Starting on page 41 of Dirac's The Principles of Quantum Mechanics, he defines $f(\xi)$ in general to be that linear operator which satisfies

$$f(\xi)|\xi'\rangle = f(\xi')|\xi'\rangle\tag {34}$$

for every eigenket $|\xi'\rangle$ of the real dynamical variable $\xi$, $f(\xi')$ being a number for each eigenvalue $\xi'$

He then defines the conjugate complex $\overline{f(\xi)}$ of ${f(\xi)}$ by the conjugate imaginary equation to (34) as

$$\langle\xi'|\overline{f(\xi)} = \overline f(\xi ')\langle\xi'|$$

holding for any eigenbra $\langle\xi'|$, $\overline f(\xi')$ being the conjugate complex function to $f(\xi')$

Why doesn't he write $\overline f(\xi ')$ as $\overline {f(\xi ')}$?

  • $\begingroup$ Looks to me that $\bar{f}(\xi')$ is the eigenvalue of the operator $\overline{f(\xi)}$ on $\langle\xi'\vert$. The difference in the lengths of the bars signifies the difference between the eigenvalue & the operator. $\endgroup$ – Kyle Kanos Oct 5 '14 at 16:59
  • $\begingroup$ @KyleKanos Dirac uses $\xi$ for a real linear operator, ' to label objects connected with eigenvalues-- $\xi'$ for an eigenvalue, $\langle\xi'|$ for an eigenbra $\endgroup$ – Physiks lover Oct 5 '14 at 18:18
  • $\begingroup$ @KyleKanos Why doesn't Dirac just substitute $\xi'$ into $\overline{f(\xi)}$ as for (34)? $\endgroup$ – Physiks lover Oct 5 '14 at 21:21
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    $\begingroup$ I am not sure the reason, that seems like a question for Dirac ;). $\endgroup$ – Kyle Kanos Oct 5 '14 at 21:31
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    $\begingroup$ I also agree with the answer of Nikos M. below. If you want to go further into deep & unchartered territories ;-), this link might be useful. As a gist, since $\xi$ is an observable, its eigenvalues $\xi^{'}$ must be real. In that case, its easy to see $\overline{f(\xi^{'})} = \overline{f}(\xi^{'})$ $\endgroup$ – jayann Oct 6 '14 at 8:52

It is just a matter of definition (dont be led too astray by this). Well a complex conjugate form acts on the dual space of the space where the normal (non-conjugate) form acts (in your example the eigenbra space). Of course for Hilbert spaces, which are usually self-dual, the difference is almost none. The rest is just a matter of notational definiton.


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