Question

Explain briefly the time independent schrodinger wave equation?

Answer 1

Let we consider the time - independent Schrödinger's equation
e.g., ${\displaystyle i\hbar{\frac{\partial}{\partial{i}}\vert\Psi(\mathbf{r}.t)\rangle={\hat{H}}\vert\Psi(\mathbf{r}.t)\rangle}$
For finding time independent Schrödinger's equation , we have to take the potential in the Hamiltonian is assumed to be time independent . e.g., V = V(x) , we calculate the solution of this equation by using the method of separation of variables . then we get
$\hat{H}\vert\Psi\rangle=E\vert\Psi\rangle$
This equation is known as time independent Schrödinger's equation.

Answer 2

[ This question needs high level of elaboration , because it comes upto M.Sc Level or Even higher then that. ]

Schrodinger wave equation is derived in three category 

(i)Time Independent (ii)For a free particle (iii) Time dependent 

Time Dependent Schrodinger Equation
Time dependent Schrodinger equation is comes through the separation of the time & position variables, {separable Wave function }

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{Refers to the Attachment For fully solved derivation } {Two pages are attached there.}

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Hope it Helps. :-)