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Schrödinger equation

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The Schrödinger equation is a partial differential equation whose solution is the wave equation, which describes the probability density of a given particle over space. The general form of the Schrödinger equation is

i \hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat H \Psi(\mathbf{r},t)

where i is the imaginary unit, ħ is the reduced Planck constant, Ψ is the wave function, and Ĥ is the Hamiltonian operator (representing the total energy of the system). In steady-state systems (where the wave equation does not depend on time, such as in an atomic or molecular orbital) this simplifies to

E \Psi = \hat H \Psi

where E is a constant. This is known as the time-independent Schrödinger equation.

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