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Schrodinger equation (view source)
Revision as of 19:06, December 13, 2009
, 19:06, December 13, 2009no edit summary
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}+V(x)\psi=i\hbar\frac{\partial \psi}{\partial t}
</math>
</math>
===Derivation===
The quickest and easiest way to derive Schrodinger's equation is to understand the Hamiltonian ''operator'' in quantum mechanics. In classical mechanics, the total energy of a system is given by
<math>
E = \frac{p^2}{2m} + V(x)</math>
where p is the momentum of the particle and V(x) is its potential energy. Applying the quantum mechanical ''[[operator]]'' for momentum:
<math>
p = \frac{\hbar}{i}\frac{\partial}{\partial x} </math>
and subbing into the classical mechanical form for energy, we get the same Hamiltonian operator in quantum mechanics:
<math>
\hat H = \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x) </math>
from which Schrodinger's equation and the eigenvalue problem <math> \hat H\Psi = E\Psi</math> can be easily seen.
===Eigenvalue problems===
===Eigenvalue problems===