- Email: [email protected]
Macroscopic detection of deformed QM by the harmonic oscillator
Accepted Manuscript Macroscopic detection of deformed QM by the harmonic oscillator Michael Maziashvili
PII: DOI: Reference:
S0003-4916(17)30173-2 http://dx.doi.org/10.1016/j.aop.2017.06.007 YAPHY 67414
To appear in:
Annals of Physics
Received date : 6 April 2017 Accepted date : 17 June 2017 Please cite this article as: M. Maziashvili, Macroscopic detection of deformed QM by the harmonic oscillator, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Macroscopic detection of deformed QM by the harmonic oscillator Michael Maziashvili∗ School of Natural Sciences and Engineering, Ilia State University, 3/5 Cholokashvili Ave., Tbilisi 0162, Georgia Based on the nonperturbative analysis, we show that the classical motion of harmonic oscillator derived from the deformed QM is manifestly in contradiction with observations (to the first order in deformation parameter it has been pointed out in [9]). For this reason, we take an alternate way for estimating the effect and discuss its possible observational manifestations in macrophysics. PACS numbers: 04.60.Bc
Motivated by the paper [1] (see also [2]), suggesting a way for observing the Planck length deformed QM via the optomechanical experiments, we take an attempt to discuss further this question. This paper provides the possible experimental basis for observing the corrections to the QM, say, of the form (we employ units in which: c = ~ = 1) i h b PbX = i 1 + ßPb 2 , X, X
(1)
by the interference effect when the part of the light entering the optomechanical device is subject to the oscillation of the moving mirror-wall of the device, under assumption that the latter undergoes ”quantum motion” with respect to the modified prescription (1). In Eq.(1) ß ≃ lP2 , where lP (≃ 10−33 cm) is the Planck length. It it naturally expected that the gravitational corrections to the processes probing the length scale ℓ(≫ lP ) should be suppressed by some powers of the ratio lP /ℓ [3]. The Eq.(1) can be interpreted in much the same way. Namely, one may hold that because of quantum fluctuations of the background metric - any length scale, ℓ ≫ lP , acquires uncertainty that on the general grounds can be parameterized as: ≃ lPα ℓ1−α . Hence the position uncertainty √ of αthe particle gets increased, δx → δx + ßα/2 δx1−α , leading to the modified uncertainty relation δxδp & 1 + ßδp . The Eq.(1) corresponds
to the specific case: α = 2. Another case of interest might be: α = 1/2 [4]. We take this judgment as a guiding principle for our further discussion. It can be stated simply as follows: the size of the gravitational corrections depends primarily on the length scale probed by the system. Returning to the discussion of [1, 2], the system under consideration consists of a monochromatic electromagnetic field confined in a mirror box - one of the walls of which is vibrating non-relativistically. The vibrating wall, which is assumed to be parallel to the (Y, Z) plane, is moving along the X axis. This system can be modeled by the Hamiltonian b2 b2 mω 2 X P b , b = ωph b + − gb a +b aX H a +b a+ 2m 2
(2)
where the three-dimensional momentum P (instead of the one-dimensional one) is introduced purely for technical b stand for convenience, ωph is the photon frequency (b a+ , b a are photon creation and annihilation operators) and ω and X 1 the vibrating mirror-wall frequency and position operator, respectively . The coupling parameter g can be elucidated in terms of input parameters in the following way. Because of photon gas pressure, which is equal to the one third of the photon gas energy density, under assumption that the amplitude of mirror vibrations is much smaller then the cavity length, that is, X ≪ l, there is in fact a constant force acting in outward direction on the moving wall FX =
ωph nph s , 3ls
where s denotes its surface area and l stands for the cavity length. Hence the coupling parameter: g = ωph /3l. b Yb , Z, b P b operators in terms of the standard One can construct a particular representation for the deformed X, b = x b = zb, b , for an arbitrary α (α 6= 1) [4]: X position and momentum operators, x b, yb, zb, p b, Yb = yb, Z ∗ Electronic 1
address: [email protected] Detailed derivation of this Hamiltonian in one-dimensional case can be found in [5, 6].
