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the wigner function @xcite , which was introduced in 1932 , is up to now the most commonly used phase space quasi - distribution representation of physical systems in momentum and position representations .
this function fully characterizes the quantum state and gives basic physical intuition about the investigated state
. moreover , there are some simple physical properties that make this function unique @xcite .
nevertheless , over the years many others quasi - distributions have been introduced and studied . especially well - known are functions from the glauber and cahill s - parameterized class of distributions @xcite containing the wigner function , the glauber - sudarshan @xmath0-representation @xcite , and the husimi or the @xmath1-representation @xcite .    for many systems the wigner function is well known and has been carefully studied , but there are some elementary quantum mechanical states , for which this phase space representation is not known .
an example of such system is the hydrogen atom .
no exact analytical formula of the wigner function is known even for the 1s state . in the literature
one can find a limited number of papers devoted to this subject @xcite , and all the published results have been achieved using methods of approximation .
it is the purpose of this short communication to present an analytical phase space representation of the hydrogen atom using a lesser - known kirkwood  rihaczek ( k - r ) distribution function @xcite .
its form allows us to obtain analytical results , and as we have shown in @xcite this distribution , like the wigner function , fully characterizes a quantum state .
in general , the problem of defining a joint probability function in phase space lies in the operator ordering .
because different orderings of position @xmath2 and momentum @xmath3 operators are not equivalent , the association of the quantum operator with classical functions has no unique solution .
the association of classical phase space functions @xmath4 with quantum operators is shown by : @xmath5 the fourier decomposition of the dirac delta functions makes it possible to rewrite the above equation in the form @xmath6 for every operator ordering there exists a corresponding probability quasi - distribution in phase space .
the wigner distribution function is obtained by taking a quantum average of the above formula with the respect to the wigner - weyl ordering : @xmath7 in which @xmath8 and @xmath9 operators are put in the same exponent .    the k 
r distribution function corresponds to an ordering called the anti - standard ordering , that is obtained by putting all @xmath10 operators on the left of all @xmath11 operators : @xmath12 the complex conjugation of the k 
r function corresponds to the standard ordering ( all operators @xmath11 are on the left followed by @xmath9 operators ) , and the real part of the k 
r distribution function is obtained by a symmetric superposition of the anti - standard and the standard ordering : @xmath13 .
obviously , the wigner - weyl ordering and the anti - standard ordering are not equivalent , so they lead to different quasi - distributions . but both of these orderings are intuitive and simple ones , and appropriate phase space distribution functions have the same marginal properties .
the definition of the famous wigner quasi - distribution function was firstly presented as : @xmath14 although for most one dimensional systems formula ( [ zwig ] ) allows to evaluate the wigner function relatively easily , in three dimensions this problem is more complicated .
integrals become quite cumbersome and sometimes impossible to handle analytically . as we have mentioned before , an analytical formula for the wigner function
is not known even for 1s state of hydrogen atom .
that is why , in order to investigate the hydrogen atom in phase space we shall use the k 
r distribution .
this function was introduced by kirkwood @xcite just one year after wigner introduced his function .
then , in 1968 , the same function was rediscovered by rihaczek @xcite in the context of signal time - frequency distributions and is known by his name in signal transmitting theory .
the definition of the k 
r function in terms of anti - standard ordering has a simple form : @xmath15 elsewhere @xcite we have presented an extensive analysis of the k  r distribution and its comparison with the wigner distribution function . here
we will only mention the main properties of the k  r function .
similar to the wigner function , the k - r distribution has the correct marginal properties : @xmath16 hydrogen atom in phase space ----------------------------    the definition of the k - r distribution ( [ zrih ] ) allows one to evaluate analytical formulas of this distribution for many complicated quantum mechanical states , for which the analytical expression of the wigner function is not accessible .    in the following
we will present analytical results obtained for states of the hydrogen atom .
the wave functions of a non - relativistic hydrogen atom are well known , both in position and momentum representation .
the schrdinger equation with coulomb potential can be separated in the spherical polar coordinates ( see , e.g. @xcite ) .
