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What makes expectation values real?

  1. Feb 7, 2015 #1
    If you have some wave function of some particle, say...


    And you calculate the expectation value of momentum, say...


    What ensures that that spatial integral is real valued?

    Separately, all the components of the integral are complex valued
  2. jcsd
  3. Feb 7, 2015 #2


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    The operators, representing observables, are usually assumed to be self-adjoint, i.e., ##\hat{p}=\hat{p}^{\dagger}##. Now you have
    $$\langle \psi|\hat{p} \psi \rangle^*=\langle \psi|\hat{p}^{\dagger} \psi \rangle = \langle \psi|\hat{p} \psi \rangle,$$
    which implies that the expectation value is real.
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