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 m>=0 and e=1 for m<0. 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>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>1
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n > 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’
, l’equazione in f si puo’ scrivere
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