Hamiltonian operator

See also: Hamiltonian

In Mechanical quantum, the operator of Hamilton , Hamiltonian operator or quite simply Hamiltonian is the mathematical operator which makes it possible to describe the evolution of a quantum system during time in the representation of Schrödinger by the equation:

I \ hbar \ frac {D \ mid \ psi \ rangle} {D T} = \ hat {H} \ mid \ psi \ rangle

where $\ mid \ psi \ rangle$ is the Fonction of wave system, and $\ hat {H}$ the Hamiltonian operator. In a stationary state:

$\ mid \ psi (T) \ rangle = e^ {- I \ frac {E T} {\ hbar}} \ mid \ psi (0) \ rangle,$

where $E$ is the energy of the stationary state. One easily sees that a stationary state is a clean Vecteur of the Hamiltonian operator, with energy like Eigenvalue. The Hamiltonian being a square operator, energies obtained are real.

In the representation of Heisenberg, the states are independent of time, and the operators are dependant on time. The Hamiltonian operator intervenes then in the equation of evolution of the operators:

$I \ hbar \ frac {D \ hat {has}} {dt} = I \ hbar \ frac {\ partial \ hat {has}} {\ partial T} +$

where $\ partial \ partial t$ indicates a derivation compared to an explicit dependence compared to time and $= \ hat {has} \ hat {H} - \ hat {H} \ hat {has}$ is the switch of the operators $\ hat {has}$ and $\ hat {H}$ .

One passes from the representation of Schrödinger to the representation of Heisenberg by means of the Opérateur of evolution.

In the case not-relativist, the Hamiltonian operator can be obtained starting from the Hamiltonian of the traditional Mécanique by the principle of correspondence. If $H (p, Q)$ is the traditional Hamiltonian, the quantum Hamiltonian is obtained in substituent with the traditional variables $p$ (impulse) and $q$ (coordinated) the operators $\ hat {p}$ and $\ hat {Q}$ . It is sometimes necessary to symmetrize the Hamiltonian thus obtained to make sure of the hermiticity of the Hamiltonian. Indeed, the principle of correspondence always makes it possible to obtain the traditional Hamiltonian starting from the quantum Hamiltonian by replacing the operators by numbers, but several quantum operators, differing only by the order from the operators (who do not commutate) can lead to the same quantum variable.

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