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

(8)