in an atomic orbital arises not only from the electron angular momentum but also from the electron spin, expressed in the spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
.
Since each electron has a magnetic moment in a magnetic field, it will be subject to a torque which tends to make the vector parallel to the field, a phenomenon known as Larmor precession.
See also
* Quantum number
In quantum physics and chemistry, 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 ...
** Azimuthal quantum number
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 that describe ...
** Principal quantum number
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.
A ...
** Spin quantum number
In atomic physics, the spin quantum number is a quantum number (designated ) which describes the intrinsic angular momentum (or spin angular momentum, or simply spin) of an electron or other particle. The phrase was originally used to describe t ...
** Total angular momentum quantum number
* Electron shell
* Basic quantum mechanics
* Bohr atom
* Schrödinger equation
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 th ...
References
{{electron configuration navbox
Atomic physics
Rotational symmetry
Quantum numbers