2
1 h α/2 i 1−α b = p b 1 − (α − 1) ßb P p2 .
When α = 2, one arrives at a well known result of [7]. It is worth noticing that the case α > 1 (α < 1) corresponds to the assumption that the length fluctuation δℓ = ßα/2 ℓ1−α decreases (increases) with increasing of ℓ. A specific feature for the deformed QM with α > 1 is that in this case the state vectors are represented by the cutoff Fourier representation2, but it is less important for us as we are dealing with the objects having the localization with much greater than the Planck length. By using this particular representation, the Hamiltonian (2) gets modified as
b = ωph b H a +b a+
2 h α/2 i 1−α b 2 1 − (α − 1) ßb p2 p
2m
The Heisenberg equations of motion read
+
mω 2 x b2 − gb a +b ax b. 2
b a˙ = −iωph b a + ig b ax b , pb˙x = −mω 2 x b + gb a+ b a , pb˙y = pb˙z = 0 , 1+α 2 i h h α/2 i 1−α α/2 1−α α/2 1 − (α − 1) ßb p2 p2 p2 αb px ßb pbx 1 − (α − 1) ßb + . x b˙ = m m
(3)
(4) (5)
The equations for yb(t) and zb(t) have plainly the same form as Eq.(5) with pbx replaced by pby and pbz , respectively. The solution for b a(t) can immediately be written as
b a(t) = b a(0) exp −iωph t + ig
Zt 0
dτ x b(τ ) .
(6)
From this solution it is clear that b a+ (t)b a(t) = b a+ (0)b a(0), which enables one to rewrite the equation for pbx (t) in the form ˙ = −mω 2 x pb b + gb a+ (0)b a(0) . x
(7)
Let us now specify the initial state of the system. For this purpose, we take the coherent state for the photon gas b a(0)|ζi = ζ|ζi, and the minimum uncertainty state3 ! 2 y2 [x − x(0)] 1 z2 p exp − ψ(x, y, z) = exp ipx (0)x − . − √ 4 4(δx)2 4(δy)2 4(δz)2 (2π)3 δxδyδz for the moving mirror. For averaged quantities with respect to these states one gets the following equations p˙x = −mω 2 x + g|ζ|2 , py = pz = 0 , y = z = 0 , 1+α 2 h α/2 i 1−α α/2 i 1−α α/2 h 1 − (α − 1) ßp2x αpx ßp2x px 1 − (α − 1) ßp2x + . x˙ = m m These equations can be derived from classical Hamiltonian
2 3
The interested reader may be referred to [8]. Here we mean the uncertainty relation between the standard position and momentum operators.
(8)
3
H =
2 h α/2 i 1−α p2x 1 − (α − 1) ßp2x
2m
+
mω 2 x2 − g|ζ|2 x . 2
(9)
Let us notice that by adding a linear term to the potential of oscillator - the potential has still the pure oscillator form with the equilibrium position shifted. So, this term in Eq.(9) can merely be dropped as in the paper [9]. The question of experimental detection can be addressed merely by measuring the velocity x˙ for x = 0. The conservation of energy allows one to estimate the value of momentum px for x = 0, 2 h α/2 i 1−α p2x 1 − (α − 1) ßp2x
2m
=
mω 2 x2max , 2
(10)
and then by substituting this value of px in (8) - to determine the velocity at this point. It is almost obvious that the deviation from the standard velocity becomes appreciable when the momentum approaches ß−1/2 (Planck energy scale). Let us consider two specific cases α = 2 and α = 1/2. In the former case, assuming |px | is approaching ß−1/2 (|px | < ß−1/2 ) and usig Eq.(10), one obtains for the velocity at x = 0 m|x| ˙ = |px |
1 + ßp2x (1 − ßp2x )3
1 = (mωxmax ) ß + 2 ≃ 2ß (mωxmax )3 . px 3
Now let us turn to the case α = 1/2. With no loss of generality, we can absorb the factor 1 − α in ßα/2 (for ß is defined up to the numerical factor of order unity). Then under assumption ßp2x ≃ 1 at x = 0, from Eq.(10) for for the velocity (at x = 0) one obtains √ 1/4 1/4 3 24 mωxmax ≃ 4/ ß , m|x| ≃ √ . 1 + 2 ßp2x ˙ = |px | 1 + ßp2x ß
