Hermitian Operators?

  • Thread starter Axiom17
  • Start date
  • #1
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Homework Statement



I have some operators, and need to figure out which ones are Hermitian (or not).

For example:

1. [tex]\hat{A} \psi(x) \equiv exp(ix) \psi(x)[/tex]

Homework Equations



I have defined the Hermitian Operator:

[tex]A_{ab} \equiv A_{ba}^{*}[/tex]

The Attempt at a Solution



I just don't know where to start with this :uhh:
 
Last edited:

Answers and Replies

  • #2
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One good way is to see if something is hermitian is if

[tex]\langle \psi | A | \psi \rangle= \langle \psi |A| \psi \rangle^*[/tex]

if A is hermitian then the equality will hold.
 
  • #3
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Ok but I don't understand the calculations to do :uhh:
 
  • #4
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The way you expressed the opeator A_{ab}=A*_{ba} is a matrix notation, useful when acting on a set of vectors like v_{b}. But in your problem, how does the operator act on the wavefuntion Psi(x)? Can you re-express your definition above for Hermiticity in this specific case?
 
  • #5
70
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Umm I have this:

[tex]\langle a|\hat{A}|b \rangle = \int dV \psi_{a}^{*}(r)r^{2}\psi_{b}(r)=[\langle b|\hat{A}|a \rangle]^{*}[/tex]

But I don't know if that's any use :grumpy:
 
  • #6
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Still don't get this :frown:
 
  • #7
1,860
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So, for example

[tex]\hat{A} \psi(x) \equiv exp(ix) \psi(x)[/tex]

[tex]\langle \psi | A | \psi \rangle= \int \psi^*(x) exp(ix) \psi(x)[/tex]

[tex]\langle \psi | A | \psi \rangle^*= \int \psi^*(x) exp(ix)^* \psi(x)=\int \psi^*(x) exp(-ix) \psi(x) \neq \int \psi^*(x) exp(ix) \psi(x)[/tex]

The operator is not hermitian.
 

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