<p>To address systems with spatially and temporally varying particle masses, we construct the Hamiltonian using Born-Jordan quantisation. This approach also includes systems influenced by electromagnetic fields as a specific case. By introducing a transformation based on a freely chosen action function, we can separate the time-dependent Schrödinger equation into two distinct components: the classical Hamilton–Jacobi equation and a complementary quantum term. The classical part yields Hamilton’s principal function, which, when promoted to an operator, governs the quantum dynamics. Ultimately, by applying established methods from quantum mechanics, we investigate the relationship between quantum and classical dynamics. We particularly emphasise the role of classical trajectories, mediated through Hamilton’s principal function, in shaping quantum behaviour.</p>