What is Hermitian, Operator or Eigenfunction?

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What is Hermitian, Operator or Eigenfunction?

Post by Confusions allowed on Mon Sep 12, 2016 6:54 pm

What is considered Hermitian, the operators (x, -i*(h/2π)*d/dx, H^, etc.) or the Wave Function (ψ) itself? Many texts suggest that there are Hermitian Operators, but no one says that the wave function is hermitian.
By the way, a Hermitian operator, as given in various sources, satisfies the property :
(Integral,-inf, inf)fA^g = (Integral,-inf,inf)gA^f, i.e., A^ = A^*.
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Re: What is Hermitian, Operator or Eigenfunction?

Post by manish1012 on Mon Sep 12, 2016 10:01 pm

Operators are Hermitian not the wave-function.
A notation:


<ψ₁|O|ψ₂>        ≡ ∫ {ψ₁*(O ψ₂)} dx from -∞ to ∞.


There is a quantity called Hermitian Conjugate of any operator .Let us for the moment denote it as .It satisfies
<ψ₁ | O ψ₂> = < ψ₁ | ψ₂>


Now if  = 
Then  is called a Hermitian Operator.

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