Various theories of quantum gravity predict the existence of a minimum length scale, which leads to the modification of the standard uncertainty principle to the Generalized Uncertainty Principle (GUP). In this paper, we study two forms of the GUP and calculate their implications on the energy of the harmonic oscillator and the hydrogen atom more accurately than previous studies. In addition, we show how the GUP modifies the Lorentz force law and the time-energy uncertainty principle.
1. Introduction
Developing a theory of quantum gravity is currently one of the main challenges in theoretical physics. Various approaches predict the existence of a minimum length scale [1, 2] that leads to the modification of the Heisenberg Uncertainty Principle
(1)ΔxΔp≥ℏ2,
and to the Generalized Uncertainty Principle (GUP) [3, 4]
(2)ΔxΔp≥ℏ2(1+β(Δp)2+ζ),
where β=β0lP2/ℏ2,β0 is a dimensionless constant usually assumed to be of order unity, lP≡ℏG/c3 is the Planck length lP≃1.616×10-35m, and ζ may depend on 〈p〉 but not on Δp. The second term on the RHS above is important at very high energies/small length scales (i.e., Δx~lP).
In this paper, we study two forms of the GUP. The first (GUP1) [5, 6] is
(3)ΔxiΔpi≥ℏ2[1+β((Δp)2+〈p〉2)+2β((Δpi)2+〈pi〉2)],
which follows from the modified commutation relation [6]:
(4)[xi,pj]=iℏ(δij+β(p2δij+2pipj)).
The second (GUP2) [7, 8] is
(5)ΔxΔp≥ℏ2[1-2α〈p〉+4α2((Δp)2+〈p〉2)].
which follows from the proposed modified commutation relation [7]:
(6)[xi,pj]=iℏ(δij-α(pδij+pipjp)+α2(p2δij+3pipj)),
where α=α0lP/ℏ=α0/MPc and α0 is a constant usually assumed to be of order unity. In addition to a minimum measurable length, GUP2 implies a maximum measurable momentum.
The commutation relation (4) admits the following representation in position space [9, 10]:
(7)xi=x0i,pi=p0i(1+βp02),
where x0i,p0i satisfy the canonical commutation relation [x0i,p0j]=iℏδij. This definition modifies any Hamiltonian near the Planck scale to [9, 10]
(8)H=p022m+V(r)+βmp04+β22mp06.
Similarly, (6) admits the definition [7, 8]
(9)xi=x0i,pi=p0i(1-αp0+2α2p02),
leading to the perturbed Hamiltonian
(10)H=p022m+V(r)-αmp03+5α22mp04-2α3mp05+2α4mp06.
The aim of this paper is to study the impact of GUP1 and GUP2 on the energy of the harmonic oscillator and hydrogen atom more accurately than previous studies. In addition, we show how the GUP modifies the Lorentz force law and the time-energy uncertainty principle.
2. Harmonic Oscillator
The harmonic oscillator is a good model for many systems, so it is important to calculate its energy accurately to compare it with future experiments. Recently a quantum optics experiment was proposed [11] to probe the commutation relation of a mechanical oscillator with mass close to the Planck mass.
The effect of GUP1 on the eigenvalues of the harmonic oscillator was calculated exactly in [12]. The effect of GUP2 was considered in [8] to first and second order for the ground energy only. In this section, we consider first and second order corrections to all energy levels for both GUPs to compare them, and we use the ladder operator method, which is simpler than the other methods.
2.1. GUP1-First Order
The momentum p0 can be expressed using the ladder operators [13, Page 49] as
(11)p0=iℏmω2(a†-a),
where a† is the raising operator: a†|n〉=n+1|n+1〉 and a is the lowering operator: a|n〉=n|n-1〉. Thus, the change in energy to first order due to H′=βp04/m+β2p06/2m is
(12)ΔEn(GUP1)(1)=〈n|H′|n〉=βm(ℏmω2)2〈n|(a†-a)4|n〉-β22m(ℏmω2)3〈n|(a†-a)6|n〉.
