The temporal component gives: The spatial components give: where is the Hamiltonian. This is actually related to the 4-wavevector being equal the negative 4-gradient of the phase from above. To get the HJE, one first uses the Lorentz scalar invariant rule on the 4-momentum: But from the minimal coupling rule: So: Breaking into the temporal and spatial components: where the final is the relativistic Hamilton–Jacobi equation. As a component of the Schrödinger relations in quantum mechanics The 4-gradient is connected with quantum mechanics. The relation between the 4-momentum and the 4-gradient gives the Schrödinger QM relations. The temporal component gives: The spatial components give: This can actually be composed of two separate steps. First: which is the full 4-vector version of: The (temporal component) Planck–Einstein relation The (spatial components) de Broglie matter wave relation Second: which is just the 4-gradient version of the wave equation for complex-valued plane waves The temporal component gives: The spatial components give: As a component of the covariant form of the quantum commutation relation In quantum mechanics (physics), the canonical commutation relation is the fundamental relation between canonical conjugate quantities (quantities which are related by definition such that one is the Fourier transform of another). According to: Taking the spatial components, Since , Since , And, relabeling indices gives the usual quantum commutation rules: As a component of the wave equations and probability currents in relativistic quantum mechanics The 4-gradient is a component in several of the relativistic wave equations: In the Klein–Gordon relativistic quantum wave equation for spin-0 particles (ex. Higgs boson): In the Dirac relativistic quantum wave equation for spin-1/2 particles (ex. electrons): where are the Dirac gamma matrices and is a relativistic wave function. is Lorentz scalar for the Klein–Gordon equation, and a spinor for the Dirac equation. It is nice that the gamma matrices themselves refer back to the fundamental aspect of SR, the Minkowski metric: Conservation of 4-probability current density follows from the continuity equation: The 4-probability current density has the relativistically covariant expression: The 4-charge current density is just the charge () times the 4-probability current density: As a key component in deriving quantum mechanics and relativistic quantum wave equations from special relativity Relativistic wave equations use 4-vectors in order to be covariant. Start with the standard SR 4-vectors: 4-position 4-velocity 4-momentum 4-wavevector 4-gradient Note the following simple relations from the previous sections, where each 4-vector is related to