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Unformatted text preview: The Hydrogen Electron Cloud In spherical coordinates for three-dimensional problems, Schrödinger’s equation becomes ( r Ψ r ) r / r + ( sin θ Ψ θ ) θ / r sin θ + Ψ φφ / r sin θ = -2m( E – U) Ψ / , where every incidence of a subscript r , θ , or φ indicates a partial derivative by that variable. For instance, Ψ φφ means ∂ ² Ψ / ∂ φ ² . 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 U involved here is simply the electric potential energy, -q /4 π ε ₀ r , where q is the charge of the electron. This is a partial differential equation that can be broken down through a process called separation of variables into three component equations based on functions R , Θ , and Φ that are interrelated through constants we will temporarily call k ₁ and...
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This note was uploaded on 04/05/2012 for the course PHYS 131 taught by Professor Tibbets during the Spring '11 term at Cuyamaca College.
- Spring '11