User:TheScientist

The Schrödinger equation
The Schrödinger equation takes several different forms, depending on the physical situation. This section presents the equation for the general case and for the simple case encountered in many textbooks.

General quantum system
For a general quantum system:


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

where
 * $$i\ $$ is the imaginary unit
 * $$\Psi(\mathbf{r},\,t)$$ is the wave function, which is the probability amplitude for different configurations of the system.
 * $$\scriptstyle \hbar$$ hbar is the Reduced Planck's constant, (Planck's constant divided by $$2\pi$$), and it can be set to a value of 1 when using natural units.
 * $$\scriptstyle \hat H$$ is the Hamiltonian operator.

Single particle in three dimensions
For a single particle in three dimensions:


 * $$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = -\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t)$$

where
 * $$\mathbf{r} = (x,y,z) $$ is the particle's position in three-dimensional space,
 * $$\Psi(\mathbf{r},t)$$ is the wavefunction, which is the amplitude for the particle to have a given position r at any given time t.
 * $$m$$ is the mass of the particle.
 * $$V(\mathbf{r})$$ is the time independent potential energy of the particle at each position r.