Schrödinger equation

{c}
{tag=Important partial differential equation}
{wiki}

The <partial differential equation> of <special relativity>[non-relativistic] <quantum mechanics>.

Experiments explained:
* via the <Schrödinger equation solution for the hydrogen atom> it predicts:
  * <spectral line> basic lines, plus <Zeeman effect>
* <Schrödinger equation solution for the helium atom>: perturbative solutions give good approximations to the energy levels
* <double-slit experiment>: I think we have a closed solution for the max and min probabilities on the measurement wall, and they match experiments

Experiments not explained: those that the <Dirac equation> explains like:
* <fine structure>
* <spontaneous emission> coefficients

To get some intuition on the equation on the consequences of the equation, have a look at:
* <Schrödinger equation simulations>
* <solutions of the Schrodinger equation>

The easiest to understand case of the equation which you must have in mind initially that of the <Schrödinger equation for a free one dimensional particle>.

Then, with that in mind, the general form of the <Schrödinger equation> is:
$$
i\hbar\pdv{\psi(\vv{x}, t)}{t} = \hat{H}[\psi(\vv{x}, t)]
$$
{title=Schrodinger equation}
where:
* $\hbar$ is the <reduced Planck constant>
* $\psi$ is the <wave function>
* $t$ is the time
* $\hat{H}$ is a <linear operator> called the <Hamiltonian>. It takes as input a function $\psi$, and returns another function. This plays a role analogous to the Hamiltonian in <classical mechanics>: determining it determines what the physical system looks like, and how the system evolves in time, because we can just plug it into the equation and solve it. It basically encodes the total energy and forces of the system.

The $\vv{x}$ argument of $\psi$ could be anything, e.g.:
* we could have preferred <polar coordinates> instead of linear ones if the potential were symmetric around a point
* we could have more than one particle, e.g. <solutions of the Schrodinger equation for two electrons>, which would have e.g. $x_1$ and $x_2$ for different particles. No matter how many particles there are, we have just a single $\psi$, we just add more arguments to it.
* we could have even more generalized coordinates. This is much in the spirit of <Hamiltonian mechanics> or <generalized coordinates>
Note however that there is always a single magical $t$ time variable. This is needed in particular because there is a time <partial derivative> in the equation, so there must be a corresponding time variable in the function. This makes the equation explicitly <non-relativistic>.

The general <Schrödinger equation> can be broken up into a trivial time-dependent and a <time-independent Schrödinger equation> by separation of variables. So in practice, all we need to solve is the slightly simpler <time-independent Schrödinger equation>, and the full equation comes out as a result.

Existence and uniqueness: https://mathoverflow.net/questions/212913/existence-and-uniqueness-for-two-dimensional-time-dependent-schr%C3%B6dinger-equation
