2 Corrected typo in title; tried to make title even more informative; retagged;

# 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.

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.