Schrodinger equation in plain English A mathematician once said that every equation is a sentence that can be expressed in English.
Can you restate Schrodinger's equation solely in English, perhaps for the position and momentum of an electron in a hydrogen atom, identifying what each term stands for along the way?
Here's an example of what we're after:
$$2+2=4$$
"If you take two of anything, or even just the number 2, and add two of the same, you never have to count them to answer the question how many you have, because whenever you do, the answer will always be 4."
 A: Treating the nonrelativistic case only.
Time-independent Schrodinger equation:
$\hat{H}\psi=E\psi$
With math terms allowed:
"The eigenvalues of the Hamiltonian $(\hat{H})$ are the energies $(E)$ of stationary states $(\psi)$."
Without math terms:
"The function responsible for evolving a quantum system in time $(\hat{H})$ has states $(\psi)$ which remain the same up to a constant scaling factor $(E)$ under its action, and the scaling constant is the energy of the respective state."
Time-dependent Schrodinger equation (for a single particle moving in an electric potential):
$i\hbar\frac{\partial}{\partial t}\psi=(\frac{\hbar^2}{2m}\nabla^2+V)\psi$
With math terms allowed:
"Up to various constants $(i\hbar)$, the wavefunction $(\psi)$ satisfies the diffusion equation $(\frac{\partial \psi}{\partial t}=\frac{\hbar^2}{2m}\nabla^2\psi)$ with an extra term $(+V\psi)$ consisting of the potential energy $(V)$ multiplied by the wavefunction."
Without math terms:
"The rate of change $(\frac{\partial}{\partial t})$ of the wavefunction $(\psi)$ at any point is proportional to a measure of its curvature $(\nabla^2\psi)$ at that point multiplied by a constant $(\frac{\hbar^2}{2m})$, added to the potential $(V)$ applied to the system multiplied by the wavefunction itself."
Time-dependent Schrodinger equation (general):
$i\hbar\frac{\partial}{\partial t}\psi=\hat{H}\psi$
With math terms allowed:
"Up to various constants, $(i\hbar)$, the time derivative of the wavefunction $(\frac{\partial}{\partial t}\psi)$ is given by the action of the Hamiltonian $(\hat{H})$ on the wavefunction."
Without math terms:
"The rate of change of the wavefunction at a point $(\frac{\partial}{\partial t}\psi)$ is proportional to the action of the function responsible for evolving a quantum system in time $(\hat{H})$ on the wavefunction."
