“Hermitian Operator”的概念、定义、习题解题方法及翻译-数学辞典

单词

Hermitian Operator

释义

Hermitian Operator

A Hermitian Operator is one which satisfies

(1)

As shown in Sturm-Liouville Theory, if

is Self-Adjoint and satisfiesthe boundary conditions

(2)

then it is automatically Hermitian. Hermitian operators have Real Eigenvalues,Orthogonal Eigenfunctions, and the correspondingEigenfunctions form a Complete set when

is second-order andlinear. In order to prove that Eigenvalues must be Real andEigenfunctions Orthogonal, consider

(3)

Assume there is a second Eigenvalue

such that

(4)

(5)

Now multiply (3) by

and (5) by

(6)

(7)

(8)

Now integrate

(9)

But because

is Hermitian, the left side vanishes.

(10)

If Eigenvalues

and

are not degenerate, then

, so theEigenfunctions are Orthogonal. If theEigenvalues are degenerate, the Eigenfunctions are not necessarilyorthogonal. Now take

(11)

The integral cannot vanish unless

, so we have

and the Eigenvaluesare real.

For a Hermitian operator ,

(12)

In integral notation,

(13)

Given Hermitian operators

and

(14)

Because, for a Hermitian operator

with Eigenvalue

(15)

(16)

Therefore, either

. But

Iff

, so

(17)

for a nontrivial Eigenfunction. This means that

, namely that Hermitian operators produce Real expectation values. Every observable must therefore have a corresponding Hermitian operator. Furthermore,

(18)

(19)

since

. Then

(20)

For

(i.e.,

(21)

For

(i.e.,

(22)

Therefore,

(23)

so the basis of Eigenfunctions corresponding to a Hermitian operator are Orthonormal. Given two Hermitian operators

and

(24)

the operator

equals

, and is therefore Hermitian, only if

(25)

Given an arbitrary operator


			(26)

is Hermitian.


			(27)

is Hermitian. Similarly,

(28)

is Hermitian.

Define the Hermitian conjugate operator by

(29)

For a Hermitian operator,

. Furthermore, given two Hermitian operators

and


			(30)

(31)

By further iterations, this can be generalized to

(32)

See also Adjoint Operator, Hermitian Matrix, Self-Adjoint Operator,Sturm-Liouville Theory

References

Arfken, G. ``Hermitian (Self-Adjoint) Operators.'' §9.2 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 504-506 and 510-516, 1985.

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