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  1. According to the Quantum Mechanics, can we write $\langle q|p\rangle = e^{ipq}$?

  2. If so then how?

  3. And if we transfer to integrate formulation then how it will look like?

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Yes, this notation is very common. Consider q and p as quantum numbers designating eigenstates of coordinate and positional operators. The integral representation reads: $$\langle q|p\rangle = \int d \mathbf{q}' \delta(\mathbf{q}-\mathbf{q}') e^{i\mathbf{p} \mathbf{q}'}$$

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Just one important formal remark: Note, that $|q \rangle$ and $|p \rangle$ are so-called generalized eigenstates of position and momentum operator respectively, which means that they are not elements of a Hilbert space (i.e. aren't square-integrable). However, they have a rigorous mathematical meaning in the sense of Gelfand triples, so this fact doesn't make any problems in calculations. – AGP Apr 19 '13 at 16:56

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