Complex Dynamics - Advanced System Dynamics in Complex by Péter Érdi

By Péter Érdi

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The integral of f (z) equals 2πi times the sum of residues of f (z) at the poles enclosed by c. Cauchy’s Theorem and integral formulas are special cases of this result. It is used for evaluation of various definite integrals of both real and complex functions. 9) is used in inverse Laplace transform. If F (s) = L{f (t)} = ∞ −st e f (t) dt, 0 then L−1 {F (s)} is given by f (t) = L−1 {F (s)} = = 1 2πi est F (s) ds c Res [est F (s)] at poles of F (s) where c ⊂ C is the so–called Bromwich contour .

3 Soft Introduction to Quantum Dynamics In this section we give a soft introduction to quantum dynamics (based on two ‘classical’ experiments), to be used in the following chapters. 11 shows a photon source which we assume emits photons one at a time. There are two slits, A and B, and a screen behind them. The photons arrive at the screen as individual events, where they are detected separately, just as if they were ordinary particles. The curious quantum behavior arise in the following way [Pen97].

Any initial state will decay to the ground state (of zero energy) as time tends to infinity. The corresponding coordinates in phase space (normal modes) are complex as well. This suggests that the equations are of Hamiltonian form, but with a complex Hamiltonian. It is not difficult to verify that this is true directly. The real part of the Hamiltonian is a harmonic oscillator, although with a shifted frequency; the imaginary part is its constant multiple. If we pass to the quantum theory in the usual way, we get a non–Hermitian Hamiltonian operator.

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