For characterizing the size of the effect - it might be useful to look at the ratio |x|/ωx ˙ max , which in the standard case is equal to 1. Such a big deviation at the macroscopic scales is not natural4 that leads to the opinion to revise the whole discussion from the very outset. For this purpose, let us look at the well known QM description of the classical motion of harmonic oscillator [10]. The wave function
ψ(t, x) =
r 4
mω(x − x(t))2 iωt ipx (t)x(t) mω , exp ipx (t)x − − − π 2 2 2
(11)
where x(t) and px (t) obey the classical equations for harmonic oscillator, p˙ x (t) = −mω 2 x(t) , x(t) ˙ =
px (t) , m
solves the Schr¨odinger equation for harmonic oscillator. It is plain to see that: px (t) = hψ(t)|b px |ψ(t)i, x(t) = hψ(t)|b x |ψ(t)i. On the other hand, this wave function corresponds to the minimum uncertainty state: δx = p √ 1/ 2mω, δpx = mω/2. The classicality of motion means that the quantum fluctuations should be smaller as compared to the classical quantities. It is convenient to characterize the degree of classicality in terms of the energy p2x (t) mω 2 x2 (t) + 2m 2 p2x (t) mω 2 x2 (t) ω = + + . 2m 2 2
hψ(t)|Oscillator Hamiltonian|ψ(t)i = +
4
mω 2 (δx)2 (δpx )2 + 2m 2
This point can easily be missed in the framework of perturbative treatment [9] the validity of which requires ßp2
α/2
≪ 1.
4 Expanding |ψ(t)i into the Eigenfunktionen, |ji, of harmonic oscillator Hamiltonian ∞ iωt |b(t)|2 X bj (t) √ |ji , − |ψ(t)i = exp − 2 2 j! j=0
(12)
where b(t) =
mωx(t) + ipx (t) √ , 2mω
the energy can be expressed in terms of the mean occupation number, n, as p2x (t) mω 2 x2 (t) + = ωn . 2m 2 Therefore, the motion is essentially√classical if n ≫ 1. This discussion tells us that there is an oscillating particle localized within the region δx ∼ 1/ mω and for estimating the gravitational corrections to this system with respect to physics underlying the Eq.(1), one has to use this length scale and look for the corrections in the form (lP /δx)2 . So that, the size of the gravitational correction depends primarily on δx. Certainly, the moving mirror√in optomechanical device with the parameters given in [1], m ≃ MP , ω ≃ 105 Hz, is not localized within the region 1/ MP ω ∼ 10−14 cm but for the moment let us forget about it and ask what might be the size of the gravitational corrections to the classical oscillator in the framework of the above discussion. As it was discussed in the beginning of the paper, because of fluctuating background the oscillator gets modified in such a way as to respect the modified uncertainty relation α o √ o 1n 1n α/2 δxδpx ≥ ß/δx = 1+ 1 + (ßmω) . 2 2 Let us continue of the previous discussion and ascribe this modification to the momentum operator: h √in theαspirit i ß/δx . It makes sense because for the problem under consideration δx is constant. Thus, we are pbx → pbx 1 + just saying that the momentum for oscillating particle, which is localized within the region δx, is increased by the α √ ß/δx . The modified Hamiltonian factor: 1 + h
1+
√ α i2 ß/δx pb2x 2m
+
m′ ω ′2 x b2 mω 2 x b2 pb2 = x′ + , 2 2m 2
m m′ = h √ α i2 , 1+ ß/δx
h √ α i ω′ = ω 1 + ß/δx ,
in view of the above discussion, tells us that the classical motion gets modified as
x(t) = x(0) cos(ω ′ t) , px (t) = −m′ ω ′ x(0) sin(ω ′ t) .
