Making a complex exponent
Making a complex exponential ansatz transforms the Hamilton-Jacobi equation into a variant of this equation, which is a diffusion equation with imaginary diffusion constant. Its electronic form is used in the Born-Oppenheimer approximation and can be in turn solved using the Hartree-Fock method. Solutions to this equation satisfy local conservation of probability and must be normalizable. The Klein-Gordon and Dirac equations are Lorentz-invariant alterations of it. The time-independent form of this equation contains an energy eigenvalue E. For 10 points, name this equation written "i h-bar d psi d t equals H psi," an equation from quantum mechanics whose solutions are wavefunctions.
Making a complex exponential ansatz transforms the Hamilton-Jacobi equation into a variant of this equation, which is a diffusion equation with imaginary diffusion constant.
Schrödinger equation