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The exponential of the derivative translates a function 𝑓(𝑥) by an amount εpic.twitter.com/mwQ1aLpOHt
We define the exponential of linear operator (i.e. derivative, matrix, etc.) through its Taylor Series The proof then follows directly:https://twitter.com/InertialObservr/status/1145396940122669057?s=20 …
In physics, this is why we call the derivative operator the "generator of translations" pic.twitter.com/iqJoydnyltI tried to interpret this intuitively but am struggling because it relies on the spooky action at a distance enjoyed by analytic functions and is false for general infinite-dimensional functions. Pretty cool though.
that's true .. I think my Physicist is showing .. we damn near always assume analyticity and function/finite number or arguments
Think of this as expressing wave functions and operators in the position representation, and consider translations https://en.m.wikipedia.org/wiki/Translation_operator_(quantum_mechanics) …
This is actually why we call momentum the generator of translations on QM
Is this what e^At (where A is a matrix of differential equation) is all about?
similar idea — matrix exponentials are defined through power series
I might be wrong, but I think Prof @robertghrist talked about this theorem on @myfavethm!! So cool
Yep! Anybody know if this theorem has a name?
I have a copy of this book from the 1800s!
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