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the generalized uncertainty principle ( gup ) is a common feature of all promising candidates of quantum gravity .
string theory , loop quantum gravity , black hole physics , and noncommutative geometry ( based on a deeper insight to the nature of quantum spacetime at planck scale ) all support the need for a necessary modification of the standard heisenberg principle .
it has been shown that measurements in quantum gravity are governed by the generalized uncertainty principle @xcite .
there are some evidences from string theory and black holes physics ( based on some gedanken experiments ) , that lead authors to re - examine the usual uncertainty principle of heisenberg @xcite .
these evidences have origin on the quantum fluctuation of the background spacetime metric and are related to the very nature of spacetime in quantum gravity era . the introduction of this idea has drown attention and many authors considered various problems in the framework of the generalized uncertainty principle ( see for instance @xcite ) .
these investigations have revealed some new features of the very nature of spacetime : spacetime is not commutative at planck scale and it has a foam - like structure in this scale , it seems that gravity is not a fundamental interaction of the nature ( it may be induced by the residual effects of fundamental quantum fields on the vacuum , with the lagrangian playing the role of an elastic stress ) , constants of the nature are not really constant and the very notion of locality in position space representation breaks down in planck scale .
therefore , it seems that a reformulation of quantum theory is required in order to incorporate gravitational effects in planck scale phenomena .
these issues have been the topics of a wide range of researches in recent years .    in this paper , we consider the problem of a particle of mass @xmath0 that is bouncing vertically and elastically on a smooth reflecting floor in earth s gravitational field .
we solve this problem in the presence of a minimal length within the gup framework .
first , we give an overview to the gup formalism and in this way we obtain a generalized schrdinger equation . after studying this equation in the momentum space , we find the modified eigenstates and energy spectrum of this system .
we compare the outcomes of our study with the results obtained from elementary quantum mechanics .
a potential application of the present study in the spontaneous decay of an excited state for ultra cold neutrons bouncing above a perfect mirror in the earth s gravitational field is discussed finally .
we note that modification to the decay rate due to existence of a minimal length studied here , becomes important at or above the planck energy .
although this modification is too small to be measurable at present , we speculate on the possibility of extracting measurable predictions in the future .
quantum mechanics with modification of the usual canonical commutation relations has been investigated intensively in the last few years ( see @xcite and references therein ) . such works which are motivated by several independent streamlines of investigations in string theory and quantum gravity , suggest the existence of a finite lower bound to the possible resolution @xmath1 of spacetime points .
the following deformed commutation relation has attracted much attention in recent years @xcite @xmath2=i\hbar(1+\beta p^2),\ ] ] and it was shown that it implies the existence of a minimal resolution length @xmath3 @xcite .
this means that there is no possibility to measure coordinate @xmath4 with accuracy smaller than @xmath5 .
since in the context of the string theory the minimum observable distance is the string length , we conclude that @xmath6 is proportional to this length . if we set @xmath7 , the usual heisenberg algebra is recovered .
the use of the deformed commutation relation ( [ com ] ) brings new difficulties in solving the quantum problems .
a part of difficulties is related to the break down of the notion of locality and position space representation in this framework @xcite .
the above commutation relation results in the following uncertainty relation @xmath8 where @xmath9 and @xmath10 are positive quantities which depend on the expectation value of the position and the momentum operators .
in fact , we have @xmath11 where @xmath12 is the planck mass and @xmath13 is of the order of unity .
we expect that these quantities are only relevant in the domain of the planck energy @xmath14 .
therefore , in the low energy regime , the parameters @xmath9 and @xmath10 are irrelevant and we recover the well - known heisenberg uncertainty principle . these parameters , in principle , can be obtained from the underlying quantum gravity theory such as string theory .
note that @xmath4 and @xmath15 are symmetric operators on the dense domain @xmath16 with respect to the following scalar product @xcite @xmath17 moreover , the comparison between eqs .
( [ com ] ) and ( [ gup ] ) shows that @xmath18 .
now , let us define @xmath19 where @xmath20 and @xmath21 obey the canonical commutation relations @xmath22=i\hbar$ ] .
one can check that using eq .
( [ x0p0 ] ) , eq .
( [ com ] ) is satisfied to @xmath23 .
also , from the above equation we can interpret @xmath21 as the momentum operator at low energies ( @xmath24 ) and @xmath15 as the momentum operator at high energies .
now , consider the following form of the hamiltonian : @xmath25 which using eq .
( [ x0p0 ] ) can be written as @xmath26 where @xmath27 and @xmath28 .    in the quantum domain ,
this hamiltonian results in the following generalized schrdinger equation in the quasi - position representation @xmath29 where the second term is due to the generalized commutation relation ( [ com ] ) .
this equation is a @xmath30th - order differential equation which in principle admits @xmath30 independent solutions .
therefore , solving this equation in @xmath20 space and separating the physical solutions is not an easy task . in the next section ,
for the case of a bouncing particle , we find the energy spectrum and the corresponding eigenstates up to the first order of the gup parameter .
consider a particle of mass @xmath0 which is bouncing vertically and elastically on a reflecting hard floor so that @xmath31 where @xmath32 is the acceleration in the earth s gravitational field .
the hamiltonian of the system is @xmath33 which results in the following generalized schrdinger equation @xmath34 this equation , for @xmath7 is exactly solvable and the solutions can be written in the form of the airy functions .
