The Hydrogen Electron Cloud In spherical coordinates for three-dimensional problems, Schrödinger’s equation becomes ( rΨr)r/ r+ (sinθΨθ)θ/ rsinθ+ Ψφφ/ rsinθ= -2m(E– U)Ψ/ , where every incidence of a subscript r, θ, or φindicates a partial derivative by that variable. For instance, Ψφφmeans ∂²Ψ/∂φ². Obviously, you are not expected to memorize this, or even fully comprehend it, but it gives a sense of how complicated the problem becomes even for a simple spherical hydrogen atom where quantum mechanics is involved. The potential Uinvolved here is simply the electric potential energy, -q/4πε₀r, where qis the charge of the electron. This is a partial differential equationthat can be broken down through a process called separation of variablesinto three component equations based on functions R, Θ, and Φthat are interrelated through constants we will temporarily call k₁and k₂: ( rRr)r= – ( 2mEr/ + mqr/2
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Atomic orbital, Partial differential equation, Hydrogen Electron Cloud