Applying the raising and lowering operators and simplifying
(13)ΔEn(GUP1)(1)=3β0lP24mω2(2n2+2n+1)+5β02lP4m2ω316ℏ(4n3+6n2+8n+3).
Therefore, the relative change in energy is
(14)ΔEn(GUP1)(1)En=3β0lP2mω4ℏ(2n2+2n+1)n+1/2+5β02lP4m2ω216ℏ2(4n3+6n2+8n+3)n+1/2.
The first term in (13) differs from that derived in [12] by a factor of three because instead of the commutation relation (4) they use the relation [x,p]=iℏ(1+βp2).
2.2. GUP1-Second Order
The second order correction can be calculated using second order perturbation theory [13, Page 256]
(15)ΔEn(GUP1)(2)=∑m≠n|〈m|H′|n〉|2En(0)-Em(0),H′=βmp04.
Expanding and neglecting terms with equal number of a and a†(16)〈m|H′|n〉=βm(ℏmω2)2〈m|×(a†a†a†a†-a†a†a†a-a†a†aa†hhhhh-a†aa†a†-a†aaa-aa†a†a-aa†aahhhhlh-aaa†a-aaaa†+aaaa)|n〉.
Applying the raising and lowering operators:(17)〈m|H′|n〉=βm(ℏmω2)2×((n+1)(n+2)(n+3)(n+4)δm,n+4-(4n+6)(n+1)(n+2)δm,n+2-(4n-2)n(n-1)δm,n-2+n(n-1)(n-2)(n-3)δm,n-4).Because of the delta functions and the orthogonality of the eigenfunctions, squaring the above expression means squaring each term individually. After simplifying and dividing by En(18)ΔEn(GUP1)(2)En=-m2ω2lP4β028ℏ2(34n3+51n2+59n+21)n+1/2.
2.3. GUP2-First Order
For GUP2, H′=(-α/m)p03+(5α2/2m)p04-(2α3/m)p05+(2α4/m)p06. The p03 and p05 terms do not contribute to first order because they are odd functions. The first order correction for the p04 and p06 terms is the same as (14) with β→5α2/2 and β2→4α4:
(19)ΔEn(GUP2)p04(1)En=15lp2α02mω8ℏ(2n2+2n+1)n+1/2+5α04lP4m2ω24ℏ2(4n3+6n2+8n+3)n+1/2,
which agrees with the expression derived in [8] when n=0.
2.4. GUP2-Second Order
The second order correction for the p03 term can be calculated using the same method that led to (18)
(20)ΔEn(GUP2)p03(2)=∑m≠n|〈m|H′|n〉|2En(0)-Em(0),H′=-αmp03,(21)〈m|H′|n〉=iαm(ℏmω2)3/2×((n+1)(n+2)(n+3)δm,n+3hhhhl-3(n+1)n+1δm,n+1+3nnδm,n-1hhhhl-n(n-1)(n-2)δm,n-3).
Squaring and substituting in (20)
(22)ΔEn(GUP2)p03(2)=α2m2(ℏmω2)3×[(n+1)(n+2)(n+3)-3ℏω+9(n+1)3-ℏωhhhhh+9n3ℏω+n(n-1)(n-2)3ℏω].
Simplifying and dividing by En(23)ΔEn(GUP2)p03(2)En=-mωlP2α028ℏ(30n2+30n+11)n+1/2,
which agrees with the expression derived in [8] when n=0.
The second order correction for the p04 term is the same as (18) with β→5α2/2:
(24)ΔEn(GUP2)p04(2)En=-25m2ω2lP4α0432ℏ2(34n3+51n2+59n+21)(n+1/2).
Adding (14) and (18) we get for GUP1
(25)ΔEn(GUP1)En=3β0lP2mω4ℏ(2n2+2n+1)n+1/2-3β02lP4m2ω216ℏ2(16n3+24n2+26n+9)n+1/2.
