Multiplicity (redirected from Multiple roots of a polynomial)

multiplicity

Physics the number of levels into which the energy of an atom, molecule, or nucleus splits as a result of coupling between orbital angular momentum and spin angular momentum

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multiplicity [‚məl·tə′plis·əd·ē]

(mathematics)

A root of a polynomial ƒ(x) has multiplicitynif (x - a)ⁿis a factor of ƒ(x) andnis the largest possible integer for which this is true.

The geometric multiplicity of an eigenvalue λ of a linear transformationTis the dimension of the null space of the transformationT - λI, whereIdenotes the identity transformation.

The algebraic multiplicity of an eigenvalue λ of a linear transformationTon a finite-dimensional vector space is the multiplicity of λ as a root of the characteristic polynomial ofT.

(physics)

In a system having Russell-Saunders coupling, the quantity 2S+1, whereSis the total spin quantum number.

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Multiplicity 

the number of possible spatial orientations of the total spin of an atom or molecule. According to quantum mechanics, multiplicity κ is equal to 2S + 1, where S is the spin quantum number. For systems with an odd number N of electrons, 5 = 1/2, 3/2, 5/2, . . . , and the multiplicity is even (κ = 2,4, 6,. . . ). Quantum states such as doublets, quartets, and sextets are possible &>r them. If N is even, then S = 0, 1, 2, . . ., and the multiplicity is odd (κ = 1, 3, 5,. . . ), and singlet, triplet, quintet, and other odd states are possible. For example, for systems with one electron (the H atom, the He⁺ ion; S = 1/2, κ = 2), only doublet states occur; for systems with two electrons (the He atom and the H₂ molecule), singlet states (S = 0, κ = 1; the spins of the electrons are antiparallel) and triplet states (S = 1, κ = 3; the electron spins are parallel) occur. For N electrons the maximum multiplicity (κ = N + 1) corresponds to parallel orientation of their spins.

The multiplicity determines the degree of degeneracy of the levels of the atom or molecule. The 25 + 1 quantum states that correspond to an energy level with a given S differ in the values of the projection of the total spin and are characterized by the quantum number M_s = S, S— 1,. . ., —S, which determines the magnitude of the projection. As a result of spin-orbital interaction, an energy level may split into κ = 2S + 1 sublevels (multiplet splitting, which leads to the splitting of spectral lines).

The values of multiplicity for the quantum states of atoms and molecules are determined by the electrons in open shells, since electron spins are compensated in closed shells. For the energy levels of alkali metals with one outer electron, κ = 2, just as for the H atom; for the energy levels of complex atoms with filling p-, d-, and f- shells, the multiplicity may be high (up to 11). Multiplicities of κ = 1 for the ground level and κ = 1 and 3 for excited energy levels are characteristic of chemically stable molecules, which usually have an even number of electrons. For free radicals with one electron with uncompensated spin, a multiplicity of κ = 2 is typical.

M. A. EL’IASHEVICH