Mean position of a parti le submitted to a

Mean position of a parti le submitted to a potential barrier D. Mer ier, V. Régnier ? Abstra t A one-dimensional Klein-Gordon problem, whi h is a physi al model for a quantum parti le submitted to a potential barrier, is studied numeri ally : using a variational formulation and a Newmark numeri al method, we ompute the mean position and standard deviation of the parti le as well as their time evolution. Key words Klein-Gordon equation, Newmark method, mean and standard devia- tion, kernel smoothing, linear regression. AMS 35A15, 65M06, 65M12, 81Q05, 81Q10. 1 Introdu tion It has been well-known for a few years now that in quantum me hani s a parti le an limb up a step even if it has not enough energy a priori and it will be ree ted then with a delay. In lassi al me hani s it would just try and go ba k to its position. Everything happens here as if the parti le ould go through a wall ( f. Fig. 5.1 in [17?) ! This phenomenon is alled tunnel ee t and has been a subje t of interest for physi ians and mathemati ians : the delay has been measured by the physi ists A.

• iennes


• tion


• initial boundary


• tly supported initial


• exa tly


• gaussian wave


• tral theory


• mer ieruniv-valen


• delay whi


• numeri al