Question

I want a solution to the whole matrix, not just the diagonal, in detail until I reach the final solution!

In the discussion of the intermediate Zeeman effect, you learned that both the Zeeman effect and the fine structure effect must be treated on an equal footing. That is, one forms the Hamiltonian matrix of i' = H + Hy then diagonalises it to obtain the perturbed energy states. As an example, we looked into the n = 2 states of the hydrogen atom. To solve this problem, one starts by forming the coupled states li m > from the uncoupled states 1 l m > 18 m, >, then uses them to obtain the Hamiltonian matrix of H'. The resultant states and Hamiltonian matrix are given in Griffith, p. 281 (note that the Hamiltonian matrix in this reference needs to be multiplied by -1). In class, we calculated one or two of the matrix elements. Your task is to calculate all coupled states and all the Hamiltonian matrix elements.

Answer 1

Solutions Green (54-B 0 0 Mahrix W' (ilet) 0 0 0 0 lo -23 0 0 O O O (+& B 0 0 0 0 T 000 oooooo collam ī w imattis (let) 0 0 B 54-48 o ol 1 0 B hos Scanned witho CamScanner 54+5B/

marked four sub-m In above matrix which takes the I have form WE / [A] a here matrx AC / 57-6 0 0 0 .R-17-es ; B=1 lip po 01 Ep 57-4B O 54+ / B/ and OF M::: lo o 7-280 O O O Y+QB ! To o eß o= ro oool Too oo lz Null matrix and 0 To find the solution we find Eigen values of w-mat via. Matix Iden propesties are :- Property 1-: For any diagonal metrix, the diagonal elements are its sigen valus 3.- =169) the the rigen values of P are 1 and a iedal, and a from this property the eigen values of w are the rigen values of matrix A & matmx B Eigen values of Matra A = (57-oo lo 5Y+ß 0 0 lepas melo 07-26 o joo O Y + 2ß! 2 Eigen values of A are A = 54-B, st+ P, X- pf & (+28 (cs by using property & mentioned above]

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This rigenvalues of Matha w' includes Eigen values of Matria A & matrix B together, all regen vales of matre wi di=58-8 d3=8-28 de = 5%+ß N4=Y+qß Ng = 3x- & + 5 4x²+2 x 8 + 22 h A6 = 37- & - J42+2 (B + p d7 = 37+&+ 548²-2 18 + 8 2 2 and HD (38+ £) - 3422=32p Feat

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