Final answer:
To set up the Lagrangian function in spherical coordinates (r, θ, φ), one must use the conversion from Cartesian coordinates and understand that the associated wave function Ψ can be expressed as a product of three functions of r, θ, and φ, labelled by quantum numbers.
Step-by-step explanation:
To set up the Lagrangian function in spherical coordinates (r, θ, φ), it is essential to understand the mapping of these coordinates from the Cartesian system. In spherical coordinates, r represents the radius vector from the origin to the point, θ (theta) is the polar angle measured from the positive z-axis, and φ (phi) is the azimuthal angle in the x-y plane from the positive x-axis.
The relationship between spherical and rectangular coordinates is represented as: x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ). This system is helpful especially in cases like finding solutions to Schrödinger's wave equation with a spherically symmetric potential U(r). In such scenarios, the solution to the equation can be expressed as a product of three functions: Ψ(r, θ, φ) = R(r)Y(θ)Z(φ), where R, Y, and Z are functions of r, θ, and φ respectively. These are labelled by quantum numbers n, l, and m.