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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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