Editing Hamiltonian (quantum mechanics) (section)

==Schrödinger equation==

{{Main|Schrödinger equation}}
The Hamiltonian generates the time evolution of quantum states. If <math> \left| \psi (t) \right\rangle</math> is the state of the system at time <math>t</math>, then

<math display="block"> H \left| \psi (t) \right\rangle = i \hbar {d\over\ d t} \left| \psi (t) \right\rangle.</math>

This equation is the [[Schrödinger equation]]. It takes the same form as the [[Hamilton–Jacobi equation]], which is one of the reasons <math>H</math> is also called the Hamiltonian. Given the state at some initial time (<math>t = 0</math>), we can solve it to obtain the state at any subsequent time. In particular, if <math>H</math> is independent of time, then

<math display="block"> \left| \psi (t) \right\rangle = e^{-iHt/\hbar} \left| \psi (0) \right\rangle.</math>

The [[Matrix exponential|exponential]] operator on the right hand side of the Schrödinger equation is usually defined by the corresponding [[Exponential function#Formal definition|power series]] in <math>H</math>. One might notice that taking polynomials or power series of [[unbounded operator]]s that are not defined everywhere may not make mathematical sense. Rigorously, to take functions of unbounded operators, a [[functional calculus]] is required. In the case of the exponential function, the [[continuous functional calculus|continuous]], or just the [[holomorphic functional calculus]] suffices. We note again, however, that for common calculations the physicists' formulation is quite sufficient.

By the *-[[homomorphism]] property of the functional calculus, the operator

<math display="block"> U = e^{-iHt/\hbar} </math>

is a [[unitary operator]]. It is the ''[[time evolution]] operator'' or ''[[propagator]]'' of a closed quantum system. If the Hamiltonian is time-independent, <math>\{U(t)\}</math> form a [[Stone's theorem on one-parameter unitary groups|one parameter unitary group]] (more than a [[C0 semigroup|semigroup]]); this gives rise to the physical principle of [[detailed balance]].