2 Corrected typo in title; tried to make title even more informative; retagged;
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What does a the commutation relationCanonical Commutation Relation (CCR) tell me about the positionoverlap between Position and momentumMomentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:

  1. $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
  2. $|p\rangle$ is an eigenvector of an operator $P$ with eigenvalue $p$.
  3. $Q$ and $P$ are Hermitian.
  4. $[Q,P] = i \hbar$.

I'm asking because books and references I've looked in tend to assume that $Q$ is a differential operator when viewed in the $P$-basis, then show that it satisfies the commutation relation. (I haven't read one yet that proves that the form given for $Q$ is the only possible one.) I've heard, though, that we can work purely from the hypothesis of the commutation relation and prove the properties of $Q$ and $P$ from it.

What does a the commutation relation tell me about the position and momentum bases?

I'm curious whether I can find $\langle q | p \rangle$ knowing only the following:

  1. $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
  2. $|p\rangle$ is an eigenvector of an operator $P$ with eigenvalue $p$.
  3. $Q$ and $P$ are Hermitian
  4. $[Q,P] = i \hbar$

I'm asking because books and references I've looked in tend to assume that $Q$ is a differential operator when viewed in the $P$-basis, then show that it satisfies the commutation relation. (I haven't read one yet that proves that the form given for $Q$ is the only possible one.) I've heard, though, that we can work purely from the hypothesis of the commutation relation and prove the properties of $Q$ and $P$ from it.

What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following:

  1. $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
  2. $|p\rangle$ is an eigenvector of an operator $P$ with eigenvalue $p$.
  3. $Q$ and $P$ are Hermitian.
  4. $[Q,P] = i \hbar$.

I'm asking because books and references I've looked in tend to assume that $Q$ is a differential operator when viewed in the $P$-basis, then show that it satisfies the commutation relation. (I haven't read one yet that proves that the form given for $Q$ is the only possible one.) I've heard, though, that we can work purely from the hypothesis of the commutation relation and prove the properties of $Q$ and $P$ from it.

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What does a the commutation relation tell me about the position and momentum bases?

I'm curious whether I can find $\langle q | p \rangle$ knowing only the following:

  1. $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$.
  2. $|p\rangle$ is an eigenvector of an operator $P$ with eigenvalue $p$.
  3. $Q$ and $P$ are Hermitian
  4. $[Q,P] = i \hbar$

I'm asking because books and references I've looked in tend to assume that $Q$ is a differential operator when viewed in the $P$-basis, then show that it satisfies the commutation relation. (I haven't read one yet that proves that the form given for $Q$ is the only possible one.) I've heard, though, that we can work purely from the hypothesis of the commutation relation and prove the properties of $Q$ and $P$ from it.