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## Main Question or Discussion Point

Hi all, I've posted a little bit here in the past but I don't think anyone's going to really remember me from those. I hope to come back more frequently now, especially since I've now learned how to use the TEX feature. There was a post with members asking somewhat similar questions a couple months back but not many answers were given, so I apologize for starting a new post rather than reviving the old one.

Anyhow, on the most recent discussions in the following thread on sciforums.com, at http://www.sciforums.com/showthread.php?t=78957&page=10", we're trying to derive the momentum operator as the generator of spatial translations. In classical Hamiltonian mechanics we know that choosing p, the canonical momentum, as the generator of an infinitesimal canonical transformation leads to an infinitesimal translation of the system in space. The official result for how functions of x change due to this transformation (in 1 dimension) is given by [tex]f(x+dx)=f(x)+dx\{f,p\}=f(x)+dx\frac{\partial f}{\partial x}[/tex].

Here, [tex]\{u,v\}[/tex] refers to the Poisson bracket, which is evaluated in general, for 1-dimensional systems, as [tex]\{u,v\}=\frac{\partial u}{\partial x}\frac{\partial v}{\partial p}-\frac{\partial u}{\partial p}\frac{\partial v}{\partial x}[/tex]. The correspondence between classical Poisson brackets and quantum commutators is [tex][u,v]=i\hbar\{u,v\}[/tex].

The Hamiltonian can be used in an infinitesimal canonical transformation as the classical generator of inifinitesimal time evolution, and this method carries straight over from Poisson brackets into QM. Similarly, can anyone show me a way to carry the classical generator of translations over to QM? I.e. can anyone here derive from the correspondence principle that the the infinitesimal translation operator, [tex]\mathcal{T}(dx)[/tex] is to be represented in terms of the canonical momentum as [tex]1-\frac{i}{\hbar}\hat{p}dx[/tex]?

Anyhow, on the most recent discussions in the following thread on sciforums.com, at http://www.sciforums.com/showthread.php?t=78957&page=10", we're trying to derive the momentum operator as the generator of spatial translations. In classical Hamiltonian mechanics we know that choosing p, the canonical momentum, as the generator of an infinitesimal canonical transformation leads to an infinitesimal translation of the system in space. The official result for how functions of x change due to this transformation (in 1 dimension) is given by [tex]f(x+dx)=f(x)+dx\{f,p\}=f(x)+dx\frac{\partial f}{\partial x}[/tex].

Here, [tex]\{u,v\}[/tex] refers to the Poisson bracket, which is evaluated in general, for 1-dimensional systems, as [tex]\{u,v\}=\frac{\partial u}{\partial x}\frac{\partial v}{\partial p}-\frac{\partial u}{\partial p}\frac{\partial v}{\partial x}[/tex]. The correspondence between classical Poisson brackets and quantum commutators is [tex][u,v]=i\hbar\{u,v\}[/tex].

The Hamiltonian can be used in an infinitesimal canonical transformation as the classical generator of inifinitesimal time evolution, and this method carries straight over from Poisson brackets into QM. Similarly, can anyone show me a way to carry the classical generator of translations over to QM? I.e. can anyone here derive from the correspondence principle that the the infinitesimal translation operator, [tex]\mathcal{T}(dx)[/tex] is to be represented in terms of the canonical momentum as [tex]1-\frac{i}{\hbar}\hat{p}dx[/tex]?

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