Adjoint

The word adjoint has a number of related meanings. In linear algebra, it refers to the conjugate transpose and is most commonly denoted . 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  (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 as , or .

Given a second-order ordinary differential equation

(1)

with differential operator

(2)

where  and , the adjoint operator  is defined by

			(3)
			(4)

Writing the two linearly independent solutions as  and , the adjoint operator can then also be written

			(5)
			(6)

In general, given two adjoint operators  and ,

(7)

which can be generalized to

REFERENCES:

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.