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Unformatted text preview: Phys 852, Quantum mechanics II, Spring 2009 Time-Dependent Perturbation Theory Prof. Michael G. Moore, Michigan State University 1 The central problem in time-dependent perturbation theory: In time-independent perturbation theory, the object was to find the new eigenvalues and eigenstates when a system whose states are known is ‘perturbed’ by adding an additional term to the Hamiltonian. The main trick was to multiply the perturbation operator by λ , and then expand both the states and eigenvalues in a power series in λ . Inserting these two expansions into the energy eigenvalue equation and equating terms of equal powers of λ led to a systematic way to build up an approximate solution. At the end λ can be set to unity to match the solution to the original Hamiltonian. In time-dependent perturbation theory the main goal is to determine the time-evolution of a perturbed quantum system, with particular emphasis on calculating transition probabilities and modeling the irre- versible decay of probability from a small quantum system coupled to a very large quantum system. Formally, we want to find the time evolution of a state governed by the Schr¨odinger Equation, d dt | ψ ( t ) ) = − i planckover2pi1 ( H + V ( t )) | ψ ( t ) ) , (1) where H is the ‘bare’ Hamiltonian, whose eigenstates and eigenvalues are known, and V ( t ) is some perturbation. Experimentally, important information can be obtained by observing how a system responds when we ‘wiggle’ it or ‘kick’ it, or otherwise perturb it in a time-dependent way. While V ( t ) is thus explicitly taken as time-dependent, time-dependent perturbation theory is equally suited to the case where V is constant in time. In order to keep track of perturbation ‘order’, it is customary to introduce the perturbation parameter, λ , and start from the Hamiltonian H = H + λV ( t ) , (2) and then set λ = 1 at the end of the calculation. Generally, we will assume that the system starts in one of the unperturbed eigenstates, which we will refer to as | m ) . But the perturbation approach applies equally well to an arbitrary initial state | ψ (0) ) , i.e. s a superposition of bare eigenstates. The goal is to find | ψ ( t ) ) , the state at some later time t . In principle, one can simply start by inserting a perturbation expansion for | ψ ( t ) ) into Eq. (1), and start turning the crank. However, we can make things much easier on ourselves by (a) computing the perturbed propagator, which can then be used to propagate any initial state; and (b) switching to the interaction picture, where the calculation are much cleaner. Thus we will first briefly review the tranformation between the Schr¨odinger and Interaction pictures....
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- Spring '11
- mechanics, initial state, Time-dependent perturbation theory, Schr¨dinger picture