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Angular momentum quantum number

An angular quantum momentum number is any of the quantum numbers that quantize an angular momentum. They express an angular momentum as an integer multiple of $\hbar / 2$ (the reduced Planck's constant divided by two).

Contents

1 Quantized angular momenta

2 Addition of quantized angular momenta

3 List of angular momentum quantum numbers

4 See also

Quantized angular momenta

In quantum mechanics, angular momenta of electrons (and also of other particles or systems of particles) are quantified vectors, i.e., vectors whose allowed values are not constinuous but discrete, so their projections on an arbitrary axis differ in one unit of $\hbar$ . Moreover, they can be expressed as a function of quantum numbers (e.g. the magnetic quantum number or the azimuthal quantum number). Usually boldface is used to represent the angular momentum vectors, and italics for the associated quantum numbers. Small case letters are used for the electron (or individual particle) while CAPS are used for compound systems.

Given a quantified angular momentum $\mathbf l$ , its modulus is parameterized by its associated quantum number l:

$\Vert \mathbf l \Vert = \sqrt{l \, (l+1)} \, \hbar$

where l is a non-negative integer. The z-projection of the angular-momentum is also parameterized by a second quantum number, m_l:

$l_z = m_l \, \hbar$

where m_l ranges from −l to +l in steps of one unit. This means that for a given value of l, there are 2l + 1 different values of m_l, each one representing a different "state" or orientation for the angular momentum vector.

Examples:

The orbital angular momentum is parametrized by the azimuthal quantum number l and its z-projection by the magnetic quantum number m_l.
- for l = 0 the orbital angular momentum is 0, and m_l is also 0.
- for l = 1, m_l can be −1, 0 or +1. The orbital angular momentum modulus is $\sqrt{2}\,\hbar$ , and l_z is $-\hbar$ , 0 or $+\hbar$ , which gives three possible values for the z-projection. This represents three possible orientations of the angular momentum vector relative to a arbitrary axis z.
The intrinsic angular momentum of a particle is parametrized by the spin angular momentum quantum number s, and its projection by m_s.
- for the electron, s = ½ and m_s = ±½. The intrinsic angular momentum modulus is

${\sqrt{3} \over 2} \hbar$ ,

and its two possible projections

$\pm{1 \over 2} \hbar$ ,

which correspond to the two possible states of an electron in an orbital: the "up" orientation and the "down" orientation.

Addition of quantized angular momenta

Given two quantized angular momenta l₁ and l₂, and a third angular momentum j which is their vectorial sum

$\mathbf j = \mathbf l_1 + \mathbf l_2$

then its associated quantum number j can take any integer or half-integer value that holds

$|l_1 - l_2| \le j \le l_1 + l_2$

where l₁ and l₂ are the associated quantum numbers for the corresponding angular momenta.

List of angular momentum quantum numbers

Intrinsic (or spin) angular momentum quantum number, or simply spin quantum number
orbital angular momentum quantum number
magnetic quantum number, related to the orbital momentum quantum number
total angular momentum quantum number

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