Magnetic quantum number

In atomic physics, a magnetic quantum number is a quantum number used to distinguish quantum states of an electron or other particle according to its angular momentum along a given axis in space. The orbital magnetic quantum number ( $m l$ or $m$ ^[a]) distinguishes the orbitals available within a given subshell of an atom. It specifies the component of the orbital angular momentum that lies along a given axis, conventionally called the z-axis, so it describes the orientation of the orbital in space. The spin magnetic quantum number $m s$ specifies the z-axis component of the spin angular momentum for a particle having spin quantum number $s$ . For an electron, $s$ is 1⁄2, and $m s$ is either +1⁄2 or −1⁄2, often called "spin-up" and "spin-down", or α and β.^[1]^[2] The term magnetic in the name refers to the magnetic dipole moment associated with each type of angular momentum, so states having different magnetic quantum numbers shift in energy in a magnetic field according to the Zeeman effect.^[2]

The four quantum numbers conventionally used to describe the quantum state of an electron in an atom are the principal quantum number n, the azimuthal (orbital) quantum number $\ell$ , and the magnetic quantum numbers $m l$ and $m s$ . Electrons in a given subshell of an atom (such as s, p, d, or f) are defined by values of $\ell$ (0, 1, 2, or 3). The orbital magnetic quantum number takes integer values in the range from $-\ell$ to $+\ell$ , including zero.^[3] Thus the s, p, d, and f subshells contain 1, 3, 5, and 7 orbitals each. Each of these orbitals can accommodate up to two electrons (with opposite spins), forming the basis of the periodic table.

Other magnetic quantum numbers are similarly defined, such as $m j$ for the z-axis component the total electronic angular momentum $j$ ,^[1] and $m I$ for the nuclear spin $I$ .^[2] Magnetic quantum numbers are capitalized to indicate totals for a system of particles, such as $M L$ or $m L$ for the total z-axis orbital angular momentum of all the electrons in an atom.^[2]

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^ ^a ^b Martin, W. C.; Wiese, W. L. (2019). "Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas". National Institute of Standards and Technology, Physical Measurement Laboratory. NIST. Retrieved 17 May 2023.
^ ^a ^b ^c ^d Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5.
^ Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. pp. 136–137. ISBN 0-13-111892-7. OCLC 53926857.

[NIST_2019-2] Martin, W. C.; Wiese, W. L. (2019). "Atomic Spectroscopy - A Compendium of Basic Ideas, Notation, Data, and Formulas". National Institute of Standards and Technology, Physical Measurement Laboratory. NIST. Retrieved 17 May 2023.

[Atkins_1991-3] Atkins, Peter William (1991). Quanta: A Handbook of Concepts (2nd ed.). Oxford University Press, USA. p. 297. ISBN 0-19-855572-5.

[4] Griffiths, David J. (2005). Introduction to quantum mechanics (2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. pp. 136–137. ISBN 0-13-111892-7. OCLC 53926857.

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