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QUANTIZED ANGULAR MOMENTA

In quantum mechanics, Angular Momenta of Electron s (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 \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, ''ml'':
:l_z = m_l \, \hbar
where ''ml'' ranges from −''l'' to +''l'' in steps of one unit. This means that for a given value of ''l'', there are 2''l'' + 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 \sqrt{2}\,\hbar, and ''lz'' 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 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

    • ::{\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