√ In the case when the localization width of the oscillating body, ℓ, is much bigger than δx = 1/ 2mω, one could make a straightforward generalization of this result x(t) = x(0) cos
h √ α i ß/ℓ ωt . 1+
Obviously, it is extremely hard to detect such a deviation from standard α for the realistic input parameters. √picture −8 ß/ℓ becomes ∼ 10−12 for α = 1/2 and Even if one takes an extreme value: ℓ ∼ 10 cm, the correction term
∼ 10−50 for α = 2. Summarizing, we see that a classical motion derived from the nonperturbative treatment of quantum oscillator on the basis of deformed QM (1) is almost obviously in conflict with ”macrophysics”. This point has already been stressed in [8], in the framework of somewhat different nonperturbatie analysis, and also indicated in [11] - in the framweork of perturbative treatment. The inconsistent classical limit derived from the Eq.(1) is connected with the fact that the gravitational correction term in this equation is written in terms of the momentum instead of the length scale probed by the particle. The momentum of classical bodies are usually high but their localization widths are not as small as 1/p. An alternative treatment (in the end of the paper), which has a natural motivation as it is stated in the beginning, is to write the gravitational correction term to the QM in terms of the wave-packet localization width. This way one arrives at the ”reasonably small” corrections, which in the case α = 1/2 may be possibly within the range of observations.
5 Acknowledgments
Useful comments from Zurab Silagadze and David Vitali are kindly acknowledged. The research was partially supported by the Shota Rustaveli National Science Foundation under contract number 31/89.
[1] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim and C. Brukner, Nature Phys. 8, 393 (2012) [arXiv:1111.1979 [quant-ph]]. [2] I. Pikovski, PhD Thesis ”Macroscopic quantum systems and gravitational phenomena”, (URL: http://inspirehep.net/record/1365384). [3] J. F. Donoghue and B. R. Holstein, J. Phys. G 42, no. 10, 103102 (2015) [arXiv:1506.00946 [gr-qc]]. [4] M. Maziashvili, Phys. Rev. D 86, 104066 (2012) [arXiv:1206.4388 [gr-qc]]. [5] G. T. Moore, J. Math. Phys. 11, 2679 (1970). [6] C. K. Law, Phys. Rev. A 51, 2537 (1995). [7] A. Kempf and G. Mangano, Phys. Rev. D 55, 7909 (1997) [hep-th/9612084]. [8] M. Maziashvili and L. Megrelidze, PTEP 2013, no. 12, 123B06 (2013) [arXiv:1212.0958 [hep-th]]. [9] M. Bawaj et al., Nature Communications 6, 7503 (2015) [arXiv:1411.6410 [gr-qc]]. [10] L. D. Landau and E. M. Lifshitz, ”Kvantovaia Mekhanika” (Moscow, Nauka, 1989). [11] Z. K. Silagadze, Phys. Lett. A 373, 2643 (2009) [arXiv:0901.1258 [gr-qc]].
PII: DOI: Reference:
S0003-4916(17)30173-2 http://dx.doi.org/10.1016/j.aop.2017.06.007 YAPHY 67414
To appear in:
Annals of Physics
Received date : 6 April 2017 Accepted date : 17 June 2017 Please cite this article as: M. Maziashvili, Macroscopic detection of deformed QM by the harmonic oscillator, Annals of Physics (2017), http://dx.doi.org/10.1016/j.aop.2017.06.007 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
*Manuscript Click here to view linked References
Macroscopic detection of deformed QM by the harmonic oscillator Michael Maziashvili∗ School of Natural Sciences and Engineering, Ilia State University, 3/5 Cholokashvili Ave., Tbilisi 0162, Georgia Based on the nonperturbative analysis, we show that the classical motion of harmonic oscillator derived from the deformed QM is manifestly in contradiction with observations (to the first order in deformation parameter it has been pointed out in [9]). For this reason, we take an alternate way for estimating the effect and discuss its possible observational manifestations in macrophysics. PACS numbers: 04.60.Bc
Motivated by the paper [1] (see also [2]), suggesting a way for observing the Planck length deformed QM via the optomechanical experiments, we take an attempt to discuss further this question. This paper provides the possible experimental basis for observing the corrections to the QM, say, of the form (we employ units in which: c = ~ = 1) i h b PbX = i 1 + ßPb 2 , X, X
