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Proving the adjoint nature of operators using Hermiticity

  1. Jun 25, 2015 #1
    How can the fact that ##\hat x## and ##\hat p## are Hermitian be used to prove that ##\hat x - \frac{i}{m \omega} \hat p## and ##\hat x + \frac{i}{m \omega} \hat p## are adjoints of each other?
     
  2. jcsd
  3. Jun 25, 2015 #2

    RUber

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    Adjoints have the property that ##\langle A, u \rangle =\langle u, A^* \rangle##. In the simplest case, assume that x and p are purely real. Then you should be able to show that the conjugate matrices satisfy the required properties.
     
  4. Jun 25, 2015 #3

    Fredrik

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    Prove the rules ##(A+B)^\dagger =A^\dagger+B^\dagger## and ##(cA)^\dagger=c^*A^\dagger##, and then use them.

    Hint: ##\langle x,Ay\rangle =\langle A^\dagger x,y\rangle## for all ##x,y## such that ##y## is in the domain of A, and ##x## is in the domain of ##A^\dagger##.
     
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