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The Semi-classical Approximation - an elementary introduction
Uzy Smilansky
Department of Physics o Complex Systems, The Weizmann Institute of Science, Rehovot 76100, Israel
Abstract These lectures are intended to introduce the Semi ClassicalApproximation and some of its intricacies in a consistent andtransparent way. The main topics to be covered are • The semi classical approximation for the quantum evolution operator • Semi-classical spectral theory: • The trace formula and some of its applications. (For systems in 1-d(
It is hoped that the ideas and tools presented here will provide asolid jumping-board for further studies and applications.
“Putting quantum flesh on classical bones” (W.H. Miller)
a) The buckyball carbon-70;b) The pancake-shaped biomolecule tetraphenylporphyrin (TPP) C44H30N4;c) The fluorinated fullerene C60F48. (atomic mass of 1632 units )
Probing the limits of the quantum worldM. Arndt, K. Hornberger, and A. Zeilinger,Physics World (March 2005) 35-40
Motivation (if necessary) : A two slits experiment with heavy molecules
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Preliminaries: The quantum evolution operator
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Feynman Path Integral representation The Democracy of Paths:
sin®=Ry
0 dx
Hcl (p;q) =1
2mp2+ V(q)r = a
t’’
q’
t’
q’’
Scl [q(t)]=Rt
00
t0 L[q(t); :q(t); t]dta = ba = b
q(t)
Richard P. Feynman and Albert R. Hibbs, Quantum Mechanics and Path Integrals(McGraw-Hill, New York, 1965).
g= h
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Back to the propagator and the path integral
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Note: The boundary conditions do not determine a classical path uniquely! (examplesbelow) However, there is no conflict with the uncertainty principle: the path is notprescribed by the simultaneous values of the position and the momentum.
V(q)
qq’ q’’
Tmax
Example: Several trajectories which satisfy the same boundary conditions.
Tmax : If t”-t’>Tmax direct transition becomes classically forbidden
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Classical bones: dynamics (trajectory), action, stability Quantum flesh : transition amplitude, interference, the quantum scale: ~
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q(t’’; p’)
p’
q’’
δp=2¼h / δq’’
p’i p’2
Classical transition probability
δq’’
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t
caustics
Classically forbidden domain
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End of lecture I
Appendix
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