Angular momentum quantum number
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In atomic physics, 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 (the reduced Planck's constant divided by two).
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[edit] 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 continuous but discrete, so their projections on an arbitrary axis differ in one unit of . 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 , its modulus is parameterized by its associated quantum number l:
where l is a non-negative integer. The z-projection of the angular-momentum is also parameterized by a second quantum number, ml:
where ml 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 ml, 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 ml.
- for l = 0 the orbital angular momentum is 0, and ml is also 0.
- for l = 1, ml can be −1, 0 or +1. The orbital angular momentum modulus is , and lz is , 0 or , which gives three possible values for the z-projection. This represents three possible orientations of the angular momentum vector relative to an arbitrary axis z.
- The intrinsic angular momentum of a particle is parametrized by the spin angular momentum quantum number s, and its projection by ms.
- for the electron, s = ½ and ms = ±½. The intrinsic angular momentum modulus is
-
- ,
and its two possible projections
-
- ,
which correspond to the two possible states of an electron in an orbital: the "up" orientation and the "down" orientation.
[edit] Addition of quantized angular momenta
Given a quantized total angular momentum which is the sum of two individual quantized angular momenta and ,
the quantum number j associated with its magnitude can range from | l1 − l2 | to l1 + l2 in integer steps where l1 and l2 are quantum numbers corresponding to the magnitudes of the individual angular momenta.
[edit] 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
[edit] See also
- Principal quantum number, not an angular momentum quantum number, but controls the possible values of the azimuthal quantum number.