(1)
by the interference effect when the part of the light entering the optomechanical device is subject to the oscillation of the moving mirror-wall of the device, under assumption that the latter undergoes ”quantum motion” with respect to the modified prescription (1). In Eq.(1) ß ≃ lP2 , where lP (≃ 10−33 cm) is the Planck length. It it naturally expected that the gravitational corrections to the processes probing the length scale ℓ(≫ lP ) should be suppressed by some powers of the ratio lP /ℓ [3]. The Eq.(1) can be interpreted in much the same way. Namely, one may hold that because of quantum fluctuations of the background metric - any length scale, ℓ ≫ lP , acquires uncertainty that on the general grounds can be parameterized as: ≃ lPα ℓ1−α . Hence the position uncertainty √ of αthe particle gets increased, δx → δx + ßα/2 δx1−α , leading to the modified uncertainty relation δxδp & 1 + ßδp . The Eq.(1) corresponds
to the specific case: α = 2. Another case of interest might be: α = 1/2 [4]. We take this judgment as a guiding principle for our further discussion. It can be stated simply as follows: the size of the gravitational corrections depends primarily on the length scale probed by the system. Returning to the discussion of [1, 2], the system under consideration consists of a monochromatic electromagnetic field confined in a mirror box - one of the walls of which is vibrating non-relativistically. The vibrating wall, which is assumed to be parallel to the (Y, Z) plane, is moving along the X axis. This system can be modeled by the Hamiltonian b2 b2 mω 2 X P b , b = ωph b + − gb a +b aX H a +b a+ 2m 2
(2)
where the three-dimensional momentum P (instead of the one-dimensional one) is introduced purely for technical b stand for convenience, ωph is the photon frequency (b a+ , b a are photon creation and annihilation operators) and ω and X 1 the vibrating mirror-wall frequency and position operator, respectively . The coupling parameter g can be elucidated in terms of input parameters in the following way. Because of photon gas pressure, which is equal to the one third of the photon gas energy density, under assumption that the amplitude of mirror vibrations is much smaller then the cavity length, that is, X ≪ l, there is in fact a constant force acting in outward direction on the moving wall FX =
ωph nph s , 3ls
where s denotes its surface area and l stands for the cavity length. Hence the coupling parameter: g = ωph /3l. b Yb , Z, b P b operators in terms of the standard One can construct a particular representation for the deformed X, b = x b = zb, b , for an arbitrary α (α 6= 1) [4]: X position and momentum operators, x b, yb, zb, p b, Yb = yb, Z ∗ Electronic 1
address: [email protected] Detailed derivation of this Hamiltonian in one-dimensional case can be found in [5, 6].
2
1 h α/2 i 1−α b = p b 1 − (α − 1) ßb P p2 .
When α = 2, one arrives at a well known result of [7]. It is worth noticing that the case α > 1 (α < 1) corresponds to the assumption that the length fluctuation δℓ = ßα/2 ℓ1−α decreases (increases) with increasing of ℓ. A specific feature for the deformed QM with α > 1 is that in this case the state vectors are represented by the cutoff Fourier representation2, but it is less important for us as we are dealing with the objects having the localization with much greater than the Planck length. By using this particular representation, the Hamiltonian (2) gets modified as
b = ωph b H a +b a+
2 h α/2 i 1−α b 2 1 − (α − 1) ßb p2 p
2m
The Heisenberg equations of motion read
+
mω 2 x b2 − gb a +b ax b. 2
b a˙ = −iωph b a + ig b ax b , pb˙x = −mω 2 x b + gb a+ b a , pb˙y = pb˙z = 0 , 1+α 2 i h h α/2 i 1−α α/2 1−α α/2 1 − (α − 1) ßb p2 p2 p2 αb px ßb pbx 1 − (α − 1) ßb + . x b˙ = m m
(3)
(4) (5)
The equations for yb(t) and zb(t) have plainly the same form as Eq.(5) with pbx replaced by pby and pbz , respectively. The solution for b a(t) can immediately be written as
b a(t) = b a(0) exp −iωph t + ig
Zt 0
dτ x b(τ ) .
