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Quantum translation operator

Let $T_\epsilon=e^{i \mathbf{\epsilon} P/ \hbar}$ an operator. Show that $T_\epsilon\Psi(\mathbf r)=\Psi(\mathbf r + \mathbf \epsilon)$. Where $P=-i\hbar \nabla$. Here's what I've gotten: ...
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Does an expression for $$\exp\left(t z^{i}\partial_{z}^{j} \right) f(z) = ?$$ exist? For j=1 we have the usual expression for translation and scaling $$\exp\left( t \partial_z\right) f(z) = ... 0answers 66 views How to express e^{yS^2}(f(x)) in closed form where \frac{d}{dx}=S$$ f(x)+\frac{y.f'(x)}{1!}+\frac{y^2 f^{''}(x)}{2!}+\cdots=e^{yS}(f(x))=f(x+y) \text{ where }\frac{d}{dx}=S$$is a operator$$ f(x)+\frac{y.f''(x)}{1!}+\frac{y^2 ...
Is it possible to define a Taylor expansion for matrices ? Can I use functional derivative ? More precisely I have to calculate something like : $\ln(A+B)$ using a Taylor expansion, where $A$ and $B$ ...
Is there a formula similar to $f(x+a) = e^{a\frac{d}{dx}}f(x)$ to express $f(\alpha\cdot x)$?
Using the Taylor expansion $$f(x+a) = \sum_{k=0}^\infty \frac{a^k}{k!}\frac{d^k }{dx^k}f(x)$$ one can formally express the sum as the linear operator $e^{a\frac{d}{dx}}$ to obtain f(x+a) = ...