# Tag Info

2

The set of all possible elements of the form $e^{i\alpha}D(x,p)$ with $\alpha, x,p \in \mathbb R$ verifying the commutation relations you wrote in addition to: $$D(x,p)^* = D(-x,-p)\:,\quad D(0,0)=I$$ is a group and it is called Heisenberg group, it is homeomorphic (diffeomorphic) to $U(1) \times \mathbb R^2$ but not isomorphic as a Lie group. It is a real ...

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We can realize the displacement operator as $$\tag{1}\hat{D}(x,p)~=~e^{x\hat{P}+p\hat{X}},$$ where the elements $\hat{X}$, $\hat{P}$ and ${\bf 1}$ generates the Heisenberg algebra $$\tag{2} [\hat{X},\hat{P}]=i{\bf 1}.$$ These elements can be realizes as differential operators in the Schrödinger representation. (See also the Stone-von Neumann theorem.) ...

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One thing to note is that it is not just the crystal, but a crystal which is coherently pumped by a strong laser field in which one sees SHG. In this setting, the pump creates a periodic modulation of the index of refraction (thus the requirement for a non liner medium) which effectively acts as a phase grating. One way of thinking about SHG is that it is ...

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