This thesis is concerned with a concept of geometrising time evolution of quantum systems. This concept is inspired by the fact that the Legendre transform expresses dynamics of a classical system through first-order Hamiltonian equations. We consider, in this thesis, coherent state transforms with a similar effect in quantum mechanics: they reduce certain quantum Hamiltonians to first-order partial differential operators. Therefore, the respective dynamics can be explicitly solved through a flow of points in extensions of the phase space. This, in particular, generalises the geometric dynamics of a harmonic oscillator in the Fock-Segal-Bargmann (FSB) space. We describe all Hamiltonians which are geometrised (in the above sense) by Gaussian and Airy beams and exhibit explicit solutions for such systems.