Editing Equations of motion (section)

===Quantum theory===

In quantum theory, the wave and field concepts both appear.

In [[quantum mechanics]] the analogue of the classical equations of motion (Newton's law, Euler–Lagrange equation, Hamilton–Jacobi equation, etc.) is the [[Schrödinger equation]] in its most general form:

<math display="block">i\hbar\frac{\partial\Psi}{\partial t} = \hat{H}\Psi \,,</math>

where {{math|''Ψ''}} is the [[wavefunction]] of the system, {{math|''Ĥ''}} is the quantum [[Hamiltonian operator]], rather than a function as in classical mechanics, and {{math|''ħ''}} is the [[Planck constant]] divided by 2{{pi}}. Setting up the Hamiltonian and inserting it into the equation results in a wave equation, the solution is the wavefunction as a function of space and time. The Schrödinger equation itself reduces to the Hamilton–Jacobi equation when one considers the [[correspondence principle]], in the limit that {{math|''ħ''}} becomes zero. To compare to measurements, operators for observables must be applied the quantum wavefunction according to the experiment performed, leading to either [[wave-particle duality| wave-like or particle-like]] results.

Throughout all aspects of quantum theory, relativistic or non-relativistic, there are [[mathematical formulation of quantum mechanics|various formulations]] alternative to the Schrödinger equation that govern the time evolution and behavior of a quantum system, for instance: 
*the [[Heisenberg picture|Heisenberg equation of motion]] resembles the time evolution of classical observables as functions of position, momentum, and time, if one replaces dynamical observables by their [[operator (physics)|quantum operators]] and the classical [[Poisson bracket]] by the [[commutator]], 
*the [[phase space formulation]] closely follows classical Hamiltonian mechanics, placing position and momentum on equal footing,
*the Feynman [[path integral formulation]] extends the [[principle of least action]] to quantum mechanics and field theory, placing emphasis on the use of a Lagrangians rather than Hamiltonians.