1 L10 – QM in 3 dimensions. 2 V(r): separation of variables.

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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 -

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The angular equation -

Pl are the Legendre polynomials, defined by the Rodrigues formula:

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Spherical harmonics

= (-1)m for m>=0 and =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 si puo’ scrivere

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