Schrodinger wave equation source : transtutors.com

## Schrodinger wave equation-

In quantum mechanics , the  Schrodinger wave equation is a mathematical equation that describes the changes over time of a physical system in which quantum effects such as wave particle duality are significant. These systems are called quantum systems . This equation is a central result in the study of quantum systems and its derivation was a landmark in the development of theory of quantum mechanics.

#### It was named after ERWIN SCHRODINGER.

Let us consider the vibration of a stretched string . The equation for such a wave motion is represented as,

Ψ = A sin 2π x/ λ          ————eq 1

Ψ = wave function  ;  x = displacement ;  λ = wave length ; A = amplitude of the wave

Differentiating the eq (1) with respect to ‘x’ ,

Ψ / ∂ x =A [ cos 2π x/ λ ](2π / λ )

Ψ / ∂ x = (2π A/ λ )[ cos 2π x/ λ ]          ——— eq 2

Again differentiating the eq (2) with respect to ‘x’ ,

2Ψ / ∂ x2 = (2π A/ λ )[ – sin 2π x/ λ ] (2π / λ )

2Ψ / ∂ x2 = ( – 4π2 A/ λ 2)[  sin 2π x/ λ ]        ——— eq 3

putting the value of ‘ Ψ’ from eq (1) to eq (3)

Ψ = A sin 2π x/ λ

2Ψ / ∂ x2 = ( – 4π2 / λ2)[  Asin 2π x/ λ ]

2Ψ / ∂ x2 = ( – 4π2Ψ/ λ2)         ———— eq 4

This equation is applicable to all particles of  waves like electrons , protons.

According to De Broglie equation,

λ = h / mu

1/λ = mu / h

1/λ= m2u2 / h2              ——- eq 5

putting the value of  ‘ 1/λ2 ‘ from eq (5) to eq (4)

2Ψ / ∂ x2 = ( – 4π2Ψm2u2 / h2)       ——–eq (6)

Kinetic energy = mu2/2

Total energy E = potential energy + Kinetic energy

E = V +  mu2/2

mu2 = 2 (E-V)       ———–eq 7

putting the value of  ‘ mu2 ‘ from  eq (7) to eq (6)

2Ψ / ∂ x2 = ( – 4π2 m. 2 (E-V) Ψ/ h2)       =  – 8π2m (E-V)Ψ / h2

2Ψ / ∂ x2     =  – 8π2m (E-V)Ψ / h2             ————- eq 8

This is the wave equation for the particle moving along the x- axis. Eq (8) may be extended in three directions x , y , z. Hence,

### ∂2Ψ / ∂ x2 + ∂2Ψ / ∂ y2 + ∂2Ψ / ∂ z2   + 8π2m (E-V)Ψ / h2  = 0            ————- eq 9

This is Schrodinger wave equation.

Eq (9) may also be written as,

### ∇ 2Ψ   + 8π2m (E-V)Ψ / h2  = 0            ————- eq 10

This is  also a form of Schrodinger wave equation.

2 is Laplacian operator

2/ ∂ x2 + ∂2 / ∂ y2 + ∂2 / ∂ z2  = ∇ 2

-∇ 2Ψ   = 8π2m (E-V)Ψ / h2

-∇ 2Ψ h2 / 8π2m = (E-V)Ψ

-∇ 2Ψ h2 / 8π2m = EΨ  – V Ψ

-∇ 2Ψ h2 / 8π2m + V Ψ =

### [( -∇ 2 h2 / 8π2m)+ V ] Ψ  = EΨ     ———eq 10

This is  also a form of Schrodinger wave equation.

HΨ  = EΨ

[( -∇ 2 h2 / 8π2m)+ V ] Ψ  = HΨ

### H = [ V – (∇ 2 h2 / 8π2m)              ———eq 11

H =  Hamiltonian operator , E = Eigen value

Ψ (wave function) has no physical significance, Ψ  only represents the amplitude of the electron wave. Ψ2 represents the probability of locating an electron associated with a specific energy.