One postulate that a particle (or a system of many particles) is described by a wavefunction, and this satisfies a quantum wave equation: namely the Schrödinger equation (which can be written as a non-relativistic wave equation, or a relativistic wave equation).
Other postulates change the idea of physical observables; using quantum operators; some measurements can't be made at the same instant of time (Uncertainty principles), particles are fundamentally indistinguishable.