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classically if a particle is located in one of the wells of a dswp and does not have sufficient energy to surmount the barrier in the potential , it will be forever confined to that well .
quantum mechanics shows that due to the wave - like nature of the particle , after a certain length of time , there is a non - zero probability that it will be located in the other well .
it is therefore quantum mechanically possible for a particle to pass through a barrier that it can not classically overcome .
this phenomenon is known as quantum tunneling effect .
the quantum tunneling results in the splitting of the low - lying energy levels which occur in pairs with slightly different energy values .
the transition frequency between the energy levels of each pair is associated with the emission or absorption of electromagnetic radiation . particularly in the ammonia molecule
this transition frequency for the ground - state has been measured at about @xmath4 @xcite .
this phenomenon , namely the inversion spectrum of the ammonia molecule , has been observed through infrared spectroscopy and plays a fundamental role in the principle of operation of the ammonia maser@xcite .
although inversion effect occurs in other molecules as well ( @xmath1 , @xmath2 , @xmath3 ) @xcite , @xmath0 provides a tractable vibrational system for experimental observation and exploitation , since inversion frequency falls in the microwave region .    until recently ,
oscillation of probability density in position space @xmath5 between the wells has been the usual way to approach the quantum tunneling phenomenon and consequently the inversion spectrum .
heisenberg uncertainty relation reflects this phenomenon , but information entropy offers a more sensitive approach , to study a particle moving in a non - classical way through the barrier .
information - theoretic tools , initially applied to communication systems , have been employed extensively to investigate various classical and quantum systems e.g. in physics @xcite , chemistry @xcite , biology @xcite and many other scientific branches as well .
specifically , the well - known information measures defined by shannon @xcite and fisher @xcite , have been applied with considerable success in quantum systems e.g. atoms @xcite .
shannon information entropy has been correlated fairly well with experimental data for atomic ionization potentials and dipole polarizabilities @xcite .
a comprehensive account of applications to molecules can be found in @xcite and @xcite .
another example is an information - theoretic treatment of a molecule ( @xmath6-system ) described in @xcite .    to begin with , one needs a probabilistic treatment of a system , which in fact is especially suitable and relevant for quantum systems , and then use the corresponding probability densities , @xmath7 in position space and @xmath8 in momentum space as input to the definitions of shannon information entropy and fisher information .
thus , one proceeds to the calculation of the information content of the system and investigate its related properties .
an additional merit of the probabilistic treatment is that one can calculate quantitatively , in a systematic manner , a measure of complexity of the quantum system , the so - called lmc statistical complexity @xcite .
the lmc complexity of the @xmath9 ion was studied using a simple wavefunction of coulson type , leading to a promising relation of complexity with chemical bonding @xcite .
atomic complexity has been calculated for the first time in the literature in @xcite , where another definition of complexity was employed , namely the sdl measure @xcite .
last but not least , calculations of molecular information entropies were carried out in @xcite .    in the present paper
we apply the above methods in order to study the ammonia molecule as a test bed , and in particular to assess the effect of tunneling on the shannon information entropy , the fisher information and the lmc complexity , together with heisenberg s uncertainty .
we employ a simple model for the ammonia molecule , i.e. the double square well potential ( dswp ) , which captures its essential properties required for a probabilistic treatment via the schrdinger equation .
this paper is organized as follows . in section [ dswp ] ,
we derive the time - dependent wavefunction for the ammonia molecule ( dswp ) , and we also plot the time dependence of probability density in both position and momentum spaces . in section [ iswp ] , we consider the infinite square well potential ( iswp ) , in order to compare the corresponding results of the dswp with a system , where quantum tunneling is absent . in section [ statistical measures ]
we define the relevant statistical measures and present the intrinsic relation among them . in section [ results ]
we illustrate and comment the results i.e. the time evolution of statistical measures for both ammonia molecule ( dswp ) and iswp .
the conclusions of the analysis are drawn in section [ summary and conclusions ] .
the ammonia molecule has the shape of a pyramid where the nitrogen atom is at the apex and the three hydrogen atoms form the base in the shape of an equilateral triangle @xcite .
the position of the nitrogen atom is chosen as the origin of the x axis .
manning potential @xcite constitutes a good approximation of the ammonia molecule potential , which is defined by the function    @xmath10        in the following analysis we replace the manning potential ( fig . [ manning ] ) by the simplified dswp ( fig .
[ dswp - potential ] ) .
dswp in general , is a tractable quantum system which can be solved analytically , and therefore it will be used to illustrate the main features of the actual potential .
we study the quantum motion of a  particle " representing the collective motion of the three hydrogen atoms .
the mass of this hydrogen plane is equal to @xmath11 , where @xmath12 is the mass of a hydrogen atom .
the depth is assumed to be @xmath13 , the height of the central barrier @xmath14 and the maximum distance that the hydrogen plane can take from nitrogen is @xmath15  where @xmath16 @xcite .
adjustments to the length of the wells @xmath17 and @xmath18 have been made , so that the calculated frequencies are in accordance with the ones that have been found experimentally ( @xmath19 ) @xcite .
dswp in the ammonia molecule is defined by the following piecewise function ,    @xmath20    , width=211,height=139 ]    where @xmath13 , @xmath14 , @xmath21  and @xmath22  .
the one - dimensional schrdinger equation ( se ) is :    @xmath23    by dividing the @xmath24 axis into five spatial regions , and following the standard procedure @xcite we solve analytically the corresponding second order differential equations , taking into account continuity and smoothness conditions for the wavefunction at @xmath25 and @xmath26 .
the potential is an even function of position , i.e. , @xmath27 . due to this symmetry ,
the solutions of the se are either odd or even functions of position .
we solve these equations only for @xmath28 , because only in that case the energy is less than the barrier s height , and so quantum tunneling is possible .
the symmetric ( even ) solution is :    @xmath29    while the antisymmetric one ( odd ) is :    @xmath30    where    @xmath31    in the even case we set @xmath32 , and the values of the constants @xmath33 are obtained by solving the resulting transcendental equation numerically .
working similarly in the odd case , we set again @xmath32 and find @xmath34 and @xmath35 .
the energies and phases found numerically are :    .energies and phases in ammonia molecule @xmath0 [ cols="^,^,^ " , ]     finally , disequilibrium @xmath36 and lmc complexity @xmath37 versus time are plotted in fig .
