Equation de Schrödinger

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Le function  \psi(\mathbf{x},t)=<\mathbf{x} |\psi(t)> = <\mathbf{x} | e^{-\frac{i}{\hbar} \mathcal{H} t}|\psi> obedi le equation de Schrödinger

H\psi(\mathbf{x},t)=-\frac{\hbar^2 }{2m}\nabla \psi(\mathbf{x},t)+U(\mathbf{x},t)\psi(\mathbf{x},t)=i\hbar\frac{\partial }{\partial t}\psi(\mathbf{x},t)

ubi U(\mathbf{x},t) es un energia potential. H es le operator de energia (le function de Hamilton) o operator Hamiltoniano

H =\frac{p^2}{2m}+U =-\frac{\hbar^2 }{2m}\nabla +U(\mathbf{x},t).

Le decomposition

\psi(\mathbf{x},t) = \sum_n c_n (t) \psi_n(\mathbf{x})

transforma le equation temporal de Schrödinger in le

H \psi_n(\mathbf{x},t)=E_n \psi_n(\mathbf{x},t)

equation non temporal de Schrödinger, con c_n(t) = e^{-i\frac{E_n t}{\hbar}}.

Vide etiam[modificar | modificar fonte]