Time-Independent Schrodinger Equation for the Hydrogen Atom

1 Time-Independent Schrodinger Equation for the Hydrogen Atom 

Stimulated by the wave theory of de Broglie, but well aware of the pitfalls and limitations of this theory, Erwin Schrodinger set up an equation for the hydrogen atom. The time-independent Schrodinger equation (SE) for a single electron and a nucleus with atomic number Z may be written using SI units and atomic units, respectively  

The functions Y j are the well-known spherical harmonic functions tf is a quantum number called the azimuthal quantum number m (or m ) is the magnetic quantum number and K = if(f + l)/2 is the eigenvalue and if it is inserted in Equation 2.5, we obtain 

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Calculations of the electronic structure of molecules, crystals and surfaces are often performed in atomic units. They are defined by setting the most important constants equal to unity h — eo — me — 1, where me is the electronic mass. The Coulomb law is written in electrostatic units V(r) = q/r, so that the time-independent Schrodinger equation for the hydrogen atom takes on the simple form ... 

Construct the time-independent Schrodinger equation for the hydrogen atom in terms of the coordinates and T2 of the electron and nucleus respectively. Introducing the coordinates of the centre of mass, R, and of the electron relative to the centre of mass, r, show that the Schrodinger equation may be separated into ... 

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