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Hermitian Operator


A second-order linear Hermitian operator is an operator L^~ that satisfies

 int_a^bv^_L^~udx=int_a^buL^~v^_dx.
(1)

where z^_ denotes a complex conjugate. As shown in Sturm-Liouville theory, if L^~ is self-adjoint and satisfies the boundary conditions

 v^_pu^'|_(x=a)=v^_pu^'|_(x=b),
(2)

then it is automatically Hermitian.

Hermitian operators have real eigenvalues, orthogonal eigenfunctions, and the corresponding eigenfunctions form a complete biorthogonal system when L^~ is second-order and linear.

Note that the concept of Hermitian operator is somewhat extended in quantum mechanics to operators that need be neither second-order differential nor real. Simply assuming that the boundary conditions give sufficiently strongly vanishing near infinity or have periodic behavior allows an operator L^~ to be Hermitian in this extended sense if

 intpsi^__1L^~psi_2dtau=intL^~psi_1^_psi_2dtau,
(3)

which is identical to the previous definition except that quantities have been extended to be complex (Arfken 1985, p. 506).

In order to prove that eigenvalues must be real and eigenfunctions orthogonal, consider

 L^~u_i+lambda_iwu_i=0.
(4)

Assume there is a second eigenvalue lambda_j such that

 L^~u_j+lambda_jwu_j=0
(5)
 L^~u^__j+lambda^__jwu^__j=0.
(6)

Now multiply (4) by u^__j and (6) by u_i

 u^__jL^~u_i+u^__jlambda_iwu_i=0
(7)
 u_iL^~u^__j+u_ilambda^__jwu^__j=0
(8)
 u^__jL^~u_i-u_iL^~u^__j=(lambda^__j-lambda_i)wu_iu^__j.
(9)

Now integrate

 int_a^bu^__jL^~u_i-int_a^bu_iL^~u^__j=(lambda^__j-lambda_i)int_a^bwu_iu^__j.
(10)

But because L^~ is Hermitian, the left side vanishes.

 (lambda^__j-lambda_i)int_a^bwu_iu^__j=0.
(11)

If eigenvalues lambda_i and lambda_j are not degenerate, then int_a^bwu_iu^__j=0, so the eigenfunctions are orthogonal. If the eigenvalues are degenerate, the eigenfunctions are not necessarily orthogonal. Now take i=j.

 (lambda^__i-lambda_i)int_a^bwu_iu^__i=0.
(12)

The integral cannot vanish unless u_i=0, so we have lambda^__i=lambda_i and the eigenvalues are real.

For a Hermitian operator O^~,

 <phi|O^~psi>=<phi|O^~psi>^_=<O^~phi|psi>.
(13)

In integral notation,

 intA^~phi^_psidx=intphi^_A^~psidx.
(14)

Given Hermitian operators A^~ and B^~,

 <phi|A^~B^~psi>=<A^~phi|B^~psi>=<B^~A^~phi|psi>=<phi|B^~A^~psi>^_.
(15)

Because, for a Hermitian operator A^~ with eigenvalue a,

 <psi|A^~psi>=<A^~psi|psi>
(16)
 a<psi|psi>=a^_<psi|psi>.
(17)

Therefore, either <psi|psi>=0 or a=a^_. But <psi|psi>=0 iff psi=0, so

 <psi|psi>!=0,
(18)

for a nontrivial eigenfunction. This means that a=a^|, namely that Hermitian operators produce real expectation values. Every observable must therefore have a corresponding Hermitian operator. Furthermore,

 <psi_n|A^~psi_m>=<A^~psi_n|psi_m>
(19)
 a_m<psi_n|psi_m>=a^__n<psi_n|psi_m>=a_n<psi_n|psi_m>,
(20)

since a_n=a^__n. Then

 (a_m-a_n)<psi_n|psi_m>=0
(21)

For a_m!=a_n (i.e., psi_n!=psi_m),

 <psi_n|psi_m>=0.
(22)

For a_m=a_n (i.e., psi_n=psi_m),

 <psi_n|psi_m>=<psi_n|psi_n>=1.
(23)

Therefore,

 <psi_n|psi_m>=delta_(nm),
(24)

so the basis of eigenfunctions corresponding to a Hermitian operator are orthonormal.

Define the adjoint operator A^~^| (also called the Hermitian conjugate operator) by

 <A^~psi|psi>=<psi|A^~^|psi>.
(25)

For a Hermitian operator,

 A^~=A^~^|.
(26)

Furthermore, given two Hermitian operators A^~ and B^~,

<psi_2|(A^~B^~)^|psi_1>=<(A^~B^~)psi_2|psi_1>
(27)
=<B^~psi_2|A^~^|psi_1>
(28)
=<psi_2|B^~^|A^~^|psi_1>,
(29)

so

 (A^~B^~)^|=B^~^|A^~^|.
(30)

By further iterations, this can be generalized to

 (A^~B^~...Z^~)^|=Z^~^|...B^~^|A^~^|.
(31)

Given two Hermitian operators A^~ and B^~,

 (A^~B^~)^|=B^~^|A^~^|=B^~A^~=A^~B^~+[B^~,A^~],
(32)

the operator A^~B^~ equals (A^~B^~)^|, and is therefore Hermitian, only if

 [B^~,A^~]=0.
(33)

Given an arbitrary operator A^~,

<psi_1|(A^~+A^~^|)psi_2>=<(A^~^|+A^~)psi_1|psi_2>
(34)
=<(A^~+A^~^|)psi_1|psi_2>,
(35)

so A^~+A^~^| is Hermitian.

<psi_1|i(A^~-A^~^|)psi_2>=<-i(A^~^|-A^~)psi_1|psi_2>
(36)
=<i(A^~-A^~^|)psi_1|psi_2>,
(37)

so i(A^~-A^~^|) is Hermitian. Similarly,

<psi_1|(A^~A^~^|)psi_2>=<A^~^|psi_1|A^~|psi_2>
(38)
=<(A^~A^~^|)psi_1|psi_2>,
(39)

so A^~A^~^| is Hermitian.


See also

Adjoint, Hermitian Matrix, Self-Adjoint, Sturm-Liouville Theory

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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.

Referenced on Wolfram|Alpha

Hermitian Operator

Cite this as:

Weisstein, Eric W. "Hermitian Operator." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HermitianOperator.html

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