L10

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L10 QM in 3 dimensions
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V(r) separation of variables
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The angular equation
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The angular equation - f
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The angular equation - q
Pl are the Legendre polynomials, defined by the
Rodrigues formula
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Spherical harmonics
e (-1)m for mgt0 and e1 for mlt0. The Y are
orthogonal, so
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The radial equation
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L11 The H atom
u(r) rR(r)
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Asymptotic behavior
u(r) rR(r)
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The radial solution
u(r) rR(r)
n gt l
is the q-th Laguerre polynomial.
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The radial solution energy
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Ground state
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n gt 1
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n gt 1 (continued)
Since they are eigenvectors for different
eigenvalues
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Example
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Graphs
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Angular momentum
Poiche
Poiche ,
lequazione in f si puo scrivere
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