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

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Presentation on theme: "1 L10 – QM in 3 dimensions. 2 V(r): separation of variables."— Presentation transcript:

1 1 L10 – QM in 3 dimensions

2 2 V(r): separation of variables

3 3

4 4 The angular equation

5 5 The angular equation -

6 6 P l are the Legendre polynomials, defined by the Rodrigues formula:

7 7

8 8

9 9 Spherical harmonics = (-1) m for m>=0 and =1 for m<0. The Y are orthogonal, so

10 10 The radial equation

11 11 L11 – The H atom u(r) = rR(r)

12 12 Asymptotic behavior u(r) = rR(r)

13 13 The radial solution u(r) = rR(r) n > l is the q-th Laguerre polynomial.

14 14 The radial solution: energy

15 15 Ground state

16 16 n > 1

17 17 n > 1 (continued) Since they are eigenvectors for different eigenvalues

18 18 Example

19 19 Graphs

20 20 Angular momentum Poiche Poiche, lequazione in si puo scrivere

21 21

22 22

23 23


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