The physics of 2-state systems

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Outline Symmetry Oscillations Density matrices Keywords and ReferencesResonant stabilizationFor any value of w, including zero,we can label their simultaneouseigenstates by the eigenvalues of 1(which are the degenerate pair 1),and those of σ 0 . In other words, wehave |+〉 = |1,1〉 and |−〉 = |1,−1〉.As a result, the eigenvalues ofH = E 0 +wσ 1 are E ± = E 0 ±w.There is a level crossing, i.e.,degeneracy, at w = 0.w|+>EActually we can always choose w < 0, the lowest energy state is|+〉. The energy of this state is lower than that of |1〉 and |2〉.This is true of of H2 + and benzene. This general feature ofsymmetric states is called resonant stabilization.Sourendu Gupta Quantum Mechanics 1 2013: Lecture 5
Outline Symmetry Oscillations Density matrices Keywords and ReferencesLack of symmetry: avoided crossingsTake an unperturbed HamiltonianH 0 = E 0 +∆σ 3 . The originalenergies are E 1 = E 0 +∆ andE 2 = E 0 −∆. The eigenstates |1〉and |2〉 can be distinguished, since2.01.51.00.5they have different energies. 1.0 0.5 0.5 1.0The permutation generator, S = wσ 1 +w ′ σ 2 , does not commutewith H 0 . If H = H 0 +S, then the energy levels areE ± = E 0 ± √ ∆ 2 +|W| 2 , where W = w +iw ′ . In general theenergy splitting is quadratic in |W|; although, for ∆ ≪ |W|, it isnearly linear, as in the earlier example.For ∆ ≫ |W| it may seem that the energy levels might cross, butany |W|, no matter how small, causes avoided level crossings.Sourendu Gupta Quantum Mechanics 1 2013: Lecture 5
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The physics of 2-state systems

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