(6)
From this solution it is clear that b a+ (t)b a(t) = b a+ (0)b a(0), which enables one to rewrite the equation for pbx (t) in the form ˙ = −mω 2 x pb b + gb a+ (0)b a(0) . x
(7)
Let us now specify the initial state of the system. For this purpose, we take the coherent state for the photon gas b a(0)|ζi = ζ|ζi, and the minimum uncertainty state3 ! 2 y2 [x − x(0)] 1 z2 p exp − ψ(x, y, z) = exp ipx (0)x − . − √ 4 4(δx)2 4(δy)2 4(δz)2 (2π)3 δxδyδz for the moving mirror. For averaged quantities with respect to these states one gets the following equations p˙x = −mω 2 x + g|ζ|2 , py = pz = 0 , y = z = 0 , 1+α 2 h α/2 i 1−α α/2 i 1−α α/2 h 1 − (α − 1) ßp2x αpx ßp2x px 1 − (α − 1) ßp2x + . x˙ = m m These equations can be derived from classical Hamiltonian
2 3
The interested reader may be referred to [8]. Here we mean the uncertainty relation between the standard position and momentum operators.
(8)
3
H =
2 h α/2 i 1−α p2x 1 − (α − 1) ßp2x
2m
+
mω 2 x2 − g|ζ|2 x . 2
(9)
Let us notice that by adding a linear term to the potential of oscillator - the potential has still the pure oscillator form with the equilibrium position shifted. So, this term in Eq.(9) can merely be dropped as in the paper [9]. The question of experimental detection can be addressed merely by measuring the velocity x˙ for x = 0. The conservation of energy allows one to estimate the value of momentum px for x = 0, 2 h α/2 i 1−α p2x 1 − (α − 1) ßp2x
2m
=
mω 2 x2max , 2
(10)
and then by substituting this value of px in (8) - to determine the velocity at this point. It is almost obvious that the deviation from the standard velocity becomes appreciable when the momentum approaches ß−1/2 (Planck energy scale). Let us consider two specific cases α = 2 and α = 1/2. In the former case, assuming |px | is approaching ß−1/2 (|px | < ß−1/2 ) and usig Eq.(10), one obtains for the velocity at x = 0 m|x| ˙ = |px |
1 + ßp2x (1 − ßp2x )3
1 = (mωxmax ) ß + 2 ≃ 2ß (mωxmax )3 . px 3
Now let us turn to the case α = 1/2. With no loss of generality, we can absorb the factor 1 − α in ßα/2 (for ß is defined up to the numerical factor of order unity). Then under assumption ßp2x ≃ 1 at x = 0, from Eq.(10) for for the velocity (at x = 0) one obtains √ 1/4 1/4 3 24 mωxmax ≃ 4/ ß , m|x| ≃ √ . 1 + 2 ßp2x ˙ = |px | 1 + ßp2x ß
For characterizing the size of the effect - it might be useful to look at the ratio |x|/ωx ˙ max , which in the standard case is equal to 1. Such a big deviation at the macroscopic scales is not natural4 that leads to the opinion to revise the whole discussion from the very outset. For this purpose, let us look at the well known QM description of the classical motion of harmonic oscillator [10]. The wave function
ψ(t, x) =
r 4
mω(x − x(t))2 iωt ipx (t)x(t) mω , exp ipx (t)x − − − π 2 2 2
(11)
where x(t) and px (t) obey the classical equations for harmonic oscillator, p˙ x (t) = −mω 2 x(t) , x(t) ˙ =
px (t) , m
solves the Schr¨odinger equation for harmonic oscillator. It is plain to see that: px (t) = hψ(t)|b px |ψ(t)i, x(t) = hψ(t)|b x |ψ(t)i. On the other hand, this wave function corresponds to the minimum uncertainty state: δx = p √ 1/ 2mω, δpx = mω/2. The classicality of motion means that the quantum fluctuations should be smaller as compared to the classical quantities. It is convenient to characterize the degree of classicality in terms of the energy p2x (t) mω 2 x2 (t) + 2m 2 p2x (t) mω 2 x2 (t) ω = + + . 2m 2 2
hψ(t)|Oscillator Hamiltonian|ψ(t)i = +
4
mω 2 (δx)2 (δpx )2 + 2m 2
This point can easily be missed in the framework of perturbative treatment [9] the validity of which requires ßp2
α/2
≪ 1.
