Question

write the Schrodinger time independent equation ?
with mathmetical prove? ​

Answer 1

To support uncertainty principle E. Schrodinger utilized de Brogue's concept of matter wave and incorporated the wavelength expression into general classical wave equation.By this hi develop a wave equation for a moving particle which is known as Schrodinger's wave equation.At different point in space around nucleus.

on considering the oscillation along the axis x-axis the equation of standing wave is

$\psi = a \sin( \frac{2\pi \: x }{ \ \lambda })$

Now

differentiating with respect to X we get

$\frac{ d\psi}{dx} = \frac{2\pi}{ \lambda} a \cos( \frac{2\pi \: x}{ \lambda} )$

Again taking the second derivative

$\frac{ {d }^{2} \psi }{d {x}^{2} } = \frac{2\pi}{ \lambda} a \sin( \frac{2\pi \: x}{ \lambda} \times \frac{2\pi}{ \lambda} ) \\ \frac{ {d }^{2} \psi }{d {x}^{2} } = \frac{4 {\pi}^{2} }{ \lambda} \psi$

Extending the relation to three dimension we get

$\frac{ {d }^{2} \psi }{d {x}^{2} } + \frac{ {d }^{2} \psi }{d {y}^{2} } + \frac{ {d }^{2} \psi }{d { z }^{2} } = - \frac{4 {\pi}^{2} }{ { \lambda}^{2} }$

on putting lambda = h/mv we get

${ \delta}^{2} \psi \: + \frac{8 {\pi}^{2}m}{ h}(E - V ) \psi = 0$

:-)