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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 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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