Why are there so many Schrödinger equations? Why are there so many Schrödinger equations? I have been looking through many pages about the Schrödinger equation, yet every page I seem to find has a different variation of it. Many are similar, But I don't know what to trust. Are they all just simplified versions of each other? Or are they all different? What is the original equation?


*

*Here's a video which begins, "finally we've arrived at the Schrödinger equation" and displays
$$
i\hbar \frac{\partial\Psi}{\partial t} = \frac{-\hbar^2}{2m}\frac{\partial^2\Psi}{\partial t^2} + V(x)\Psi
$$

*Here's another from in an image:


*A third version from another image:
$$
i\hbar\frac{\partial}{\partial t}\Psi = \hat H\Psi
$$
 A: At heart, there is only one Schrödinger equation, and it reads
$$
i\hbar\frac{\partial \Psi}{\partial t}= \hat{H}\Psi.
\tag1
$$
Here the crucial part is $\hat H$: it is called the hamiltonian and it describes the energy of the system. In general it is a linear operator and it acts on $\Psi$, and it can include things like derivatives and so on. Any equation of the form $(1)$ can rightfully be called a Schrödinger equation (modulo a technical concern - you normally require $\hat H$ to have a technical property called self-adjointness).
The thing with this form, though, is that it applies to many different possible systems, and because of this the actual form of the hamiltonian can take many different shapes. As such, you can write the Schrödinger equation for, say, the internal states of a decaying nucleus, or the magnetic interactions between different atoms in a material, or the evolution of light, and those can look very different and have very different properties.
The images you refer to all have the Schrödinger equation in the form
$$
i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V(x)\Psi,
$$
which reflects the dynamics of a single particle of mass $m$ in one dimension under the action of a potential energy $V(x)$. In addition to this, one very often looks for solutions of the form $\Psi=\Psi(x,t) = \psi(x) e^{-iEt/\hbar}$, which reduces the main equation to the subsidiary form
$$
E\psi = -\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2} + V(x)\psi,
$$
which is generally known as the stationary or time-independent Schrödinger equation; this looks structurally different to the time-dependent Schrödinger equation but in essence it describes a special case of the same dynamics.
Finally, it's important to note that in different presentations the specific form of the equations, including things like the constants and symbols in use and such, can vary. These are typically easy to sort through, and if two equations have the same global structure then they are considered to be the same equation. If this kind of thing causes you trouble, then buckle down and use it for practice, because there will be plenty more from where that came from.
