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Unformatted text preview: ),
which you might have encountered elsewhere. The are of the form
of Legendre polynomials times e±imφ.
(Actually solving the equations is too much math for us.)
Monday, April 1, 2013 The Hydrogen Atom For hydrogen, like the spherical
box, we have spherical
symmetry, but the potential
energy is the Coulomb potential
energy between opposite
U(r) = -e2/4πε0r.
The radial wave function R(r)
must go to 0 at large r, since the
electrons are bound to the
protons, but does not necessarily
go to 0 at the origin. The
angular functions must be ﬁnite
and periodic in φ, with a period
of 2π. The wave functions are
characterized by 3 quantum
numbers ... Monday, April 1, 2013 The Principal Quantum Number
n = 1, 2, 3, ...: The wave function is of the form of a polynomial
series in r (∑i=0,...n-1 ciri) (some ci are 0) times an exponential
exp(-r/na0). Here a0 is the Bohr radius, 0.053 nm. The resulting
energies are the same as in the Bohr model, En = -13.6 eV / n2.
n=2 n=3 examples of
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