In non-relativistic Quantum Physics are there equations other than the Schrödinger equations that can be used to model wave functions?

Question

The general form for the time-dependent Schrödinger equations is $$i\hbar\frac{\partial\Psi}{\partial{t}}=\hat{H}\Psi$$ with $\hat{H}$ being the Hamiltonian operator, which as I understand it describes the energy of the system, and I know that $$\frac{\partial\Psi}{\partial{t}}$$ is the partial derivative of the wave function with respect to time. The general form of the time-independent Schrödinger equation is $$\hat{H}\Psi=E\Psi$$ with $E$ being the energy eigenvalue of the system.

I haven't been able to find any equations other than the Schrödinger equations that can be used for modeling wave functions, in non-relativistic Quantum Mechanics, that can be used for modeling wave functions of particles, although I'm not sure if this is because I don't know what to look for or if it's because there are no equations other than the Schrodinger Equation that can be used for modeling wave functions in non-relativistic Quantum Mechanics.

My question is are equations other than Schrödingers equations that can be used to model the wave functions of particles, which would use a different set of terms for modeling the wave functions? If so what are some of the equations that can be used to model wave functions of particles other than the Schrödinger equation?

Answer 1

There is no alternative to the Schrödinger equation in its general form as $$ \mathrm{i}\hbar\partial_t \lvert \psi\rangle = H\lvert \psi\rangle.$$ It's not a property of any specific system, but it is an evolution equation for all quantum systems. Just like Hamilton's equations in classical Hamiltonian mechanics, or Newton's $F(x) = m\ddot{x}$, this is the generic evolution equation for all systems the theory is concerned with, no exceptions. Systems that do not obey the Schrödinger equation are not systems described by standard quantum mechanics.

However, like classical mechanics, you can find equivalent formulations to it that may be better suited to the task at hand, but then you usually don't deal with wavefunctions as the evolving entities anymore:

In the Heisenberg picture, the observables and not the states are time-dependent, and the evolution equation for a (not explicitly time-dependent) observable $A$ is $$ -\mathrm{i}\hbar\partial_t A = [H,A].$$

In statistical quantum mechanics, states are not wavefunctions or vectors, but density matrices $\rho$ obeying the von Neumann equation: $$ \mathrm{i}\hbar \partial_t \rho = [H, \rho]$$

There are plenty more formulations one can explore, but none of them is an alternative to the Schrödinger equation - they are either equivalent, or generalizations of it.