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This question is somewhat related to (but not by any means the same as) the question I asked recently.

In his Lectures on Quantum Theory, Isham essentially says (reference given below) that if an operator $\hat{d}_x$ satisfies $[\hat{x}, \hat{d}_x] = i\hbar 1\!\!1$ (where $\hat{x}$ is the familiar operator corresponding to position), then "by appealing to the classical limit of the theory, or by requiring consistency with the results of elementary wave mechanics", we can conclude $\hat{d}_x = \hat{p}_x$.

Questions:

  1. If Isham is building the axiomatic framework for QM, how can he conclude this by demanding consistency with elementary wave mechanics which is just a special case of his more general formalism, and which also lacks a proper formalism (in his book at least)?

  2. Also, wouldn't appealing to the classical limit of the theory be circular if the theory has not been developed in the first case? Or is it a forcing condition we apply on the theory that we're building to account for the observed classical phenomena?

  3. In any case, can someone clearly answer whether $[\hat{x}, \hat{d}_x] = i\hbar 1\!\!1$ is assumed in the foundations, or is it derivable, or is it imposable as some classical limit (in the sense of the latter part of the second question)?

Reference: Section 7.2.2 Displaced Observers and the Canonical Commutation Relations, page 137 (bottom), point 10.

Link to my copy of Isham's book.

Qmechanic
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  • Regarding your point 3, do you mean can someone answer whether $[x,p]=i\hbar$ is an assumption or derivable? Because what you wrote is clearly derived by Isham as you mention in your previous question. –  May 21 '20 at 21:22
  • @DvijD.C. In the previous, I ask how he so-called "derived" what he claims. Sorry for the late reply though, I slept... :) – Atom May 22 '20 at 04:56
  • Normally, the Heisenberg/Born relation is a cornerstone QM postulate, comporting with the vision of Dirac's PhD dissertation on the correspondence principle. I should not be familiar with what Isham's book assumes. – Cosmas Zachos May 22 '20 at 14:59

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