HERMITIAN PROPERTY OF OPERATORS

It's very important to note that a quantum mechanical operator satisfied the following conditions known as Hermitian condition.

        If an operator  Â has two eigen function ψ and ɸ , and if

∫ψ ( Â ɸ ) dτ = ∫ ( Âψ). ɸdτ................ (1)

When ψ and ɸ are real ,

(or)

∫ ψ* ( Âɸ ) dτ = ∫ ( Â ψ)*. ɸ dτ  ......... (2)

When ψ and ɸ are complex, ψ* is the complex conjugate of ψ and dτ is the volume element of space in which the function is defined, then the operator  is called " Hermitian Operator"

PROPERTIES OF HERMITIAN OPERATOR :

                          There are two types of very important characteristics of a Hermitian operator.

(1) The eigen values are real positive or negative.

(2) Eigen function corresponding to different eigen values are orthogonal to each other.

These two important theorems about the Hermitian operators can be proved easily as follows :

Eigen values of a Hermitian operator are experimentally real :

 Let  be a Hermitian operator,

       ψ is eigen function,

       λ the eigen value.

Then, 

   Â ψ = λ ψ and also ( Â ψ)* = λ* ψ*

Multiplying the first equation by ψ* and ψ, then integrating.

∫ψ* Â ψ dj = ∫ψ* λψ dj = λ ∫ ψ* ψ dj  and

∫ ψ ( Â ψ)* dτ = ∫ ψ λ* ψ * dτ = λ* ∫ ψ ψ* dτ

But since  is Hermitian,

  ∫ ψ* ( Â ψ) dτ = ∫ ψ ( Âψ)* dτ

Therefore, 

  λ ∫ ψ* ψ dτ = λ* ∫ ψ ψ* dτ .................. (3)

(or)

         λ = λ*, i.e., λ is real values. 

Eigen function of a Hermitian operator corresponding to different types of eigen values are orthogonal :

Let ψ₁ and ψ₂ be two eigen functions of a Hermitian operator  corresponding to two eigen values λ₁ and λ₂ respectively. The conditions of orthogonality is,

           ∫ ψ₁ ψ₂ dτ = 0,

(or)   

           ∫ ψ₁ ψ₂* dτ = 0

(or)

          ∫ ψ₁* ψ₂ dτ = 0.

The eigen value equations are :

             Â ψ₁ = λ₁ ψ₁    ....................(4)

            Â ψ₂ = λ₂ ψ₂    ......................(5)

Multiplying equation (4) by ψ₂* on integrating,

∫ ψ₂* Â ψ₁ dτ = ∫ ψ₂* λ₁ ψ₁ dτ = λ₁ ∫ ψ₂* ψ₁ dτ

 But since  is Hermitian,

∫ ψ₂* Â ψ₁ dτ = ∫ ψ₁ ( Â ψ₂ )* dτ = 

                 ∫ ψ₁ ( λ₂ψ₂ )* dτ

              = λ₂* ∫ ψ₁ ψ₂* dτ

              = λ₂ ∫ ψ₁ ψ₂* dτ

Thus, λ₁ ∫ ψ₂* ψ₁ dτ = λ₂ ∫ ψ₁ψ₂* dτ

(or) 

 (λ₁ −λ₂) ∫ ψ₁ ψ₂* dτ = 0

But since λ₁ ≠ λ₂, ∫ ψ₁ ψ₂* dτ = 0

If , However,  λ₁ λ₂ (i.e.,ψ₁ and ψ₂ are degenerate), then the integral ∫ ψ₁ ψ₂*dτ

Need not be zero. 

(i. e) ψ₁ and ψ₂ may not be orthogonal.