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B R H H H=RBH=RB Martin Čížek Charles University, Prague t = 0 t > 0 Introduction to Scattering Theory
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Time evolution: Formal solution: Formal Scattering Theory - Intro t = 0 t > 0
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Formal Theory: Asymptotes Hamiltonian: t = 0 t < 0 Definition: in-asymptote
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Formal Theory: Asymptotes t = 0 t > 0 Definition: out-asymptote
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Asymptotic condition The theory is said to fulfill the asymptotic condition if for every exists for which: + the same for t → +∞ Møller wave operators:
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Asymptotic completeness The theory is said to be asymptoticly complete if Ω + ( H ) = Ω - ( H ) = R (orthogonal complement to B ) B R H H H=RBH=RB
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Scattering operator B R H H In asymptotesOut asymptotes Unitarity: S + S = S S + =1
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Energy conservation Intertwining relations: Corollary i.e. we can define “On-Shell T-matrix” S 1 + remainder
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The Cross Section
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It can be shown, that the procedure does not depend on the shape of ψ in =φ(p), provided φ(p) is sharply peaked around p 0 Key assumption is
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Rotation described by Rotational invariance: [H,R(α)] = [H 0, R(α)] = 0 Consequence: [R(α),Ω ± ] = [R(α),S] = 0 Symmetries: Rotational Invariance S-matrix is diagonal in basis formed by common eigenvectors of operators H 0, J 2, J 3
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Symmetries: Rotational Invariance
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More on symmetries Parity invariance: Time reversal:
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Resolvent – Green’s operator Significance: 1. Green’s function for time-independent Schrodinger equation 2. Fourier transform of evolution operator Resolvent (Lippmann-Schwinger) equation: G=G 0 +G 0 VG
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Green and Møller It is possible to show: i. e.
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Green’s and T-operator Lippmann-Schwinger for T: Born series
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Scattering and T-operators Recall:
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Stationary scattering states Recall: We apply: Spatial representation: Lippmann-Schwinger eqution
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Partial wave expansions Free wave (just question to expand |p in terms of |E,l,m ) Stationary scattering state: Note:
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Partial waves – integral eqation
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Normalized / Regular solution Jost function
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Analytic properties φ l,p (r) is analytic function of p, λ Jost function is analytic in λ Jost function is analytic in p in upper half-plain Zeros of Jost function = poles of s-matrix: Re p Im p bound states virtual state resonances physical sheet
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Rieman surface for the energy Re E E=p 2 /2m
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S, T, K - matrix Recall: (1-S) and 2πiT was the same “on shell” more generally T(z)=V+VG(z)V Def: M=i(1-S)(1+S) -1 i. e. S=(1+iM)(1-iM) -1 K-matrix (reaction or Heitler’s matrix): note: it is possible to define it in analogy for T with standing-wave G
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Literature J. R. Taylor: Scattering Theory, R. G. Newton: Scattering Theory of Waves and Particles, P. G. Burke: Potential Scattering in Atomic Physics
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