Question

Derive Schrodinger’s wave equation for a free particle in free space

Answer 1

The operator for kinetic energy is:

$$p = -i\hbar\dfrac{\partial}{\partial x}$$

and for full energy is:

$$E = i\hbar\dfrac{\partial}{\partial t}$$

Using the analogy with the classical case (where for a free particle $E = \dfrac{p^2}{2m}$) and applying these operators to a wavefunction $\Psi$, obtain the Schrodinger’s wave equation for a free particle:

$$-\dfrac{\hbar^2}{2m}\dfrac{\partial^2 \Psi}{\partial x^2} = i\hbar\dfrac{\partial\Psi}{\partial t}$$

More detailed derivation see hee: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/Schr2.html.