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QUANTUM NUMBERS In the quantum mechanical description of microscopic systems, quantities such as the total energy, orbital angular momentum and the intrinsic spin angular moments of particles may have only certain discrete values (Cohen-Tannoudji et al., 1977). The whole and fractional real numbers that are used to enumerate these possible values are called quantum numbers. If the particular quantity is a constant of the motion (that is, if it does not depend explicitly on time and commutes with the Hamiltonian), its associated quantum number is called a good quantum number. The good quantum numbers in the simple theory (excluding fine and hyperfine interactions) of the hydrogen atom are: the principal quantum number n associated with the energy En= -me 4 /2h 2n2 , the orbital (or azimuthal) quantum number l associated with the total orbital angular momentum (squared) L 2 = l(l + 1)h 2 , the magnetic quantum number m associated with the z-component of the orbital angular momentum Lz = mh, and the spin quantum number m, associated with the intrinsic spin angular momentum of the electron Sz = m,h. The good quantum numbers of the hydrogen atom are useful in the approximate treatment of many-electron
atoms and ions and in describing their atomic orbitals and energy shells. The allowed n = 1, 2, 3, etc. energy levels correspond to the K, L, M, etc. shells. Each shell can contain a maximum of 2n 2 electrons. The allowed values of l = 0, 1, 2, 3, etc. have associated s, p, d, f, etc. subshells (or orbitals), each containing a maximum of 2(21 + 1) electrons. The maximum value of l for a given shell is (n- 1). The allowed values of m for a given orbital are -l, -(l-1), ... -1, 0, 1, ... (l-1), l and m, has allowed values of -1/2 or + 1/2 for each electron in any shell or subshell. D. G. Rancourt
Bibliography Cohen-Tannoudji, C., Diu, B. and Laloe, F. (1977) Quantum Mechanics (two volumes). New York: Wiley, 1525 pp.
Cross references Atomic mass unit, Avogadro constant and mole Atomic number Elementary particles Nucleosynthesis Periodic Table
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