logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables – LegendreEquations

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.

2. If the unknown function u depends on variables ρ,θ ,φ , weassume there is a solution of the form u = R(ρ)T(θ)P(φ).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables ρ,θ ,φ , we

assume there is a solution of the form u = R(ρ)T(θ)P(φ).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables ρ,θ ,φ , we

assume there is a solution of the form u = R(ρ)T(θ)P(φ).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables ρ,θ ,φ , we

assume there is a solution of the form u = R(ρ)T(θ)P(φ).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.

5. Solutions of the ordinary differential equations we obtainmust typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables ρ,θ ,φ , we

assume there is a solution of the form u = R(ρ)T(θ)P(φ).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables ρ,θ ,φ , we

assume there is a solution of the form u = R(ρ)T(θ)P(φ).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If f (ρ) = g(θ ,φ), then f and g must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u

1. For constant f , this is an eigenvalue equation for theLaplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.

2. The time independent Schrodinger equation

− h2m

∆φ +Vφ = Eφ describes certain quantummechanical systems, for example, the electron in a

hydrogen atom. m is the mass of the electron, h =h

2π,

where h is Planck’s constant, V(ρ) is the electric potentialand E is the energy eigenvalue.

3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the

Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.

2. The time independent Schrodinger equation

− h2m

∆φ +Vφ = Eφ describes certain quantummechanical systems, for example, the electron in a

hydrogen atom. m is the mass of the electron, h =h

2π,

where h is Planck’s constant, V(ρ) is the electric potentialand E is the energy eigenvalue.

3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the

Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.

2. The time independent Schrodinger equation

− h2m

∆φ +Vφ = Eφ describes certain quantummechanical systems, for example, the electron in a

hydrogen atom. m is the mass of the electron, h =h

2π,

where h is Planck’s constant, V(ρ) is the electric potentialand E is the energy eigenvalue.

3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

The Equation ∆u = f (ρ)u1. For constant f , this is an eigenvalue equation for the

Laplace operator, which arises, for example, in separationof variables for the heat equation or the wave equation.

2. The time independent Schrodinger equation

− h2m

∆φ +Vφ = Eφ describes certain quantummechanical systems, for example, the electron in a

hydrogen atom. m is the mass of the electron, h =h

2π,

where h is Planck’s constant, V(ρ) is the electric potentialand E is the energy eigenvalue.

3. The equation ∆u = f (ρ)u had already been investigated inelectrodynamics when its importance for the states of anelectron in a hydrogen atom became clear.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)

∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP

+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP

+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′

+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′

+1

ρ2 sin2(φ)RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P

= f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R

+2ρR′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R

+P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P

+cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P

+1

sin2(φ)T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T

= ρ2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0.

(QM: Laguerre polys.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Radial Part)∂ 2u∂ρ2 +

2ρ

∂u∂ρ

+1

ρ2∂ 2u∂φ 2 +

cos(φ)ρ2 sin(φ)

∂u∂φ

+1

ρ2 sin2(φ)∂ 2u∂θ 2 = f (ρ)u

R′′TP+2ρ

R′TP+1

ρ2 RTP′′+cos(φ)

ρ2 sin(φ)RTP′+

1ρ2 sin2(φ)

RT ′′P = f (ρ)RTP

ρ2 R′′

R+2ρ

R′

R+

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)

Bring all terms that depend on ρ to the right side:

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= ρ

2f (ρ)−ρ2 R′′

R−2ρ

R′

R,

Both sides must be constant.

ρ2f (ρ)−ρ

2 R′′

R−2ρ

R′

R=−λ , or

ρ2R′′+2ρR′−

(λR+ρ

2f (ρ))

R = 0. (QM: Laguerre polys.)Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

T

Both sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic.

Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.

So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Azimuthal Part)

P′′

P+

cos(φ)sin(φ)

P′

P+

1sin2(φ)

T ′′

T= −λ

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+

T ′′

T= −λ sin2(φ)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = −T ′′

TBoth sides must be constant.

−T ′′

T= c leads to T ′′+ cT = 0.

But T must be 2π-periodic. Thus c = m2, where m is anonnegative integer.So the function T must be of the formT(θ) = c1 cos(mθ)+ c2 sin(mθ).

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.