[ fig : nh3-d - c ] .
as it is expected , @xmath36 is maximized when @xmath38 is minimized ( fig .
[ fig : nh3-shannon - net-1 ] ) and vice - versa . complexity on the other hand , reaches its relative minimum values , when either @xmath38 or @xmath36 are taking their extreme values .
0.32 , title="fig:",width=211,height=139 ]    0.32 , title="fig:",width=211,height=139 ]      we illustrate below the graphs of the same statistical measures for iswp .
heisenberg uncertainty relation in fig .
[ fig : iswp - hp - all ] shows the same behavior as in dswp in fig .
[ fig : nh3-hp - all ] .    0.32 , b ) in momentum space @xmath39 and c)@xmath40,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath39 and c)@xmath40,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath39 and c)@xmath40,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath41 and c)@xmath42,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath41 and c)@xmath42,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath41 and c)@xmath42,title="fig:",width=211,height=139 ]    on the other hand , while the results for both @xmath43 and @xmath41 are similar to those of the dswp case ( figs .
[ fig : iswp - shannon - position ] and [ fig : iswp - shannon - momentum ] ) , the net shannon information entropy in fig .
[ fig : iswp - shannon - net ] presents an interesting fluctuation ( local minimum value ) when the particle is in the middle of the oscillation ( @xmath44 ) .
this behavior is the result of the mutual overlap of the probability density curves in position space @xmath5 , which can be seen in fig .
[ fig : iswp - oscillation - position ] .
0.32 , b ) in momentum space @xmath45 and c)@xmath46,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath45 and c)@xmath46,title="fig:",width=211,height=139 ]    0.32 , b ) in momentum space @xmath45 and c)@xmath46,title="fig:",width=211,height=139 ]    contrary to the net shannon information entropy @xmath38 , which reflects the features of the probability density in position space @xmath5 , the net fisher information @xmath47 reflects the behavior of the probability density in momentum space @xmath48 .
it is clear from fig .
[ fig : iswp - oscillation - momentum ] that curves of @xmath48 overlap with each other when the particle is predominately in the left or in the right side of the well ( @xmath49 ) , which results in a local maximum value .
the importance of shannon and fisher information entropies is now obvious , since these two measures manage to differentiate dswp from iswp case , providing a more sensitive analysis of the system .    finally , in fig .
[ fig : iswp - d - c ] we plot disequilibrium and lmc complexity versus time .
0.35     0.35
we have found that when a particle tunnels through a barrier in a dswp , the heisenberg product of uncertainties and information entropy form a periodic function , which take their minimum values when the particle is located in one well , and their maximum when the particle is in the middle of the penetration region . despite the absence of the tunneling effect in iswp case
, heisenberg relation shows qualitatively the same behavior as in dswp case . on the other hand , information entropies , disequilibrium and lmc
complexity show slight but crucial qualitative differences , formulating a criterion , whether the tunneling effect is present or not in a quantum - mechanical system .
our results also verify the main differences between shannon and fisher information entropies .
fisher s information reflects the localization characteristics of the tunneling effect more sensitively than the shannon information entropy , while the first one resembles to an exponential rise , and the latter to a logarithmical one .
moreover , shannon s information entropy @xmath38 reflects the behavior of probability density in position space @xmath5 , while the fisher information the behavior of probability density in momentum space @xmath48 .
this can be observed from the results in both dswp and iswp cases . in this sense ,
shannon information entropy is complementary to fisher information .    in conclusion
, it is worth mentioning that generally dswp and ammonia molecule in particular , are of great interest in the field of quantum computation and information .
recent experimental and theoretical research @xcite has shown that dswp can be used to make quantum logic gates for ultracold atoms confined in optical lattices , while the ammonia molecule provides a nice vibrational system , which can be a promising candidate for achieving molecular quantum computation .
gadre , reviews of modern quantum chemistry , a celebration of the contributions of robert g. parr , vol .
1 , world scientific , singapore , 2002 , ch . information theoretical approaches to quantum chemistry , pp
. 108147 .
quantum tunneling is the quantum - mechanical effect where a particle tunnels through a classically forbidden region . 
 double square well potential ( dswp ) is a system where this phenomenon is feasible . 
 numerous phenomena can be illustrated by considering motion in a pair of wells that are separated by a barrier of finite height and width . 
 the energy level splitting , resulting from barrier penetration , is the reason of the so - called inversion spectrum , which is an example of quantum tunneling . out of several molecules 
 ( @xmath0 , @xmath1 , @xmath2 , @xmath3 ) where this inversion phenomenon occurs , ammonia molecule @xmath0 provides a nice physical realization of a vibrational system with a dswp . 
 the main goal of the present work is to examine the implications of quantum tunneling on information entropy measures ( shannon s and fisher s ) and statistical complexity .    
 pacs numbers : 89.70.cf , 33.20.-t , 03.65.xp