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Hydrogen - The Hydrogen Electron Cloud In spherical...

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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 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 and k : ( r R r ) r = – ( 2m E r / + mq r/2
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