In quantum mechanics, the wavefunction $|\psi(t)\rangle$ of a system with a constant Hamiltonian $H$, evolves according to:
$$\tag{1}|\psi(t)\rangle = e^z |\psi(0)\rangle,$$$$\tag{1}\label{1}\lvert\psi(t)\rangle = e^z \lvert\psi(0)\rangle,$$ where $z$ is the complex matrix:
$$\tag{2} z = \frac{\textrm{i}}{\hbar}Ht. $$$$\tag{2}\label{2} z = \frac{\textrm{i}}{\hbar}Ht. $$
This is simply because in quantum mechanics, the wavefunction evolves according to the Schroedinger equation:
$$ \frac{\textrm{d}}{\textrm{d}t}|\psi(t)\rangle = \frac{\textrm{i}}{\hbar}H |\psi(t)\rangle\tag{3}, $$$$ \frac{\textrm{d}}{\textrm{d}t}\lvert\psi(t)\rangle = \frac{\textrm{i}}{\hbar}H \lvert\psi(t)\rangle\tag{3}\label{3}, $$
and Eq. 1\ref{1}, with $z$ chosen according to Eq. 2\ref{2}, is the solution to Eq. 3\ref{3}.
