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logo1 Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation

Separation of Variables – Legendre Equations

Bernd Schr¨

• der

Bernd Schr¨

• der

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 Variables

• 1. Solution technique for partial differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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).

Bernd Schr¨

• der

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 Variables

• 1. 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 with several ordinary differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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 with several ordinary differential equations.

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

Bernd Schr¨

• der

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 Variables

• 1. 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 with several 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 useful results for the partial differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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 with several 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 useful results for the partial differential equations.

• 6. Some very powerful and deep theorems can be used to

formally justify the approach for many equations involving the Laplace operator.

Bernd Schr¨

• der

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?

Bernd Schr¨

• der

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 really awesome functional analysis.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 the

Laplace operator, which arises, for example, in separation

• f variables for the heat equation or the wave equation.

Bernd Schr¨

• der

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 the

Laplace operator, which arises, for example, in separation

• f variables for the heat equation or the wave equation.
• 2. The time independent Schr¨
• dinger equation

− ¯ h 2m∆φ +Vφ = Eφ describes certain quantum mechanical 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 potential and E is the energy eigenvalue.

Bernd Schr¨

• der

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 the

Laplace operator, which arises, for example, in separation

• f variables for the heat equation or the wave equation.
• 2. The time independent Schr¨
• dinger equation

− ¯ h 2m∆φ +Vφ = Eφ describes certain quantum mechanical 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 potential and E is the energy eigenvalue.

• 3. The equation ∆u = f(ρ)u had already been investigated in

electrodynamics when its importance for the states of an electron in a hydrogen atom became clear.

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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 + 2 ρ ∂u ∂ρ + 1 ρ2 ∂ 2u ∂φ 2 + cos(φ) ρ2 sin(φ) ∂u ∂φ + 1 ρ2 sin2(φ) ∂ 2u ∂θ 2 = f(ρ)u

Bernd Schr¨

• der

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 + 2 ρ ∂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

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP+ 1 ρ2 RTP′′

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP+ 1 ρ2 RTP′′ + cos(φ) ρ2 sin(φ)RTP′

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP+ 1 ρ2 RTP′′ + cos(φ) ρ2 sin(φ)RTP′ + 1 ρ2 sin2(φ)RT′′P

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP+ 1 ρ2 RTP′′ + cos(φ) ρ2 sin(φ)RTP′ + 1 ρ2 sin2(φ)RT′′P = f(ρ)RTP

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ R′TP+ 1 ρ2 RTP′′ + cos(φ) ρ2 sin(φ)RTP′ + 1 ρ2 sin2(φ)RT′′P = f(ρ)RTP ρ2 R′′ R

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ)

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side:

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side: P′′ P + cos(φ) sin(φ) P′ P + 1 sin2(φ) T′′ T = ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R ,

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side: P′′ P + cos(φ) sin(φ) P′ P + 1 sin2(φ) T′′ T = ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R , Both sides must be constant.

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side: P′′ P + cos(φ) sin(φ) P′ P + 1 sin2(φ) T′′ T = ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R , Both sides must be constant. ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R = −λ, or

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side: P′′ P + cos(φ) sin(φ) P′ P + 1 sin2(φ) T′′ T = ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R , Both sides must be constant. ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R = −λ, or ρ2R′′ +2ρR′ −

• λR+ρ2f(ρ)
• R = 0.

Bernd Schr¨

• der

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 + 2 ρ ∂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+ 2 ρ 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 + 1 sin2(φ) T′′ T = ρ2f(ρ) Bring all terms that depend on ρ to the right side: P′′ P + cos(φ) sin(φ) P′ P + 1 sin2(φ) T′′ T = ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R , Both sides must be constant. ρ2f(ρ)−ρ2 R′′ R −2ρ R′ R = −λ, or ρ2R′′ +2ρR′ −

• λR+ρ2f(ρ)
• R = 0. (QM: Laguerre polys.)

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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 + 1 sin2(φ) T′′ T = −λ

Bernd Schr¨

• der

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 + 1 sin2(φ) T′′ T = −λ sin2(φ)P′′ P +sin(φ)cos(φ)P′ P + T′′ T = −λ sin2(φ)

Bernd Schr¨

• der

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 + 1 sin2(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

Bernd Schr¨

• der

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 + 1 sin2(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.

Bernd Schr¨

• der

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 + 1 sin2(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.

Bernd Schr¨

• der

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 + 1 sin2(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.

Bernd Schr¨

• der

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 + 1 sin2(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 a nonnegative integer.

