By Sunil Bhardwaj

6491 Views

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.

#### Latest News

• Become an Instructor 4 March, 2018

Apply to join the passionate instructors who share their expertise and knowledge with the world. You'll collaborate with some of the industry's best producers, directors, and editors so that your content is presented in the best possible light..

#### More Chapters

• Spectroscopy
• Basic Quantum Chemistry
• Phase Rule
• Electrochemistry
• Colloidal State
• Chemical Thermodynamics
• Gaseous State
• Applied Electrochemistry
• Ionic Equilibria
• Nuclear Chemistry
• Solid State Chemistry
• Chemical Kinetics
• #### Other Subjects

• English
• Applied Physics
• Environmental Studies
• Physical Chemistry
• Analytical Chemistry
• Organic Chemistry
• Soft Skills
• Engineering Drawing
• General Medicine
• Mathematics
• Patente B Italia