SCHRODINGER'S EQUATION

For the wave and motions of a micro particles in motion, schrodinger's derived a fundamental equation, which is called schrodinger's equation.

                       According to Einstein and plank, the energy is mathematically defined as,

               E= mc²

               E= hʋ

De-broglie combined both the above equation.

                hʋ = mc²

                h c/λ = mc²

               λ= h/mc ............................ (1)

The above the equation is called de-broglie equation.

According to quantum mechanics, the wave associated with the particle is characterised by wave function. The wave equation used to relate, the wavefunction and frequency is,

ψ= ψ₀ sin 2πʋt............................... (2)

Here,

ψ= wave function (or) eigen function

ψ₀ = wave amplitude

υ = Frequency of the waves

t= time (in seconds)

Differentiating equation (2) with respect to time

dψ/dt = ψ₀ cos 2πυt. 2πr

Again differentiating write the time,

d²ψ/dt² = ψ₀ (-sin2πυt). 2πr. 2πr

d²ψ/dt² = − sin2πυt. 4π²υ²

d²ψ/dt² = − 4π²υ²ψ .........................(3)

For above equation explains the wave motion of particles in motion, according to quantum mechanics,

But according to classical mechanics, the wave function motion associated with particle in motion is,

            d²     = c²∇²ψ  .....................(4)

            dt²

Combine (3) and (4)

c²∇²ψ↓            = − 4π²υ²ψ   

Quantum             Classical

mechanics          mechanics

            [ ∴υ = c/λ ]

c²∇²ψ = - 4π²    c²   ψ

                          λ²

-   4π     ψ = ∇² ψ ............................ (5)

     λ²

Here the λ denotes the wavelength of wave motion of particles in motion.

Substituting the value of λ from the de-broglie's equation.

-    4π² m² c²      ψ = ∇² ψ .................. (6)

           h²

For the particles in motion, the sum of the potential energy (v) and the kinetic energy (T) will give total energy of particles (E)

                    E = T + V

     ∴ T = 1/2 mc²

        2T = mc²

X with m

2Tm = m²c² E= T+V

∴ T = E- V

2m (E-V) = m²c²  .........................(7)

Substituting (7) in (6)

  - 4π² 2m (E−V)     ψ = ∇²ψ

          h²

∇²ψ +   8π²m   (E− V) ψ = 0 ........ (8)

              h²

This equation is on of the form of schrodinger's equation.

Here,

∇² = laplacian operator,

[ ∇² =  ∂²  +  ∂²  +  ∂ ² ]

           ∂x²    ∂y²    ∂z²

ψ= Displacement of the particles in motion (or) wave function.

m = mass of the particle 

h = plank's constant

E = total energy of the particle 

V = potential energy of the particle.