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Geometric amplitude factors in adiabatic quantum transitions

The exponentially small probability of transition between two quantum states, induced by the slow change over infinite time of an analytic hamiltonian Ĥ = H(δt). Ŝ ( where δ is a small adiabatic parameter and Ŝ is the Vector spin -½ operator), contains an additional factor exp{ᴦg} of purely geometric origin (that is, independent of δ and ħ). For ᴦg to be non-zero, Ĥ must be complex hermitian rather than real symmetric, and the hamiltonian curve H(ז) must not lie in a plane through the origin nor be a helix identical (up to rigid motion) with its time reverse. An expression is given for ᴦg, as an integral from the real t axis to the complex time of degeneracy of the two states. Explicit examples are given. The geometric effect could be observed in experiments with spinning particles.