Problem 3: Adiabatic Limit

Re-consider the exponential switch perturbation, but now we will focus on extremely small values of the switch rate, a. This will lead us to an important theorem in

Quantum Mechanics called the Adiabatic Theorem.

The Adiabatic Theorem states that a system will stay in its evolving eigenstate provided the Hamiltonian is changed sufficiently slowly. Test this claim using the

parameters of [1c]: x = 0.1 w₁ and a = 0.001 w₁₁. You already have the numerical and perturbative solutions to this transition. Now generate a third plot by directly

calculating the occupation of the lowest eigenstate of H = Ho + V in the basis of Ho. This should allow you to calculate the occupation probability of the excited state

of Ho as a function of time. Plot this probability along with your numerical and perturbative results, and comment on what you find.

Aside: This is not the same notion of "adiabatic" as in thermal systems.

Vaik

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