Second order differential equations, like the Schrödinger Equation, can be solved by separation of variables.
These separated solutions can then be used to solve the problem in general.
Assume that we can factorize the solution between time and space.
Plug this into the Schrödinger Equation.
Put everything that depends on "x" on the left and everything that depends on "t" on the right.
Since we have a function of only "x" set equal to a function of only "t", they both must equal a constant. In the equation above, we call the constant "E", (with some knowledge of the outcome). We now have an equation in "t" set equal to a constant
which has a simple general solution,
and an equation in "x" set equal to a constant
which depends on the problem to be solved (through "V(x)"
The "x" equation is often called the Time Independent Schrödinger Equation.
Here, "E" is a constant. The full time dependent solution is.
