Overview
The Schrodinger equation is an equation that describes particles at the quantum level. It was first postulated by Erwin Schrodinger in 1925. It describes the dynamics of a wave function {% \Psi(x,t) %} which is used to calculate probabilities of the particle being observed in various states.
Equation
The Schrodinger equation is given as
{% i \hbar \Psi(x,t) = [- \frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + V(x,t)] \Psi(x,t) %}
The function {% \Psi(x,t) %} can have multiple interpretations, but the common intrepretation is that
{% |\Psi(x,t)|^2 dx %} is the probability of finding the particle in the interval {% dx %} at {% x %}.
(see probabilistic interpretation)
Operator Formalism
The Schrodinger Equation can be restated in terms of operators as in the following:
{% i \hbar \Psi(x,t) = \hat{H} \Psi(x,t) %}
where {% \hat{H} %} is the Hamiltonian operator,
{% \hat{H} = - \frac{\hbar ^2}{2m} \frac{\partial ^2}{\partial x^2} + V(x,t) %}
or restated using the momentum operator {% \hat{p} %}
{% \hat{H} = - \frac{\hat{p}^2}{2m} + V(x,t) %}