moreover , the energy eigenvalues are related to the zeros of the airy function . however
, for @xmath35 , the situation is quite different . because , we need to solve a forth order differential equation and eliminate the unphysical solutions . on the other hand , because of the linear form of the potential , this equation can be cast into a first order differential equation in the momentum space .
since the later form is much easier to handle , we define a new variable @xmath36 and rewrite above equation in the momentum space , namely @xmath37 where @xmath38 is the inverse fourier transform of @xmath39 and the prime denotes the derivative with respect to @xmath21 .
it is straight forward to check that this equation admits the following solution @xmath40.\ ] ] since @xmath9 is a small quantity , we can expand the above solution up to the first order of @xmath9 as @xmath41 now , using the fourier transform , we can find the solution in the position space up to a normalization factor @xmath42+\frac{4}{15}\beta m^2g\left(x-\frac{e}{mg}\right)\nonumber\\ & \times&\left\{4\mbox{ai}\left[\alpha\left(x-\frac{e}{mg}\right)\right]+ \left(x-\frac{e}{mg}\right)\right.\nonumber\\ & \times&\left.\mbox{ai}'\left[\alpha\left(x-\frac{e}{mg}\right)\right]\right\},\end{aligned}\ ] ] where @xmath43 and the prime denotes derivative with respect to @xmath20 . finally ,
since the potential is infinite for @xmath44 , we demand that the wave function should vanish at @xmath45 .
this condition results in the quantization of the particle s energy , namely @xmath46=0.\end{aligned}\ ] ] to proceed further and for the sake of simplicity , let us work in the units of @xmath47 . in this set of units ,
the energy eigenvalues are the minus of the roots of the following algebraic equation @xmath48=0.\ ] ] so , the energy eigenvalues will be quantized and result in the following eigenfunctions @xmath49,\end{aligned}\ ] ] where @xmath50s should satisfy eq .
( [ root ] ) .
figure [ fig1 ] shows the resulting normalized ground state and first excited state eigenfunctions for @xmath51 .
moreover , the calculated values of the energy eigenvalues for the first ten states are also shown in table [ tab1 ] .
these results show that the presence of @xmath9 increases the energy levels in agreement with the functional form of @xmath52 .
in other words , existence of a minimal length results in a positive shift in the energy levels of quantum bouncer .
as a potential application of our analysis , we note that quantization of the energy of _ ultra cold neutrons _ bouncing above a mirror in the earth s gravitational field has been demonstrated in an experiment few years ago @xcite .
this effect demonstrates quantum behavior of the gravitational field if we consider the spontaneous decay of an excited state in this experiment as a manifestation of the planck - scale effect @xcite .
since the spectrum of a quantum bouncer changes in the presence of the minimal length , we expect the rate of this decay will change as a trace of quantum gravitational effects via existence of a minimal length scale .
in fact , as we have shown , the energy levels of a quantum bouncer in the gup framework attain a positive shift as given by equation ( [ zero ] ) .
the energy levels of a quantum bouncer in ordinary quantum mechanics are given by zeros of airy function as @xmath53 .
in the presence of a minimal length , the locations of zeros are given by @xmath54 . within a semi - classical analysis
, we evaluate the rate for a bouncer to make a transition @xmath55 .
the quantum quadrupole moment for the transition @xmath55 for a quantum bouncer of mass @xmath0 is given by @xmath56 @xcite .
the quantum mechanical transition rate is ( in the quadrupole approximation ) @xmath57 where @xmath58 .
now , the quadrupole matrix element for quantum bouncer in the presence of minimal length can be calculated using the generalized airy function zeros given by equation ( [ zero ] ) .
the transition probability in our framework is therefore @xmath59 where @xmath60 , @xmath61 , and @xmath62 .
so , there will be an essentially measurable difference in the transition rate of a quantum bouncer due to the presence of the extra @xmath63 in comparison with the case that we consider just the ordinary heisenberg uncertainty relation .    ) for @xmath7 ( red ) , @xmath64 ( green ) , and @xmath65 ( blue).,width=302 ]    .the first ten quantized energies of a bouncing particle in gup formalism . [
cols="^,^,^,^",options="header " , ]
in this paper , we considered the problem of a bouncing particle in a constant gravitational field in the framework of the generalized uncertainty principle .
we found the modified hamiltonian and the generalized schrdinger equation as a forth order differential equation .
we solved this equation in the momentum space and obtained the corresponding energy eigenvalues and eigenstates up to the first order of the gup parameter .
as we have expected , we found a positive shift in the energy spectrum due to the generalized commutation relation . a potential application of this analysis for transition rate of an ultra cold neutron bouncing above a mirror in the earth s gravitational field has been explained .
we emphasize that modification to the transition rate of a quantum bouncer due to existence of a minimal length becomes important at or above the planck energy .
although this modification is too small to be measurable at present , we speculate on the possibility of extracting measurable predictions in the future . at that case , this may provide a direct test of underlying quantum gravity scenario .
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in this paper we study the effects of the generalized uncertainty principle ( gup ) on the spectrum of a particle that is bouncing vertically and elastically on a smooth reflecting floor in the earth s gravitational field ( a quantum bouncer ) . 
 we calculate energy levels and corresponding wave functions of this system in terms of the gup parameter . 
 we compare the outcomes of our study with the results obtained from elementary quantum mechanics . 
 a potential application of the present study is discussed finally .    
 _ keywords _ : quantum gravity ; generalized uncertainty principle ; bouncing particle spectrum .    _ 
 pacs _ : 04.60.-m +