Bernd Schr¨

• der

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 + 1 sin2(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 a nonnegative integer. So the function T must be of the form T(θ) = c1 cos(mθ)+c2 sin(mθ).

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 =

Bernd Schr¨

• der

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 = P′′ + cos(φ) sin(φ) P′ +

• λ −

m2 sin2(φ)

• P

=

Bernd Schr¨

• der

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 = P′′ + cos(φ) sin(φ) P′ +

• λ −

m2 sin2(φ)

• P

= This equation is complicated, because it involves trigonometric functions.

Bernd Schr¨

• der

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 = P′′ + cos(φ) sin(φ) P′ +

• λ −

m2 sin2(φ)

• P

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

d dφ P

Bernd Schr¨

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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

d dφ P = d dzP d dφ z

• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• d2

dφ2P

Bernd Schr¨

• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• d2

dφ2P = d dφ d dzP

• −sin(φ)
• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• d2

dφ2P = d dφ d dzP

• −sin(φ)
• =

d dφ d dzP

• −sin(φ)
• +

d dzP

• −cos(φ)
• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• d2

dφ2P = d dφ d dzP

• −sin(φ)
• =

d dφ d dzP

• −sin(φ)
• +

d dzP

• −cos(φ)
• =

d dz d dzP

• −sin(φ)
• −sin(φ)
• +

d dzP

• −cos(φ)
• Bernd Schr¨
• der

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

d dφ P = d dzP d dφ z

• =

d dzP d dφ cos(φ)

• =

d dzP

• −sin(φ)
• d2

dφ2P = d dφ d dzP

• −sin(φ)
• =

d dφ d dzP

• −sin(φ)
• +

d dzP

• −cos(φ)
• =

d dz d dzP

• −sin(φ)
• −sin(φ)
• +

d dzP

• −cos(φ)
• =

sin2(φ) d2 dz2P−cos(φ) d dzP

• Bernd Schr¨
• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• Bernd Schr¨
• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

sin2(φ)d2P dz2 −2cos(φ)dP dz +

• λ −

m2 sin2(φ)

• P = 0

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

sin2(φ)d2P dz2 −2cos(φ)dP dz +

• λ −

m2 sin2(φ)

• P = 0
• 1−z2 d2P

dz2

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

sin2(φ)d2P dz2 −2cos(φ)dP dz +

• λ −

m2 sin2(φ)

• P = 0
• 1−z2 d2P

dz2 −2zdP dz

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

sin2(φ)d2P dz2 −2cos(φ)dP dz +

• λ −

m2 sin2(φ)

• P = 0
• 1−z2 d2P

dz2 −2zdP dz +

• λ −

m2 1−z2

• P

Bernd Schr¨

• der

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)d2P dz2 −cos(φ)dP dz +cos(φ) sin(φ) dP dz

• −sin(φ)
• +
• λ −

m2 sin2(φ)

• P = 0

sin2(φ)d2P dz2 −2cos(φ)dP dz +

• λ −

m2 sin2(φ)

• P = 0
• 1−z2 d2P

dz2 −2zdP dz +

• λ −

m2 1−z2

• P = 0

Bernd Schr¨

• der

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

Let λ be a real number and let m be a nonnegative integer. The differential equation

• 1−x2

y′′ −2xy′ +

• λ −

m2 1−x2

• y = 0

is called the generalized Legendre equation.

Bernd Schr¨

• der

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

Let λ be a real number and let m be a nonnegative integer. The differential 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.

Bernd Schr¨

• der

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

Let λ be a real number and let m be a nonnegative integer. The differential 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.

Bernd Schr¨

• der

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

Let λ be a real number and let m be a nonnegative integer. The differential 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 the equation for T(θ) in the separation of variables.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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 a nonnegative integer, because of the following:

Bernd Schr¨

• der

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 a nonnegative integer, because of the following:

• 1. For λ not of this form the solutions go to infinity as z

approaches ±1.

Bernd Schr¨

• der

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 a nonnegative integer, because of the following:

• 1. For λ not of this form the solutions go to infinity as z

approaches ±1.

• 2. z approaching ±1 corresponds to cos(φ) approaching ±1,

which corresponds to φ approaching 0 and π.

Bernd Schr¨

• der

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 a nonnegative integer, because of the following:

• 1. For λ not of this form the solutions go to infinity as z

approaches ±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 = l(l+1), the

function u would be infinite on the z-axis, which is not sensible.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations