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TDPT - Phys 852 Quantum mechanics II Spring 2008...

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Phys 852, Quantum mechanics II, Spring 2008 Time-Dependent Perturbation Theory 1/14/2008 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 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 system, with particular emphasis on calculating transition probabilities and modeling the decay of probability out of a small quantum system , when it is coupled to a much larger quantum system. Formally, we want to find the time evolution of a state governed by the Hamiltonian H = H 0 + V ( t ) , (1) where H 0 is the ‘bare’ Hamiltonian, whose eigenstates and eigenvalues are known, and V ( t ) is some perturbation. While V ( t ) is often explicitly time-dependent in order to describe a system is driven by an oscillating field, time-dependent perturbation theory is equally suited to the case where V is constant in time. In order to keep track of perturbation ‘order’, we introduce the perturbation parameter λ and start from the Hamiltonian H = H 0 + λV ( t ) , (2) where we can set λ = 1 at the end to recover the original system. We let the ‘bare’ eigenvalues be labeled E n = planckover2pi1 ω n , and let the bare eigenstates be denoted by | n ) . The defining equation for the unperturbed states is H 0 | n ) = E n | n ) . (3) It is important to note that in time-dependent perturbation theory we are not trying to find new eigenstates so there is no need to add notation to distinguish perturbed and unperturbed eigenvalues and eigenvectors. Generally, we will assume that the system starts in one of the unperturbed eigenstates, which we will refer to as | i ) . But the perturbation approach applies equally well to an arbitrary initial state | ψ (0) ) , which may be a superposition of eigenstates. The goal is to find | ψ ( t ) ) , the state at some later time t . It is easiest, but not necessary, to work in the interaction picture, so we will now briefly review the concept of a Hilbert-space transformation, focusing in particular on the connection between the Schr¨odinger and Interaction pictures. 1
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2 The Interaction Picture A Hilbert-space transformation is a rotation in Hilbert space which leaves all physical observables un- changed. Generally these transformations are used to find the most convenient basis for solving particular problem. The key to leaving observables unchanged is to rotate both the state and the operators in the same direction. The first requirement, therefore is that such transformations conserve probability, which
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