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According to Schrödinger, Atomic particles behave like waves. Then the equation of wave motion could be applied to them. Schrödinger combined two relations:
(a) The classical time-independent wave equation to describe the particle wave. $$\frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -\frac { 4{ \pi }^{ 2 } }{ { \lambda }^{ 2 } } \Psi \qquad ...(1) $$
(b) The wave property of matter as represented by de Broglie equation. $$ \lambda=\frac { h }{ mv } \qquad ...(2)$$
Lets put the value of \(\lambda\) in equation (1) ,$$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -\frac { 4{ \pi }^{ 2 } }{ { \left( \frac { h }{ m\nu } \right) }^{ 2 } } \Psi $$ $$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -4{ \pi }^{ 2 }\frac { { \left( m\nu \right) }^{ 2 } }{ { h }^{ 2 } } \Psi $$ $$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -\left( \frac { 4{ \pi }^{ 2 }m }{ { h }^{ 2 } } \right) { m\nu }^{ 2 }\Psi \qquad ... (3) $$ But, Kinetic Energy = Total Energy - Potential Energy $$ \frac { 1 }{ 2 } { m\nu }^{ 2 } = \left( E - V \right) $$ $$ \therefore { m\nu }^{ 2 } = 2\left( E - V \right) $$ Lets substitute in equation (3) $$ \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -\left( \frac { 4{ \pi }^{ 2 }m }{ { h }^{ 2 } } \right) 2\left( E - V \right) \Psi $$ $$ \boxed { \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } = -\frac { 8{ \pi }^{ 2 }m\left( E - V \right) }{ { h }^{ 2 } } \Psi } \qquad ... (4) $$ This equation is applicable for the particle of mass \( m \) and moving along x axis in one direction. But for three dimensional motion, the Schrödinger equation will be partial differential equation with variables x, y and z. $$ \boxed { \frac { { \partial }^{ 2 }\Psi }{ \partial { x }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { y }^{ 2 } } +\frac { { \partial }^{ 2 }\Psi }{ \partial { z }^{ 2 } } = -\frac { 8{ \pi }^{ 2 }m\left( E - V \right) }{ { h }^{ 2 } } \Psi } $$ This equation is known as Schrödinger equation.
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