Adding (19), (23), and (24) we get for GUP2
(26)ΔEn(GUP2)En=mωlP2α022ℏ1n+1/2-15m2ω2lP4α0432ℏ2(46n3+69n2+77n+27)n+1/2.
It is interesting to note that to O(α2), the effect of GUP2 is to add a constant shift to all energy levels.
To compare (25) and (26) with experiment, consider an ion in a Penning trap; its motion is effectively a one-dimensional harmonic oscillator [14]. The accuracy of mass determination increases linearly with charge, so let us suppose it is possible to use completely ionized lead atoms, which have an atomic number of 82. Suppose that the magnetic field in the Penning trap is B=10T. The cyclotron frequency is ωc=qB/m; substituting the value of mωc≃820e in (25) and (26) we get the results shown in Table 1 for different n.
GUP-corrections to the energy of the harmonic oscillator.
n
ΔEn(GUP1)/En
ΔEn(GUP2)/En
0
4.9×10-52β0-3.6×10-103β02
3.2×10-52α02-2.7×10-102α04
2
1.3×10-51β0-2.3×10-102β02
6.5×10-53α02-1.6×10-101α04
5
2.7×10-51β0-9.9×10-102β02
3.0×10-53α02-7.1×10-101α04
10
5.1×10-51β0-3.5×10-101β02
1.5×10-53α02-2.5×10-100α04
100
4.9×10-50β0-3.2×10-99β02
1.6×10-54α02-2.3×10-98α04
Figure 1 is a plot of (25) and (26), as a function of n. It is clear that the difference between the corrections of GUP1 and GUP2 increases with increasing n. That difference might prove useful in future experiments to differentiate between the two GUPs.
The relative change in energy due to GUP1 and GUP2 as a function of n, assuming β0=α0=1.
The best accuracy for mass determination for stable ions in a penning trap is [14] δm/m=1×10-11, which sets an upper bound on β0 when n=100 of β0<2.0×1038 and on α0 when n=1 of α0<1.7×1020. These bounds can be lowered in future experiments by using Penning traps with higher mass determination accuracy, ions with higher charge, and stronger magnetic fields.
3. Hydrogen Atom
The effect of GUP1 on the spectrum of the hydrogen atom was calculated to first order in [15] by doing the integral to find the expectation value of the perturbed Hamiltonian. In this section, we use a simpler method, adopted from [13, Page 269], to get the same result. After that, we calculate the effect of GUP2 on the spectrum of hydrogen, which, to my knowledge, was not done before.
The GUP1-corrected Hamiltonian for Hydrogen takes the form
(27)H=p022m-kr+βmp04,
where k≡e2/4πϵ0, the change in energy to first order can be found as follows:
(28)ΔEn(GUP1)=〈ψ|H′|ψ〉=βm〈p02ψ∣p02ψ〉,
where we used the hermiticity of p02=2m(En+k/r). Thus,
(29)ΔEn(GUP1)=4βm〈(En+kr)2〉=4βm(En2+2Enk〈1r〉+k2〈1r2〉).
Using the relations [13, Page 269]:
(30)〈1r〉=1n2a0,〈1r2〉=1(l+1/2)n3a02,
where a0=4πϵ0ℏ2/me2≈5.3×10-11m is the Bohr radius, (29) becomes
(31)ΔEn(GUP1)=4βmEn2(1+2kEn1n2a0+k2En21(l+1/2)n3a02).
Using a0=ℏ2/mk and En=mk2/2ℏ2n2, we obtain the relative change in energy
(32)ΔEn(GUP1)(1)En=4βmEn(4nl+1/2-3),
which agrees with the expression derived in [15]. Equation (32) is maximum when n=1,l=0 giving:
(33)ΔE1(GUP1)(1)E1≈9.3×10-49β0.
The GUP2-corrected Hamiltonian for Hydrogen takes the form
(34)H=p022m-kr-αmp03+5α22mp04.
