Time independent Schrodinger wave equation:
We know the time dependent Schrodinger wave equation:
$i \hbar \frac{\partial \psi(x,t)}{\partial t}= -\frac{\hbar^{2}}{2m} \frac{\partial^{2} \psi(x,t)}{\partial x^{2}}+ V(x) \psi(x,t) \qquad(1)$
The wave function $\psi(x,t)$ is the product of space function $\psi(x)$ and time function $\psi(t)$. So
$\psi(x,t)=\psi(x) \psi(t) \qquad (2)$
Now apply the wave function form of equation$(2)$ to time-independent Schrodinger wave equation $(1)$
$i \hbar \psi(x) \frac{\partial \psi(t)}{\partial t}= -\frac{\hbar^{2}}{2m} \psi(t) \frac{\partial^{2} \psi(x)}{\partial x^{2}}+ V(x) \psi(x) \psi(t) \qquad(3)$
Now divide the above equation $(3)$ by $\psi(x)\psi(t)$ so
$i \hbar \frac{1}{\psi(t)} \frac{\partial \psi(t)}{\partial t}= -\frac{\hbar^{2}}{2m} \frac{1}{\psi(x)} \frac{\partial^{2} \psi(x)}{\partial x^{2}}+ V(x) \qquad(4)$
The above equation is known as the separation of time-independent part and time-independent part of the wave equation. The time-independent part is known as the energy function operator. i.e
$E=i \hbar \frac{1}{\psi(t)} \frac{\partial \psi(t)}{\partial t} \qquad(5)$
So from equation $(4)$ and equation$(5)$
$E= -\frac{\hbar^{2}}{2m} \frac{1}{\psi(x)} \frac{\partial^{2} \psi(x)}{\partial x^{2}}+ V(x)$
$E \psi(x)= -\frac{\hbar^{2}}{2m} \frac{\partial^{2} \psi(x)}{\partial x^{2}}+ V(x) \psi(x)$
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$\frac{\partial^{2} \psi(x)}{\partial x^{2}}+\frac{2m}{\hbar^{2}}(E-V)\psi(x)=0$
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This is time-independent Schrodinger wave equation.
Now for a free particle i.e, there is no force acting on the particle then the potential energy of a particle will be zero i.e. $V(x)=0$. Therefore time independent Schrodinger equation can be written as:
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$\frac{\partial^{2} \psi(x)}{\partial x^{2}}+\frac{2mE}{\hbar^{2}}\psi(x)=0$
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This is time-independent Schrodinger wave equation for a free particle.