It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separating the Equation ∆u = f (ρ)u (Polar Part)

sin2(φ)P′′

P+ sin(φ)cos(φ)

P′

P+λ sin2(φ) = m2

sin2(φ)P′′+ sin(φ)cos(φ)P′+(

λ sin2(φ)−m2)

P = 0

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

This equation is complicated, because it involves trigonometricfunctions.It turns out that the substitution z = cos(φ) will simplify theequation.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitution

ddφ

P =(

ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP

=(

ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P

=d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Derivatives for the Substitutiond

dφP =

(ddz

P)(

ddφ

z)

=(

ddz

P)(

ddφ

cos(φ))

=(

ddz

P)(

− sin(φ))

d2

dφ 2 P =d

dφ

((ddz

P)(

− sin(φ)))

=d

dφ

(ddz

P)(

− sin(φ))+

(ddz

P)(

− cos(φ))

=[

ddz

(ddz

P)(

− sin(φ))](

− sin(φ))+

(ddz

P)(

− cos(φ))

= sin2(φ)d2

dz2 P− cos(φ)(

ddz

P)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)

+(

λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0

(1− z2

) d2Pdz2 −2z

dPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2

−2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P

= 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Generalized Legendre Equation

P′′+cos(φ)sin(φ)

P′+(

λ − m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −cos(φ)

dPdz

+cos(φ)sin(φ)

dPdz

(− sin(φ)

)+

(λ− m2

sin2(φ)

)P = 0

sin2(φ)d2Pdz2 −2cos(φ)

dPdz

+(

λ − m2

sin2(φ)

)P = 0(

1− z2) d2P

dz2 −2zdPdz

+(

λ − m2

1− z2

)P = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(

1− x2)

y′′−2xy′+(

λ − m2

1− x2

)y = 0

is called the generalized Legendre equation.

For nonnegative integers l, the differential equation(1− x2

)y′′−2xy′+ l(l+1)y = 0

is called the Legendre equation.Formally, both are actually families of differential equations,because m,λ and l are parameters.m is a nonnegative integer, because this is required through theequation for T(θ) in the separation of variables.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(

1− x2)

y′′−2xy′+(

λ − m2

1− x2

)y = 0

is called the generalized Legendre equation.For nonnegative integers l, the differential equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0

is called the Legendre equation.

Formally, both are actually families of differential equations,because m,λ and l are parameters.m is a nonnegative integer, because this is required through theequation for T(θ) in the separation of variables.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(

1− x2)

y′′−2xy′+(

λ − m2

1− x2

)y = 0

is called the generalized Legendre equation.For nonnegative integers l, the differential equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0

is called the Legendre equation.Formally, both are actually families of differential equations,because m,λ and l are parameters.

m is a nonnegative integer, because this is required through theequation for T(θ) in the separation of variables.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsLet λ be a real number and let m be a nonnegative integer. Thedifferential equation(

1− x2)

y′′−2xy′+(

λ − m2

1− x2

)y = 0

is called the generalized Legendre equation.For nonnegative integers l, the differential equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0

is called the Legendre equation.Formally, both are actually families of differential equations,because m,λ and l are parameters.m is a nonnegative integer, because this is required through theequation for T(θ) in the separation of variables.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre Equations

In the Legendre equation(1− x2

)y′′−2xy′+ l(l+1)y = 0,

the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:

1. For λ not of this form the solutions go to infinity as zapproaches ±1.

2. z approaching ±1 corresponds to cos(φ) approaching ±1,which corresponds to φ approaching 0 and π .

3. So, physically this would mean that for λ 6= l(l+1), thefunction u would be infinite on the z-axis, which is notsensible.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsIn the Legendre equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0,

the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:

1. For λ not of this form the solutions go to infinity as zapproaches ±1.

2. z approaching ±1 corresponds to cos(φ) approaching ±1,which corresponds to φ approaching 0 and π .

3. So, physically this would mean that for λ 6= l(l+1), thefunction u would be infinite on the z-axis, which is notsensible.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsIn the Legendre equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0,

the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:

1. For λ not of this form the solutions go to infinity as zapproaches ±1.

2. z approaching ±1 corresponds to cos(φ) approaching ±1,which corresponds to φ approaching 0 and π .

3. So, physically this would mean that for λ 6= l(l+1), thefunction u would be infinite on the z-axis, which is notsensible.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsIn the Legendre equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0,

the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:

1. For λ not of this form the solutions go to infinity as zapproaches ±1.

2. z approaching ±1 corresponds to cos(φ) approaching ±1,which corresponds to φ approaching 0 and π .

3. So, physically this would mean that for λ 6= l(l+1), thefunction u would be infinite on the z-axis, which is notsensible.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations

logo1

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Legendre EquationsIn the Legendre equation(

1− x2)

y′′−2xy′+ l(l+1)y = 0,

the parameter λ should be of the form l(l+1) with l anonnegative integer, because of the following:

1. For λ not of this form the solutions go to infinity as zapproaches ±1.

2. z approaching ±1 corresponds to cos(φ) approaching ±1,which corresponds to φ approaching 0 and π .

3. So, physically this would mean that for λ 6= l(l+1), thefunction u would be infinite on the z-axis, which is notsensible.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Legendre Equations