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For starters I may be mistaken bringing this up as a notation problem, maybe for some reason unknown to me this is not about notation at all and I simply do not understand the topic well enough yet, this uncertainty tho is precisely why I am asking this in the first place.


In Modern Quantum Mechanics Sakurai calls the eigenvectors of an operator $A$:

$$|a'\rangle,|a''\rangle,...$$

But then he goes on writing things like this one:

$$\sum _{a'}c_{a'}|a'\rangle$$

Now I get that he means to cycle over all the eigenvectors of $A$, but this seems to me a really confusing way to write it. Written like this it seems that $a'$ should cycle from 1 to $N$, but of course this is not what it means. A much better notation would be calling the eigenvectors $$|a^{(1)}\rangle,|a^{(2)}\rangle,...$$ and then to write: $$\sum _{n=1}^N c_{n}|a^{(n)}\rangle$$

This can seem a detail, but for me it gets very confusing when we start dealing with things like this: $$\sum _{a'}\sum _{a''}\langle a''|X|a'\rangle$$ in this expression seems that we are only dealing with $|a'\rangle$ and $|a''\rangle$ when in fact we are cycling on all the eigenvectors.

Is there something that I am missing? Why do we use such confusing notation?

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  • $\begingroup$ Maybe a bit confusing because of the first equation but things like $\sum_{mm'}$ are very common in mathematics. $\endgroup$
    – JamalS
    Aug 19, 2020 at 21:18
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    $\begingroup$ You're not missing anything; Sakurai just has bad notation. $\endgroup$
    – knzhou
    Aug 19, 2020 at 21:26
  • $\begingroup$ He was heavily inspired by Dirac's notation in his QM book. $\endgroup$
    – DanielC
    Aug 19, 2020 at 22:02

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Short answer: no you're not missing anything. The notation is indeed a bit vague.

However, for completeness:

As @JamalS noted in a comment, the notation is not that uncommon. However, in mathematics the "good" notation would be $\sum_{a\in A}$ where $A$ is the set over whose elements we are summing (in your case this would be the spectrum of a given operator). Your example of natural number is then the case where $A=\{1,...,N\}$. But when it is clear from the context, people (physicists) tend to drop the set A and just write the summation index.

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