The word adjoint has a number of related meanings. In linear algebra, it refers to the conjugate transpose and is most commonly denoted A^(H). The analogous concept applied to an operator instead of a matrix, sometimes also known as the Hermitian conjugate (Griffiths 1987, p. 22), is most commonly denoted using dagger notation A^| (Arfken 1985). The adjoint operator is very common in both Sturm-Liouville theory and quantum mechanics. For example, Dirac (1982, p. 26) denotes the adjoint of the bra vector <P|alpha as alpha^||P>, or alpha^_|P>.

Given a second-order ordinary differential equation


with differential operator


where p_i=p_i(x) and u=u(x), the adjoint operator L^~^| is defined by


Writing the two linearly independent solutions as y_1(x) and y_2(x), the adjoint operator can then also be written


In general, given two adjoint operators A^~ and B^~,


which can be generalized to


See also

Adjoint Curve, Adjoint Representation, Conjugate Transpose, Dagger, Hermitian Operator, Self-Adjoint, Sturm-Liouville Equation

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Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, 1985.Dirac, P. A. M. "Conjugate Relations." §8 in Principles of Quantum Mechanics, 4th ed. Oxford, England: Oxford University Press, pp. 26-29, 1982.Griffiths, D. J. Introduction to Elementary Particles. New York: Wiley, p. 220, 1987.

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Cite this as:

Weisstein, Eric W. "Adjoint." From MathWorld--A Wolfram Web Resource.

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