Path integral formulation: Difference between revisions

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In quantum mechanics, as in classical mechanics, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator (multiplied by the negative [[imaginary unit]], {{math|−''i''}}). For states with a definite energy, this is a statement of the [[de Broglie relation]] between frequency and energy, and the general relation is consistent with that plus the [[superposition principle]].

The Hamiltonian in classical mechanics is derived from a [[Lagrangian (field theory)|Lagrangian]], which is a more fundamental quantity relativein tothe context of [[special relativity]]. The Hamiltonian indicates how to march forward in time, but the time is different in different [[Frame of reference|reference frames]]. The Lagrangian is a [[Lorentz scalar]], while the Hamiltonian is the time component of a [[four-vector]]. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.

The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later (or, equivalently for infinitesimal time separations, it is a function of the position and velocity). The relation between the two is by a [[Legendre transformation]], and the condition that determines the classical equations of motion (the [[Euler–Lagrange equation]]s) is that the [[action (physics)|action]] has an extremum.