Schrodinger wave equation

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## 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 r**espect 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’ ,

∂^{2}**Ψ / ∂ x ^{2} = (2π A/ λ )[ – sin 2π x/ λ ] (2π / λ ) **

∂^{2}**Ψ / ∂ x ^{2} = ( – 4π^{2} 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 **

∂^{2}**Ψ / ∂ x ^{2} = ( – 4π^{2} / λ^{2})[ Asin 2π x/ λ ] **

∂^{2}**Ψ / ∂ x ^{2} = ( – 4π^{2}Ψ/ λ^{2}) ———— 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 ^{2 }= m^{2}u^{2} / h^{2} ——- eq 5**

putting the value of ‘ 1/**λ ^{2} ‘ from eq (5) to eq (4)**

∂^{2}**Ψ / ∂ x ^{2} = ( – 4π^{2}Ψm^{2}u^{2} / h^{2}) ——–eq (6) **

Kinetic energy =** mu ^{2}**/2

Total energy E = potential energy + Kinetic energy

E = V + ** mu ^{2}**/2

**mu ^{2} = 2 (E-V) ———–eq 7**

putting the value of ‘ **mu ^{2} ‘ from eq (7) to eq (6) **

∂^{2}**Ψ / ∂ x ^{2} = ( – 4π^{2} m. 2 (E-V) Ψ/ h^{2}) = – 8π^{2}m (E-V)Ψ / h^{2}**

∂^{2}**Ψ / ∂ x ^{2} = – 8π^{2}m (E-V)Ψ / h^{2} ————- 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,

### ∂^{2}**Ψ / ∂ x**^{2} + ∂^{2}Ψ / ∂ y^{2} + ∂^{2}Ψ / ∂ z^{2} = – 8π^{2}m (E-V)Ψ / h^{2}

^{2}+ ∂

^{2}Ψ / ∂ y

^{2}+ ∂

^{2}Ψ / ∂ z

^{2}= – 8π

^{2}m (E-V)Ψ / h

^{2}

### ∂^{2}**Ψ / ∂ x**^{2} + ∂^{2}Ψ / ∂ y^{2} + ∂^{2}Ψ / ∂ z^{2} + 8π^{2}m (E-V)Ψ / h^{2} = 0 ————- eq 9

^{2}+ ∂

^{2}Ψ / ∂ y

^{2}+ ∂

^{2}Ψ / ∂ z

^{2}+ 8π

^{2}m (E-V)Ψ / h

^{2}= 0 ————- eq 9

This is Schrodinger wave equation.

Eq (9) may also be written as,

**∇ **^{2}Ψ + 8π^{2}m (E-V)Ψ / h^{2} = 0 ————- eq 10

^{2}Ψ + 8π

^{2}m (E-V)Ψ / h

^{2}= 0 ————- eq 10

This is also a form of Schrodinger wave equation.

**∇ ^{2}** is Laplacian operator

∂^{2}**/ ∂ x ^{2} + ∂^{2} / ∂ y^{2} + ∂^{2} / ∂ z^{2} = ∇ ^{2}**

**-∇ ^{2}Ψ = 8π^{2}m (E-V)Ψ / h^{2} **

**-∇ ^{2}Ψ h^{2} / 8π^{2}m = (E-V)Ψ **

**-∇ ^{2}Ψ h^{2} / 8π^{2}m = EΨ – V Ψ**

**-∇ ^{2}Ψ h^{2} / 8π^{2}m + V Ψ = **

**EΨ**

**[( -∇ **^{2} h^{2} / 8π^{2}m)+ V ] Ψ = EΨ ———eq 10

^{2}h

^{2}/ 8π

^{2}m)+ V ] Ψ = EΨ ———eq 10

This is also a form of Schrodinger wave equation.

**HΨ ** = **EΨ **

**[( -∇ ^{2} h^{2} / 8π^{2}m)+ V ] Ψ = HΨ **

**H = [ V – (∇ **^{2} h^{2} / 8π^{2}m) ———eq 11

^{2}h

^{2}/ 8π

^{2}m) ———eq 11

H = Hamiltonian operator , E = Eigen value

** Ψ **(wave function) has no physical significance, **Ψ only represents the amplitude of the electron wave. Ψ ^{2} represents the probability of locating an electron associated with a specific energy. **