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The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the coordinates and momenta to operators. The non-relativistic Hamiltonian is
The Hamiltonian for many quantum mechanical systems can be obtained by writing down a corresponding classical Hamiltonian and promoting all of the coordinates and momenta to operators. The non-relativistic Hamiltonian is
<math>\hat{H} = \frac{\hat{p}^2}{2m} + V</math>
<math>\hat{H} = -\frac{\hat{p}^2}{2m} + V</math>
with <math>\hat{p}</math> being the [[momentum (physics)|momentum]] operator, <math>m</math> the [[mass (science)|mass]] and <math>V</math> the potential.
with <math>\hat{p}</math> being the [[momentum (physics)|momentum]] operator, <math>m</math> the [[mass (science)|mass]] and <math>V</math> the potential. Substituting in for <math>\hat{p}</math> gives
<math>\hat{H} = -\frac{\hbar^2}{2m} \nabla^2 + V</math>
The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].
The quantum mechanical Hamiltonian is of central importance to the [[Schrodinger equation]].