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phys documents (dragged) 57 - Chapter 10 Quantum physics 51...

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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 and c (1) nm = φ m | H 1 | φ n E 0 n - E 0 m if m = n . The second-order correction of the energy is then given by: E (2) n = k = n | φ k | H 1 | φ n | 2 E 0 n - E 0 k . So to first order holds: ψ n = φ n + k = n φ k | λ H 1 | φ n 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 = 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 = E (1) n α j . Normalization requires that i | α i | 2 = 1 . 10.14.2 Time-dependent perturbation theory From the Schr¨odinger equation i ¯ h ∂ψ ( t ) t = ( H 0 + λ V ( t )) ψ ( t ) and the expansion ψ ( t ) = n c n ( t ) exp - iE 0 n t ¯ h φ n with c n ( t ) = δ nk + λ c (1) n ( t ) + · · ·
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