130_notes.dvi

Now taking the Hermitian conjugate ofA†.

〈

(

A†

)†

ψ|φ〉=〈Aψ|φ〉

(

A†

)†

=A

If we take the Hermitian conjugate twice, we get back to the same operator.

Its easy to show that

(λA)†=λ∗A†

and

(A+B)†=A†+B†

just from the properties of the dot product.

We can also show that

(AB)†=B†A†.

〈φ|ABψ〉=〈A†φ|Bψ〉=〈B†A†φ|ψ〉

• 
  

  See Example 8.8.1:Find the Hermitian conjugate of the operatora+ib.*

  
  


• 
  

  See Example 8.8.2:Find the Hermitian conjugate of the operator∂x∂.*

  
  

8.3 Hermitian Operators

Aphysical variable must have real expectation values(and eigenvalues). This implies that
the operators representing physical variables have some specialproperties.

By computing the complex conjugate of the expectation value of a physical variable, we can easily
show that physical operators are their own Hermitian conjugate.

〈ψ|H|ψ〉∗=





∫∞

−∞

ψ∗(x)Hψ(x)dx





∗

=

∫∞

−∞

ψ(x)(Hψ(x))∗dx=〈Hψ|ψ〉

〈Hψ|ψ〉=〈ψ|Hψ〉=〈H†ψ|ψ〉

H†=H

Operators that are their own Hermitian Conjugate are calledHermitian Operators.

H†=H