How to identify $\hat{d}_x$ with $\hat{p}_x/\hbar$?

Question

Isham, in his Lecture on Quantum Theory, Chapter 7, Unitary Operators in Quantum Theory, Section 7.2.2 Displaced Observers and the Canonical Commutation Relations, mentions on page 137 (bottom) the following.

10. The final step is to identify the operator $\hat{d}_x$ with $\hat{p}_x/\hbar$, where $\hat{p}_x$ is the momentum along the $x$ direction. This can be done by appealing to the classical limit of the theory, or by requiring consistency with the results of elementary wave mechanics. Thus we get the result that the states assigned by $O_2$ and $O_1$ are related by

$$|\psi\rangle_a=e^{ia\hat{p}_x/\hbar}|\psi\rangle.$$

Question: I don't know how to get this from the "classical limit" or "consistency" argument. Any help?

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The following is how Isham has defined the operator $\hat{d}_x$ (and $O_1$ and $O_2$). First he defines the operator $\hat{D}(a)$ (after showing it exists) as the operator which satisfies $|\psi\rangle_a=\hat{D}(a)|\psi\rangle$, where $|\psi\rangle$ is the state of a quantum system as observed by an observer $O_1$ and $|\psi\rangle_a$ is the state of the same system as observed by an observer $O_2$ displaced along the positive $x$ direction by a distance $a$. Then he goes on to show that there exists a self-adjoint operator $\hat{d}_x$ such that $\hat{D}(a)=e^{ia\hat{d}_x}$ for all distances $a$.

Answer 1

OK, to the extent you are not stuck on exponentiation, the "consistency with the results of elementary QM" bit is obvious. QM is predicated on Born's cornerstone relation, your text's (1.16), $$ [\hat x, \hat p]=i\hbar 1\!\! 1 , $$

You may immediately verify that in the x-basis, your (7.37) relation $[\hat d_x, \hat x]=-i~1\!\! 1$ reads $$ [x,- \partial_x] f(x) = f(x), $$ for arbitrary f(x); so $\hat p \mapsto -i \hbar \partial_x$, i.e. $$ \hat x = \int dx' ~~|x'\rangle x' \langle x'|, ~~~~ \hat p = -i\hbar\int dx' ~~ |x'\rangle \partial_{x'} \langle x'|~. $$

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But... how does (1.16) comport with the classical limit of Dirac's thesis (summarized in his monumental book)? The (fraught) formal limit of operators $\hat f$ and $\hat g$ to classical versions thereof maps quantum commutators to classical Poisson brackets, $$ [\hat f, \hat g ] \leadsto \frac{\{ f,g\} }{i\hbar} \implies \\ [\hat x , \hat p ] \leadsto \{ x, p\}/i\hbar ~~~\implies \\ \{ x, p\}=1 \leadsto [\hat x , \hat p ] =i\hbar 1\!\! 1, \qquad \hat x \leadsto x , ~~~~~ \hat p \leadsto p, ~~~ 1\!\!1 \leadsto 1 ~. $$ That is to say (1.16) is dictated by Dirac's limit, and, as above, it identifies with (7.37).

• Many of these issues and limits are best illustrated in phase-space quantization, but this outranges your text and question.

Thanks for the link to your text. Many of these points, namely the essential uniqueness of this representation you are talking about (up to equivalence: basis changes) is detailed in your text's section 7.2.2——where your question came from. This is the heart of the celebrated Stone—von Neumann theorem.

$[\hat x , \hat p -\hbar \hat d_x]=0$ suffices for the identification, up to equivalence, since the difference between the two operators is then, in general, a function of $\hat x$; which can be gauged to zero, as one always does when applying the S-vN theorem: $\hat d _x$ is formally equivalent to $e^{-ig(\hat x)} \hat d_x e^{ig(\hat x)}= \hat d_x + g'(\hat x)$.