The change in energy due to the p03 term to first order is zero, because p03 is an odd parity function; thus, its integral over all space is zero.
The effect of the p04 term is the same as (32) with β→5α2/2,
(35)ΔEn(GUP2)(1)En=10α2mEn(4nl+1/2-3).
For n=1,l=0:
(36)ΔE1(GUP2)(1)E1≈2.3×10-48α02.
The second order correction for the p03 term can be found numerically, for the ground state ψ100:
(37)ΔE1(GUP2)p03(2)=∑nlm≠100∞|〈nlm|H′|100〉|2E1(0)-En(0),H′=-αmp03=αiℏ3m∇(∇2).
From selection rules [13, Page 360] 〈nlm|p|n′l′m′〉=0 except when Δm=±1,0 and Δl=±1, which means that the sum should be taken for l=1,m=-1,0,1. Summing for all states adjacent to |100〉 (e.g., up to n=10), since their contribution is greater
(38)ΔE1(GUP2)(2)=α2ℏ6m2E1∑n=2,l=1,m=0,±1n=1011-1/n2hhhhhhhhhhhhhh×|∫02π∫0π∫0∞ψnlm∇∇2(ψ100)r2hhhhhhhhhhhhhhhhhhhhhhhhh∫02π∫0π∫0∞ψnlm×sinθdrdθdϕ|2.
The gradient of the Laplacian of ψ100 in spherical coordinates is
(39)∇∇2(ψ100)=1πr2(1a03)3/2e-r/a0(2a02+2a0r-r2)r^.
Substituting in (38) and taking into consideration that r^=sinθcosϕx^+sinθsinϕy^+cosθz^ leads to
(40)ΔE1(GUP2)(2)E1≃6.2×10-52α02
which is much less than (36), and thus can be neglected; this also happens to all other states.
Figure 2 is a plot of (32) and (35) as a function of n for different l; we see that the two GUPs have almost the same effect on the spectrum of hydrogen. The best experimental measurement of the 1S-2S transition in hydrogen [16] reaches a fractional frequency uncertainty of δf/f=4.2×10-15 which sets an upper bound on β0 of β0<4.5×1033 and on α0 of α0<4.2×1016.
GUP-corrections to the spectrum of the hydrogen atom.
4. Modified Lorentz Force Law
Because the GUP modifies the Hamiltonian, one expects that any system with a well-defined Hamiltonian is perturbed [9], perhaps even classical Hamiltonians. The impact of the GUP2-corrected classical Hamiltonian on Newton's gravitational force law was examined in [17]; here, we derive a modified Lorentz force law.
For a particle in an electromagnetic field, the GUP1-modified Hamiltonian is [5]
(41)H=12m(p0-qA)2+βm(p0-qA)4+qφ,
differentiating with respect to p0, we get
(42)r˙=∂H∂p0=1m(p0-qA)+4βm(p0-qA)3.
Using inversion of series, we get
(43)p0=qA+mr˙-4β(mr˙)3+O(β2).
Substitution in ℒ=p0·r˙-H leads to
(44)ℒ=(mr˙-4β(mr˙)3+qA)·r˙-12m(mr˙-4β(mr˙)3)2-βm(mr˙-4β(mr˙)3)4-qφ.
which simplifies to:
(45)ℒ=mr˙22-βm3r˙4+qA·r˙-qφ.
Applying the Euler-Lagrange equation (d/dt)(∂ℒ/∂r˙)-(∂ℒ/∂r)=0 we obtain
(46)mr¨-12βm3r˙2r¨=q∇(A·r˙)-qdAdt-q∇ϕ.
The RHS is q(E+v×B), which means that the Lorentz force law becomes
(47)F≡mr¨=qE+v×B1-12βm2v2,
which is approximately
(48)F≃q(E+v×B)(1+12βm2v2).
