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### Is my expansion of the state $| x \rangle$ correct? [duplicate]

In my quantum mechanics textbook it says that the relation between the basis $|x\rangle$ and $|p\rangle$ is given by: $\langle p | x \rangle = \Large \frac{e^{-ip x/ \hbar}}{\sqrt{2\pi \hbar}} \, .$ ...
$\langle p | x \rangle=e^\frac{ipx}{\hbar}$ can pretty easily be derived from wave mechanics. But how can it be derived without resorting to it? I've seen it be derived from the translation operator $\... 6answers 39k views ### How to get the position operator in the momentum representation from knowing the momentum operator in the position representation? I know that $$\tag{1}\hat{p}~=~-i\hbar \frac{\partial}{\partial x}~.$$ How can I get $$\tag{2}\hat{x}~=~i\hbar \frac{\partial}{\partial p}~?$$ I think this simple and I'm just over thinking it, ... 5answers 6k views ### How does the momentum operator act on state kets? I have been going through some problems in Sakurai's Modern QM and at one point have to calculate$\langle \alpha|\hat{p}|\alpha\rangle$where all we know about the state$|\alpha\rangle\$ is that $$\... 2answers 2k views ### Proving that i\hbar\frac{\partial}{\partial \mathbf{p}} is the operator of \mathbf{x} in momentum space How can I prove that i\hbar\frac{\partial}{\partial \mathbf{p}} is the operator of \mathbf{x} in momentum space? 3answers 2k views ### Derivation of position operator in QM For the definition of the momentum operator$$\hat{P } = -i \hbar \nabla$$in quantum mechanics, as I understand you can derive this by either considering a more general definition of momentum, i.e. '... 1answer 2k views ### Normalizing Propagators (Path Integrals) In the context of quantum mechanics via path integrals the normalization of the propagator as$$\left | \int d x K(x,t;x_0,t_0) \right |^2 ~=~ 1$$is incorrect. But why? It gives the correct pre-... 1answer 3k views ### Why path integral approach may suffer from operator ordering problem? In Assa Auerbach's book (Ref. 1), he gave an argument saying that in the normal process of path integral, we lose information about ordering of operators by ignoring the discontinuous path. What did ... 2answers 6k views ### From position space to momentum space Lets say I have a state vector \left|\Psi(t)\right\rangle in a position space with an orthonormal position basis. If I now use an operator \hat{p} on this basis I will get basis which corresponds ... 2answers 2k views ### Derivation of canonical position-momentum commutator relation We know that the position-momentum commutator is fundamental in quantum mechanics, but would it be possible to derive it starting from a different set of first principles, more specifically starting (... 2answers 1k views ### What is the most general expression for the coordinate representation of momentum operator? I have a question about deriving the coordinate representation of momentum operator from the commutation relation, [x,p]= i. One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th ... 1answer 744 views ### What conservation law corresponds to this local U(1) symmetry of the CCR? It is known that canonical commutation relations do not fix the form of momentum operator. That means that if canonical commutation relations (CCR) are given by$$[\hat{x}^i,\hat{p}_j]~=~i\hbar~\...
In textbooks it seems to be taken for granted that $$\langle \mathbf{r}|\mathbf{k}\rangle ~=~ \frac{1}{\sqrt{\Omega}}\exp(i\mathbf{k}\cdot\mathbf{r}).$$ I'm sure it's obvious but is there a ...