Magnetic quantum number

The magnetic quantum number (symbol "ml") 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 (such as s, p, d, or f) are defined by values of "ℓ" (0, 1, 2, or 3). The value of "ml" can range from -"ℓ" to +"ℓ", including zero. Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each, with values of "m" 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.
Derivation.
There is a set of quantum numbers associated with the energy states of the atom. The four quantum numbers formula_1, formula_2, formula_3, and formula_4 specify the complete and unique quantum state of a single electron in an atom called its wavefunction or orbital. The Schrödinger equation for the wavefunction of an atom with one electron is a separable partial differential equation. (This is not the case for the 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 can be broken down into the product of three functions of the radius, colatitude (or polar) angle, and azimuth:
The differential equation for formula_6 can be solved in the form formula_7. Because values of the azimuth angle formula_8 differing by 2formula_9 (360 degrees in radians) represent the same position in space, and the overall magnitude of formula_6 does not grow with arbitrarily large formula_8 as it would for a real exponent, the coefficient formula_12 must be quantized to integer multiples of formula_13, producing an imaginary exponent: formula_14. These integers are the magnetic quantum numbers. The same constant appears in the colatitude equation, where larger values of formula_32 tend to decrease the magnitude of formula_16, and values of formula_3 greater than the azimuthal quantum number formula_2 do not permit any solution for formula_16.
As a component of angular momentum.
The axis used for the polar coordinates in this analysis is chosen arbitrarily. The quantum number formula_20 refers to the projection of the angular momentum in this arbitrarily-chosen direction, conventionally called the formula_21-direction or quantization axis. formula_22, the magnitude of the angular momentum in the formula_21-direction, is given by the formula:
This is a component of the atomic electron's total orbital angular momentum formula_25, whose magnitude is related to the azimuthal quantum number of its subshell formula_2 by the equation:
where formula_28 is the reduced Planck constant. Note that this formula_29 for formula_30 and approximates formula_31 for high formula_2. 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, by Otto Stern and Walther Gerlach.
The energy of any wave is its frequency 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 formula_20 refers, loosely, to the direction of the angular momentum vector. The magnetic quantum number formula_20 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 formula_20 are equivalent. The magnetic quantum number determines the energy shift of an atomic orbital due to an external magnetic field (the Zeeman effect) — hence the name "magnetic" quantum number. However, the actual magnetic dipole moment 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.
Since each electron has a magnetic moment in a magnetic field, it will be subject to a torque which tends to make the vector formula_25 parallel to the field, a phenomenon known as Larmor precession.