The Schrödinger equation is the equation that gives the time evolution of the wavefunction in quantum mechanics. It is written as , where H is the Hamiltonian operator and ψ is the wavefunction.
Single Nonrelativistic Particle
For a single nonrelativistic particle, the Hamiltonian in the position basis has the form , so the time-dependent Schrödinger equation is .
This partial differential equation can be solved by separating the variables; the overall wavefunction is a sum of terms that are the product of a time-dependent and a position-dependent function. These terms will turn out to be stationary states; that is, states that have a single well-defined energy.
Writing the wavefunction like this allows the partial differentials to be turned into total differentials. Writing , , so .
The term on the left hand side is dependent only on position, and the term on the right hand side is dependent only on time, so they must both be equal to a constant to be equal. This constant is E, the energy of the stationary state.