4 Expanding |ψ(t)i into the Eigenfunktionen, |ji, of harmonic oscillator Hamiltonian ∞ iωt |b(t)|2 X bj (t) √ |ji , − |ψ(t)i = exp − 2 2 j! j=0
(12)
where b(t) =
mωx(t) + ipx (t) √ , 2mω
the energy can be expressed in terms of the mean occupation number, n, as p2x (t) mω 2 x2 (t) + = ωn . 2m 2 Therefore, the motion is essentially√classical if n ≫ 1. This discussion tells us that there is an oscillating particle localized within the region δx ∼ 1/ mω and for estimating the gravitational corrections to this system with respect to physics underlying the Eq.(1), one has to use this length scale and look for the corrections in the form (lP /δx)2 . So that, the size of the gravitational correction depends primarily on δx. Certainly, the moving mirror√in optomechanical device with the parameters given in [1], m ≃ MP , ω ≃ 105 Hz, is not localized within the region 1/ MP ω ∼ 10−14 cm but for the moment let us forget about it and ask what might be the size of the gravitational corrections to the classical oscillator in the framework of the above discussion. As it was discussed in the beginning of the paper, because of fluctuating background the oscillator gets modified in such a way as to respect the modified uncertainty relation α o √ o 1n 1n α/2 δxδpx ≥ ß/δx = 1+ 1 + (ßmω) . 2 2 Let us continue of the previous discussion and ascribe this modification to the momentum operator: h √in theαspirit i ß/δx . It makes sense because for the problem under consideration δx is constant. Thus, we are pbx → pbx 1 + just saying that the momentum for oscillating particle, which is localized within the region δx, is increased by the α √ ß/δx . The modified Hamiltonian factor: 1 + h
1+
√ α i2 ß/δx pb2x 2m
+
m′ ω ′2 x b2 mω 2 x b2 pb2 = x′ + , 2 2m 2
m m′ = h √ α i2 , 1+ ß/δx
h √ α i ω′ = ω 1 + ß/δx ,
in view of the above discussion, tells us that the classical motion gets modified as
x(t) = x(0) cos(ω ′ t) , px (t) = −m′ ω ′ x(0) sin(ω ′ t) .
√ In the case when the localization width of the oscillating body, ℓ, is much bigger than δx = 1/ 2mω, one could make a straightforward generalization of this result x(t) = x(0) cos
h √ α i ß/ℓ ωt . 1+
Obviously, it is extremely hard to detect such a deviation from standard α for the realistic input parameters. √picture −8 ß/ℓ becomes ∼ 10−12 for α = 1/2 and Even if one takes an extreme value: ℓ ∼ 10 cm, the correction term
∼ 10−50 for α = 2. Summarizing, we see that a classical motion derived from the nonperturbative treatment of quantum oscillator on the basis of deformed QM (1) is almost obviously in conflict with ”macrophysics”. This point has already been stressed in [8], in the framework of somewhat different nonperturbatie analysis, and also indicated in [11] - in the framweork of perturbative treatment. The inconsistent classical limit derived from the Eq.(1) is connected with the fact that the gravitational correction term in this equation is written in terms of the momentum instead of the length scale probed by the particle. The momentum of classical bodies are usually high but their localization widths are not as small as 1/p. An alternative treatment (in the end of the paper), which has a natural motivation as it is stated in the beginning, is to write the gravitational correction term to the QM in terms of the wave-packet localization width. This way one arrives at the ”reasonably small” corrections, which in the case α = 1/2 may be possibly within the range of observations.
5 Acknowledgments
Useful comments from Zurab Silagadze and David Vitali are kindly acknowledged. The research was partially supported by the Shota Rustaveli National Science Foundation under contract number 31/89.
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