Final answer:
Schrödinger's equation for a hydrogen atom can be broken down into three separate components due to its spherical symmetry. The equation's solution is a product of three functions, each corresponding to a spherical coordinate, collectively describing the electron's wave function and its probability density in space.
Step-by-step explanation:
Schrödinger's Equation for the Hydrogen Atom
The Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. For a hydrogen atom, which exhibits spherical symmetry in its potential energy function U(r), Schrödinger's equation can be simplified into three separate equations for the three spherical coordinates (r, θ, and φ), a form more amenable to finding solutions for the wave function in this context.
Ignoring time-dependence due to the static potential energy function, the wave function Ψ(r, θ, φ) for the hydrogen atom can be expressed as a product of three separate functions, one for each coordinate: R(r)Ψ(θ)Ø(φ), where R is the radial component, Ψ is the polar angle component, and Ø is the azimuthal angle component.
The time-independent wave function for the hydrogen atom is crucial for understanding its physical properties, such as the distribution of the electron density around the nucleus, with the square of the wave function's magnitude giving the probability density of finding an electron at a particular point in space.