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phys documents (dragged) 57 - Chapter 10: Quantum physics...

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Chapter 10: Quantum physics 51 and this is put into the Schr¨odinger equation the result is: E (1) n = ± φ n | H 1 | φ n ² and c (1) nm = ± φ m | H 1 | φ n ² E 0 n - E 0 m if m ³ = n . The second-order correction of the energy is then given by: E (2) n = ± k ± = n | ± φ k | H 1 | φ n ² | 2 E 0 n - E 0 k . So to Frst order holds: ψ n = φ n + ± k ± = n ± φ k | λ H 1 | φ n ² E 0 n - E 0 k φ k . In case the levels are degenerated the above does not hold. In that case an orthonormal set eigenfunctions φ ni is chosen for each level n , so that ± φ mi | φ nj ² = δ mn δ ij . Now ψ is expanded as: ψ n = N ( λ ) ± i α i φ ni + λ ± k ± = n c (1) nk ± i β i φ ki + ··· E ni = E 0 ni + λ E (1) ni is approximated by E 0 ni := E 0 n . Substitution in the Schr¨odinger equation and taking dot product with φ ni gives: i α i ± φ nj | H 1 | φ ni ² = E (1) n α j . Normalization requires that i | α i | 2 =1 . 10.14.2
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