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Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separation of Variables – Legendre Equations Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separation of Variables 1. Solution technique for partial differential equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation How Deep? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation How Deep? plus about 200 pages of really awesome functional analysis. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation The Equation ∆ u = f ( ρ ) u logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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 of variables for the heat equation or the wave equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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 of variables for the heat equation or the wave equation. 2. The time independent Schr¨ odinger equation h − ¯ 2 m ∆ φ + V φ = E φ describes certain quantum mechanical systems, for example, the electron in a h = h hydrogen atom. m is the mass of the electron, ¯ 2 π , where h is Planck’s constant, V ( ρ ) is the electric potential and E is the energy eigenvalue. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

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 of variables for the heat equation or the wave equation. 2. The time independent Schr¨ odinger equation h − ¯ 2 m ∆ φ + V φ = E φ describes certain quantum mechanical systems, for example, the electron in a h = h hydrogen atom. m is the mass of the electron, ¯ 2 π , 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. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separating the Equation ∆ u = f ( ρ ) u (Radial Part) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separating the Equation ∆ u = f ( ρ ) u (Radial Part) ∂ 2 u ∂ 2 u ∂ 2 u ∂ u ∂φ 2 + cos ( φ ) ∂ u ∂ρ 2 + 2 ∂ρ + 1 1 ∂φ + = f ( ρ ) u ρ 2 sin ( φ ) ρ 2 sin 2 ( φ ) ρ 2 ∂θ 2 ρ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separating the Equation ∆ u = f ( ρ ) u (Radial Part) ∂ 2 u ∂ 2 u ∂ 2 u ∂ u ∂φ 2 + cos ( φ ) ∂ u ∂ρ 2 + 2 ∂ρ + 1 1 ∂φ + = f ( ρ ) u ρ 2 sin ( φ ) ρ 2 sin 2 ( φ ) ρ 2 ∂θ 2 ρ R ′′ TP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations

Introduction ∆ in a Spherically Symmetric Geometry Separating Spherical Coordinates Obtaining the Legendre Equation Separating the Equation ∆ u = f ( ρ ) u (Radial Part) ∂ 2 u ∂ 2 u ∂ 2 u ∂ u ∂φ 2 + cos ( φ ) ∂ u ∂ρ 2 + 2 ∂ρ + 1 1 ∂φ + = f ( ρ ) u ρ 2 sin ( φ ) ρ 2 sin 2 ( φ ) ρ 2 ∂θ 2 ρ R ′′ TP + 2 ρ R ′ TP logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Legendre Equations