I am a physics undergraduate and me and my colleagues were discussing if the exponential of an operator is just a definition or it comes up from more fundamental concepts. I would like an explanation from where it was first used or most common area of use in mathematics and the validity of this operation.
Thanks in advance!
Any time you solve an $n$ dimensional (usually $n=2$) system of ODE's with constant coefficients (see harmonic oscillator), the matrix exponential comes up in solving ODE's of the form $$ \vec{x}'(t)=A\vec{x(t)}\implies \vec{x}(t)=\exp(At)\vec{x_0} $$ Where it is safe to define $\exp(At)$ by it's series expansion.
A less tame place: the time evolution equation for a particle governed by Schroëdinger's equation is given by an operator exponential. Namely, if $$ H\psi(t)=i\hbar \frac{\partial \psi}{\partial t} $$ where $H$ is the Hamiltonian. Then the solution is given by $$ \psi(t)=\psi(0)e^{-iHt/\hbar} $$ where much care needs to be taken in what $e^{-iHt/\hbar}$ means, but we take advantage of the analogy to regular ODE (and regular matrix exponentiation) in using the same notation. In this case, this is an unbounded operator (thanks to the momentum component), so care needs to be taken in defining what the exponential means.