It is shown how the model which was introduced by Mouchet (2008 Eur. J. Phys.
29 1033) allows one to mimic the quantum tunnelling between two symmetric
one-dimensional wells

Mouchet, Amaury

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Amaury  Mouchet

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A mechanical model of tunnelling

International audienceIt is shown how the model which was introduced by Mouchet (2008 Eur. J. Phys. 29 1033) allows one to mimic the quantum tunnelling between two symmetric one-dimensional wells

HAL Université de Tours

A mechanical model of tunnellingAmaury MouchetTo cite this version:Amaury Mouchet. A mechanical model of tunnelling. European Journal of Physics, EuropeanPhysical Society, 2009, 30 (1), pp.L1. <hal-00339777>HAL Id: hal-00339777https://hal.archives-ouvertes.fr/hal-00339777Submitted on 18 Nov 2008HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.A mechanical model of tunnellingAmaury MouchetLaboratoire de Mathe´matiques et Physique The´orique,Universite´ Franc¸ois Rabelais de Tours — cnrs (umr 6083),Fe´de´ration Denis Poisson,Parc de Grandmont 37200 Tours, France.mouchet@phys.univ-tours.frNovember 18, 2008AbstractIt is shown how the model which was introduced in [1] allows to mimicthe quantum tunnelling between two symmetric one-dimensional wells.Following the publication of [1], some readers asked me to provide more de-tails on the concluding remark about tunnelling (first paragraph of § 6). Strictlyspeaking, tunnelling is a quantum phenomenon that refers to any process thatis classically forbidden (i.e. cannot be understood from the real solutions of theHamilton equations). The paradigmatic example (that can be found in manytextbooks, like [2]) is given by the oscillations of a quantum system between twoone-dimensional potential wells separated by a barrier whose maximal energy islarger than the energy of the system. As understood as early as 1927 by Hund[3], it relies on the existence of some evanescent Schro¨dinger waves connectingthe two wells. Therefore, it is perfectly justified to consider a purely classicalmechanical model to illustrate tunnelling provided it exhibits the appropriateevanescent waves (we do not intend here to compare the classical model withits quantum analogue). In optics, where evanescent waves can be easily createdwith dielectric materials, the analogue of quantum tunnelling has already beenconsidered, for instance in [4].To mimic the double-well symmetric situation, let us consider two identicaloscillators coupled to the Klein-Gordon string (as described in [1]) and locatedat x = ±a/2 (a > 0). When the oscillation frequency of the free oscillators Ω0 isbelow the “infra-red” cut-off on the Klein-Gordon string ω0, there are two stablemodes, one symmetric and one antisymmetric whose corresponding frequenciesare denoted by ω+ and ω− respectively : If the two oscillators were infinitelyfar away, we would get ω+ = ω− = ωb given by (42) of [1] but at finite distancethe string connecting the two oscillators lifts the degeneracies between the twofrequencies : the splitting ∆ωdef=ω− − ω+ 6= 0 represents the frequency of thebeating between the two oscillators. At these frequencies ω±, both below ω0,no energy transport along the string is allowed on the average (equation (9) of[1]) and however some energy can be exchanged between the two oscillators.This is similar with the quantum situation: for the quantum symmetricdouble-well potential, the true eigenstates are delocalised between the two wells1with energies E+ and E−; a state that is localised in one well only is a linearsuperposition of a symmetric and an antisymmetric state and oscillate back andforth between the two wells at a frequency (E− − E+)/~.The tunnelling splitting is expected to be exponentially small with the dis-tance between the two oscillators [5] as can be proved by the following compu-tation:For x < −a/2, we keep an evanescent wave on the string with the formD−ei(−k(x+a/2)−ωt) with k = i√ω20 − ω2 whereas for −a/2 < x < 0 we haveC+ei(k(x+a/2)−ωt) + D+ei(−k(x+a/2)−ωt). The complex coefficients C+ and D+are linearly related to D− with the matrix T (ω) given by (23-26) of [1]. The an-tisymmetric mode corresponds to an odd wavefunction that vanishes at x = 0:C+eika/2 +D+e−ika/2 = 0. The symmetric mode corresponds to an even wave-function whose spatial derivative vanishes at x = 0 : C+eika/2−D+e−ika/2 = 0.In both cases, a non-trivial solution can be found if ρ = ±e−ika. The frequenciesω± must be the solutions of the two equations respectively1 +2√ω20 − ω2κω2 − Ω2κω2 − Ω20= ∓ e−a√ω20−ω2 . (1)If the right hand side vanished (a → +∞ while keeping fixed all the otherparameters), we would recover the equation that furnishes ωb (see equation (38)of [1]). For finite a, in the case where the right hand side is small (a√ω20 − Ω20 ≫1), the first order expansion in the coupling parameter κ provides∆ω ≃ κ22Ω0√ω20 − Ω20e−a√ω20−Ω20 . (2)References[1] A. Mouchet, Eur. J. Phys. 29, 1033 (2008).[2] A. Messiah, Me´canique Quantique (2 vol.) (Dunod, Paris, 1964), englishtranslation: North-Holland (Amsterdam).[3] E. Merzbacher, Phys. Today 55, 44 (2002).[4] J. U. No¨ckel and A. D. Stone, Nature 385, 45 (1997).[5] A. Garg, Amer. J. Phys. 68, 430 (2000).2

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