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04132010a - Quantum Mechanics The Hydrogen Atom 12th April...

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Quantum Mechanics: The Hydrogen Atom 12th April 2010 I. The Hydrogen Atom In this next section, we will tie together the elements of the last several sections to arrive at a complete description of the hydrogen atom. This will culminate in the definition of the hydrogen-atom orbitals and associated en- ergies. From these functions, taken as a complete basis, we will be able to construct approximations to more complex wave functions for more complex molecules. Thus, the work of the last few lectures has fundamentally been aimed at establishing a foundation for more complex problems in terms of exact solutions for smaller, model problems. II. The Radial Function We will start by reiterating the Schrodinger equation in 3D spherical coordi- nates as (refer to any standard text to get the transformation from Cartesian to spherical coordinate reference systems). Here, we have not placed the con- straint of a constant distance separting the masses of the rigid rotor (refer to last lecture); furthermore, we will keep in the formulation the potential V ( r, θ, φ ) for generality. Thus, in spherical polar coordinates, ˆ H ( r, θ, φ ) ψ ( r, θ, φ ) = ( r, θ, φ ) becomes: " - ¯ h 2 2 μ 1 r 2 ∂r r 2 ∂r + 1 r 2 sinθ ∂θ sinθ ∂θ + 1 r 2 sin 2 θ 2 ∂φ 2 ! + V ( r, θ, φ ) # ψ ( r, θ, φ ) (1) = ( r, θ, φ ) (2) Now, for the hydrogen atom, with one electron found in ”orbits” (note the quotes!) around the nucleus of charge +1, we can include an electrostatic potential which is essentially the Coulomb potential between a positive and negative charge: 1
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V ( | r | ) = V ( r ) = - Ze 2 4 π o r where Z is the nuclear charge (i.e, +1 for the nucleus of a hydrogen atom).
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