# Schrödinger equation

The Schrödinger equation is the equation that gives the time evolution of the wavefunction in quantum mechanics. It is written as ${\displaystyle \hat H \Psi = i \frac{\partial}{\partial t} \Psi}$, where H is the Hamiltonian operator and ψ is the wavefunction.
For a single nonrelativistic particle, the Hamiltonian in the position basis has the form ${\displaystyle \hat H = \frac{-\hbar^2}{2m} \nabla^2 + V(\underline r)}$, so the time-dependent Schrödinger equation is ${\displaystyle \left[ \frac{-\hbar^2}{2m} \nabla^2 + V(\underline r) \right] \Psi = i \frac{\partial}{\partial t} \Psi}$.
Writing the wavefunction like this allows the partial differentials to be turned into total differentials. Writing ${\displaystyle \Psi = \psi (\underline r) \eta (t)}$, ${\displaystyle \left[ \frac{-\hbar^2}{2m} \nabla^2 + V(\underline r) \right] \psi \eta = i \frac{\partial}{\partial t} \psi \eta}$, so ${\displaystyle \frac{1}{\psi} \left[ \frac{-\hbar^2}{2m} \nabla^2 \psi + V(\underline r) \psi \right] = i \frac{1}{\eta} \frac{\text{d} \eta}{\text{t} t} }$.