Time independent Schrodinger equation

qpcrib The Schrodinger equation in a single dimension can be reduced to two ordinary differential equations through separation of variables. This yields one equation that is dependent on time, and one that is time independent. It is also known as the Energy Eigenvalue Equation.

$$-\frac{{\hslash }^2}{2m}\frac{\partial \Psi (x,t)}{\partial x^2}+V(x)\Psi (x,t)=i\hslash \frac{\partial \Psi (x,t)}{\partial t}$$

Above is the Schrodinger Equation. It can be solved for a free particle. See (i.e no potential energy, $V(x)$ term is zero).

When $V(x)$ is independent of time, we can separate the Schrodinger equation into

$$\frac{df(t)}{dt}=-\frac{iE}{\hslash }f(t)$$

and

$$-\frac{{\hslash }^2}{2m}\frac{\partial \psi (x,t)}{\partial x^2}+V(x)\psi (x)=E\psi (x)$$

The latter is known as the [[Time independent Schrodinger equation]], while the former can be rather easily solved using an [[ansatz]] of $F(t) =f(0)e^{-iEt/\hbar}$ , or $F(t) =f(0)e^{i\omega}$ where $E = \hbar \omega$ . 

NB:

$$|\Psi (x,t){|}^2=|\Psi (x){|}^2$$

Probability is Independent of time