I learnt that the observables are self-adjoint operators working on wave functions which live in a Hilbert space. The eigenvalues of these operators are real and appear as outcome of measurements. Which eigenvalue comes as output, the probability is given by the weights of the orthonormal expansion of wavefunction on a basis composing the eigenvectors of the operator. And the final state would be the eigenvector corresponding to the eigenvalue emitted.

My question is: I'd like to know the importance of anti-self adjoint operators in quantum mechanics. $$\langle \psi,L\phi \rangle = - \langle L\psi,\phi \rangle$$. Their eigenvalues are imaginary. Do they make any sense to the theory of quantum mechanics?

  • 1
    $\begingroup$ they are not observables, but are related to observables by a "Wick rotation", that is, if $L$ is anti-self-adjoint, the $iL$ is self-adjoint. Take the time evolution operator as an example, i.e. $e^{iHt}$, when the Hamiltonian is time-independent. You can think of $iH$ as the generator of the time evolution, but it is customary to consider $H$ as the generator, rather than $iH$. $\endgroup$ – Phoenix87 Jan 6 '15 at 12:14
  • 2
    $\begingroup$ It seems that the heart of this question (v2) boils down to semantics, conventions, and multiplication with $i$. $\endgroup$ – Qmechanic Jan 6 '15 at 13:07

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