Separation of variable

Separation of variable

Two dimensional Laplace’s equation in Cartesian Co-ordinates.

 Let us consider a wave function ψ . Where ψ = ψ ( x , y )

Then the Laplace’s equation can be written as

s ² ψ = 0

Or in other words we can write as

( ∂ ² ψ / ∂ x ² ) + ( ∂ ² ψ / ∂ y ² ) = 0  _ _ _ _ _ _ _  ( 1 )

 As ψ = ψ ( x , y )

Therefore

Ψ( x , y ) = X ( x ) Y ( y )

The equation ( 1 ) can be written as

[ ( ∂ ² / ∂ x ² ) ( X Y ) ] + [ ( ∂ ² / ∂ y ² ) ( X Y ) ] = 0

Y (∂ ² X / ∂ x ² ) + X (∂ ² Y / ∂ y ² ) = 0

( 1 / X ) ( ∂ ² X / ∂ x ² ) + ( 1 / Y ) ( ∂ ² Y / ∂ y ² ) = 0

( 1 / X ) ( ∂ ² X / ∂ x ² )  = - ( 1 / Y ) ( ∂ ² Y / ∂ y ² )

 L. H. S. is depends only on the variable ‘ x ’ and R. H. S. is depends only on variable ‘ y ’. This is true only when  

( 1 / X ) ( ∂ ² X / ∂ x ² )  = - ( 1 / Y ) ( ∂ ² Y / ∂ y ² ) = λ

Therefore

(∂ ² X / ∂ x ² ) = λ X

&

- (∂ ² Y / ∂ y ² ) = λ Y

This can be written as

( ∂ ² X / ∂ x ² ) - λ X = 0 _ _ _ _  ( 2 )

&

( ∂ ² Y / ∂ y ² ) + λ Y = 0 _ _ _ _  ( 3 )

These equations can be solved by ordinary differential equations

 Let us consider equation no (2)

(∂ ² X / ∂ x ² ) - λ X = 0

[ ( ∂ ² / ∂ x ² ) – λ ] X = 0     

D ² – λ = 0

Where D ² =  [ ( ∂ ² / ∂ x ²  )

D = ± ( λ ) ^1/2

Hence the most general form of above equation can be written as

X = C e ^{( λ ) ½ x} + D e ^{- ( λ ) ½ x} _ _ _ _ ( 4 )

Let us consider equation no ( 3 )

( ∂ ² Y / ∂ y ² ) + λ Y = 0

[(∂ ² / ∂ y ² ) + λ ] Y = 0

D ² + λ = 0

Where D ² = [ ( ∂ ² / ∂ y ² ) ]

D² =  - λ

D = ± i ( λ ) ^½

The solution of above equation can be given as

Y = A cos ( λ ) ^½ y + B sin ( λ ) ^½ y _ _ _ _ ( 5 )

Thus the general solution of Laplace’s equation can be written as follows

Ψ ( x , y ) = X ( x ) . Y ( y )   

Where X ( x ) is given by equation ( 4 ) and Y ( y ) is given by equation ( 5 )

Ψ ( x , y ) = {[ C e ^{( λ ) ½ x} + D e ^{- ( λ ) ½ x} ] [A cos ( λ ) ^½ y + B sin ( λ ) ^½ y ]}