the solutions of angular part of the equation are given by the spherical harmonics @xmath17 , and of the radial part can be expressed in terms of laguerre functions @xmath18 ( in the position representation ) : @xmath19^{\frac{1}{2 } } \left(\frac{2 z}{n}\right)^{\frac{3}{2}}\ , e^{-\frac{z r}{n } } \left(\frac{2 z r}{n}\right)^l l^{2l+1}_{n+l}\left(\frac{2
z r}{n}\right ) , \label{lag}\ ] ] or gegenbauer functions @xmath20 ( in the momentum representation ) : @xmath21^{\frac{1}{2 } } n^2\ , 2^{2(l+1)}\ , l!\ , \frac{n^l p^l}{(n^2 p^2 + 1)^{l+2}}\ , c^{l+1}_{n - l-1 } \left(\frac{n^2 p^2 - 1}{n^2 p^2 + 1 } \right ) . \label{geg}\ ] ] as usual , @xmath22 , @xmath23 , @xmath24 denote principal , orbital , and magnetic quantum numbers , respectively .    substituting the expressions ( [ lag ] , [ geg ] ) into the definition of the k - r distribution eq .
( [ zrih ] ) , and inserting the angular relation , we obtain the general formulas for k - r functions of the hydrogen atom states . in the next section
we shall present graphical results obtained for selected hydrogen quantum numbers .
it is worth noting that the absolute square of the k - r function : @xmath25 ) .
thus , one can treat it as the cornerstone of the phase space analysis of physical systems .
figures 1 to 5 present phase space k - r representation of @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 states of the hydrogen atom .
let us note that for a 3-dimensional systems the k 
r distribution is a function of 6 variables @xmath31 , so a graphical phase space representation can only present selected cross - sections .
we have decided to use the following method to represent the hydrogen atom in phase space : all the figures labelled by ( a ) , ( b ) ( later called fig .
x.a , x.b ) are auxiliary ones 
they present hydrogen atom wave functions in position and momentum representation and the arrows depicted on those figures show the directions chosen for the cross - sections .
the corresponding cross - sections of the k - r phase space functions are plotted in figures labelled by ( c ) and ( d ) ( fig .
x.c , x.d ) .
all the figures are organized as follows : fig x.a shows cross - section @xmath32 of the absolute square of hydrogen atom wave function in the position representation , i.e. @xmath33 which is the probability density of finding an electron at point @xmath34 ( and which does not depend on the value of @xmath35 ) .
x.b shows cross - section @xmath36 of the absolute square of the same wave function in the momentum representation , i.e. @xmath37 which is the probability density of finding an electron with momentum @xmath38 ( and here again , a cross - section does not depend on the value of @xmath39 ) .
the arrows depicted on the fig .
x.a , x.b show the directions chosen for the cross - sections of the corresponding k - r phase space functions that are presented in fig .
x.c , where the real part of the k  r distribution and appropriate contour plot are shown , and in fig x.d , where the absolute value of this k  r distribution and its contour plot are plotted .
let us first study the k - r function of the @xmath40 state .
well - known solutions of the radial part of the schrdinger equation are given by , in atomic unit : @xmath41 fig .
1.a shows spatial probability density @xmath42 of finding an electron at certain point , and fig .
1.b shows momentum probability density @xmath43 of finding an electron with definite momentum .
note , that those are not radial distributions and they have nonzero values at @xmath44 and @xmath45 as the wave functions were not multiplied by @xmath46 or @xmath47 factors .
the k - r distribution of @xmath26 state is given by : @xmath48 where @xmath49 denotes the angle between @xmath50 and @xmath51 : @xmath52 fig .
1.c shows the real part of eq .
( [ s1 ] ) , @xmath53 , multiplied by @xmath54 : @xmath55.\ ] ] the cross - section is chosen in the directions depicted by arrows in fig .
1.a , 1.b .
next we have displayed its contour plot .
the absolute value of eq .
( [ s1 ] ) multiplied by @xmath54 is : @xmath56\ ] ] and its contour plot is presented in fig 1.d . in this case ,
@xmath57 and @xmath58 look quite similar mainly because the wave function both in position and momentum representation decreases rapidly with the increase of @xmath50 or @xmath51 , and the oscillating @xmath59-like structure characteristic for the real part of k - r distribution is not clearly seen on the scale used to make these figures .
these oscillations are merely marked in the contour plot of @xmath53 by dashed lines which correspond to @xmath60 .
next we shall examine the k 
r distribution of 2s and 2p states .
solutions of the radial part of the schrdinger equation in momentum representation are given by : @xmath61 thus , the corresponding k 
r distribution are of the form : @xmath62 where @xmath49 is defined by eq .
