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Unformatted text preview: Introduction to Quantum and Statistical Mechanics Handout 9 Dynamics of Two-Level Systems Textbook Reading: Hangelstein, Senturia and Orlando, Chaps. 17 - 19 (mostly Chap. 17). 9.1 Overview We will continue our discussion on the finite-base approximation to a base system which is perturbed by an additional potential. However, we will restrict ourselves in the base system with only TWO eigenstates, although many important conclusions can be generalized to the base system with more than two states (but the set needs to be finite). The two-state simplification allows us to investigate deeper into the important dynamics of the coupling between the two eigenstates. We will also be able to see that the dynamics of the wave function can be entirely captured by the dynamics of the amplitude coefficients of the eigenfunctions of the time- independent Schrödinger equation, i.e., the matrix form of the Hamiltonian can be used to describe the time evolution of the amplitude coefficient vector. For a Hamiltonian that contains a known two-level system of ˆ H and a perturbation potential of V ˆ (in the previous Chapter, this is denoted as 1 ˆ H , but now we hope to explicitly express the spatial and time dependence of the potential energy, and hence V ˆ here).: V H H ˆ ˆ ˆ + = (9.1) where ˆ H has ONLY two eigenfunctions φ 1 and φ 2 , with the corresponding eigenenergy of E 1 and E 2 . In the Dirac braket notion, this is: ij j i E H E H δ φ φ φ φ = = = ; | ˆ ; | ˆ 2 2 1 1 (9.2) Because we assume the base system has ONLY two eigenstates, φ 1 and φ 2 will form a complete and orthogonal set of functions for the given boundary condition. The dynamic wave function ψ (x,t) for ˆ H can be written as: ( 29 ( 29 1 / exp ) ( / exp ) ( ) , ( 2 02 2 01 2 2 02 1 1 01 = +- +- = c c t iE x c t iE x c t x V V φ φ ψ (9.3) Now for the time-dependent Schrodinger equation for the entire H ˆ : ( 29 ( 29 ( 29 ( 29 t x V t x H t x H t x t i , ˆ , ˆ , ˆ , ψ ψ ψ ψ + = = ∂ ∂ (9.4) 1 Because all wave functions in the present system can be composed by the linear combination of the two eigenfunctions φ 1 and φ 2 , we can write: ( 29 ) ( ) ( ) ( ) ( , 2 2 1 1 x t c x t c t x φ φ ψ + = (9.5) This form looks simple, but carries an important meaning. The dynamics of the wavefunction of H ˆ can be entirely described by the time-dependent amplitude coefficients for the eigenfunctions of ˆ H , i.e., we can reformulate the Schrödinger equation for the wavefunction evolution by the appropriate differential equations for c 1 (t) and c 2 (t) !!! Notice that c 1 (t=0) and c 2 (t=0) are usually NOT c 01 and c 02 in Eq. (9.3), unless the system starts with a particular unperturbed state, i.e., V ˆ is only applied after t = 0 . This also happens when V ˆ = 0, and Eq. (9.3) becomes a special case for Eq. (9.5), with ( 29 ( 29 / exp ) ( ; / exp ) ( 2 02 2 1 01 1 t iE c t c t iE c t c = = (9.6) If we substitute Eq. (9.5) to (9.4) and multiply the bra operators |...
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