( [ q ] ) .
the real part of eq .
( [ s2 ] ) and appropriate contour plot are shown in fig . 2.c and the real part of eq .
( [ p2 ] ) in fig .
3.c . the absolute value of eqs .
( [ s2],[p2 ] ) and their contour plots are presented in fig .
2.d and fig .
3.d , respectively . the plot presented in fig .
3.d has one maximum at @xmath63 the location of which does not depend on variables @xmath64 , @xmath65 , @xmath66 , @xmath39 chosen for the cross section .
this result is in perfect agreement with what we expect for the state which has quantum numbers @xmath22 , @xmath23 satisfying the condition @xmath67 . in fig .
3.c , where the real part of the k 
r distribution is presented , we see the additional minimum . it is due to @xmath59-like oscillations of the real part of the k  r distribution .
actually , there is an infinite number of alternate maxima and minima but their amplitude decreases rapidly and only two are seen on this scale .
dashed lines depicted on this ( and every other contour - plot ) denote @xmath68 . according to the results of fig .
2.d the absolute value of the k 
r distribution for the @xmath27 state also has quite a simple form : we notice one global and three local maxima on this cross section . fig .
2.c , where the real part of the same function is presented , is much more complicated when we look at it close up ( see contour plot ) : extrema are localized on lines where @xmath59 achieves its extremal value , but on this regular structure is another one  the wave function dependence on @xmath50 and @xmath51 that is responsible for the appearance of additional zeros and sign  changes of the plotted figure .
finally we shall present an instructive example of a rydberg state .
figure [ f4 ] shows the k - r distribution for a state with @xmath69 and @xmath70 .
the real part of the k 
r distribution is shown in fig .
[ f4].c , and the absolute value is shown in fig .
[ f4].d . just as one would expect
there is one peak in the absolute value plot and similar , but modified by @xmath59 , structure in the real part of the plot . comparing the figures with that shown in fig .
[ f3].c - d we notice similarities in their structure .
obviously , peaks are located at different values of @xmath71 but there is the property that for states with @xmath67 we observe one maximum in the absolute value plot .
its location changes with the increase of @xmath22 ( it moves towards larger @xmath50 and smaller @xmath51 ) and the maximum value gets much smaller , but it is apparent this class of states has much in common .
figure [ f5 ] shows the k 
r distribution for state @xmath69 , @xmath72 .
analysis similar to that made above holds true when we compare fig .
[ f5 ] and [ f2 ] . without counting that all the scales have changed we find 4 extrema ( @xmath73 ) in the absolute value plots and similar oscillating structures in the real part plots .
it is a general and well known result that @xmath74 is equal to the number of extrema of hydrogen atom wave functions both in position and momentum representation .
we only stress that the k - r distribution renders very well this property : its absolute value has @xmath73 extrema .
we have presented a phase space representation of the hydrogen atom using the k - r distribution .
an advantage of such a representation is that the system can be treated analytically .
it is an attractive feature of the k - r distribution that as long as one knows the wave function of the system there is no need to perform any integrals to obtain this distribution value in every point of phase space .
we have described in detail the @xmath26 , @xmath27 , @xmath28 , @xmath29 , and @xmath30 states of the hydrogen atom .
plots corresponding to the k 
r functions of these states were presented and compared with the plots of probability densities in position and momentum representations .
we thank s. daffer for comments about the final draft . this work was partially supported by a kbn grant 2p03 b 02123 , and the european commission through the research training network quest .
we present a phase - space representation of the hydrogen atom using the kirkwood - rikaczek distribution function . 
 this distribution allows us to obtain analytical results , which is quite unique because an exact analytical form of the wigner functions corresponding to the atom states is not known . 
 we show how the kirkwood - rihaczek distribution reflects properties of the hydrogen atom wave functions in position and momentum representations .