Using the same method as above, the GUP2-corrected Hamiltonian takes the form [8]
(49)H=12m(p0-qA)2-αm(p0-qA)3+5α22m(p0-qA)4+qφ,
differentiating with respect to p0 and using inversion of series, we get
(50)p0=qA+mr˙+3α(mr˙)2+8α2(mr˙)3+O(α3),
leading to the Lagrangian
(51)ℒ=mr˙22+αm2r˙3+2α2m3r˙4+qA·r˙-qφ.
from which we obtain
(52)F=qE+v×B1+6αmv+24α2m2v2,
which is approximately
(53)F≃q(E+v×B)(1-6αmv).
The new term in (48) and (53) depends on mv, which means that its effect in high energy physics will be too small even at relativistic speeds. For example, in a proton-proton scattering experiment:
(54)ΔFGUP1F=12βm2v2~10-38β0.
Experimental tests of Coulomb’s law use large, but usually static, masses [18]. For example, coulomb’s torsion balance experiment measures the torsion force needed to balance the electrostatic force; Cavendish’s concentric spheres experiment, and its modern counterparts, use two or more concentric spheres, (or cubes, or icosahedra) [18] to test Gauss’s law.
To test (48) and (53) we need large masses, with moderate velocities. Suppose we have a pendulum with length R and a bob with charge q and mass m swinging above an infinite charged plane with charge density -σ; the electric field will be E=-σ/2ε0 (See Figure 3). Without the GUP effect, the bob will experience a force
(55)F0=-(mg+qσ2ε0)y^.
If θ is the angle between the vertical and the string, the equation of motion for small θ is
(56)mRθ¨≃-(mg+qσ2ε0)θ.
Thus, the angular frequency is
(57)ω02=gR+qσ2ε0mR.
A pendulum under the effect of gravitational and electrostatic forces.
However, if we used (48) for the electrostatic force, then the equation of motion will be
(58)mRθ¨≃-(mg+qσ2ε0(1+12βm2v2))θ.
The velocity can be found from conservation of energy, taking the gravitational and electrical potentials to be zero on the plane
(59)12mv2+(σq2ε0+mg)R(1-cosθ)=(σq2ε0+mg)R(1-cosθ0),
where θ0 is the initial angle, assuming it starts with zero initial velocity. Equation (59) simplifies to:
(60)v2≃(σq2mε0+g)R(θ02-θ2).
The equation of motion will be
(61)mRθ¨≃-(mg+qσ2ε0+6qσβm2ε0(σq2mε0+g)Rθ02)θ.
Thus, the angular frequency is
(62)ω12=gR+qσ2mRε0+6qσβε0(σq2ε0+mg)θ02.
And for GUP2
(63)ω22=gR+qσ2mRε0-3qσαε0θ0σq2mRε0+gR.
Using the values θ0=π/12, σ=1μC/m2, q=2μC, R=1m, m=0.1kg and g=9.807m/s2,
(64)ω0=3.307rad/sec,F0=1.094N,ω1=3.307+3.6×10-4β0,F1=1.094+2.3×10-3β0,ω2=3.307-1.4×10-2α0,F2=1.094-8.6×10-2α0.
These values, I believe, are accessible with current technology and thus can be used to set much lower bounds on the GUP parameters than the best bound [19] of α0<108 from the anomalous magnetic moment of the muon. However, the GUP might not be applicable on large scale; maybe the GUP parameters α0 and β0 are mass dependent.
5. Generalized Time-Energy Uncertainty
Suppose a light-clock consists of two parallel mirrors a distance L apart, the time a photon takes to travel from one mirror to the other is T=L/c, but length cannot be measured more accurately than the Planck length so
(65)T=L±lPc=Lc±tP,
where tP≡Gℏ/c5≃5.4×10-44sec is the Planck time. This shows that the existence of a minimal length scale limits the precision of time measurements. A more rigorous analysis using general relativity and taking into account the gravitational attraction between the photon and the mirrors leads to the same conclusion [1, 20].
The time-energy uncertainty relation can be obtained from the position-momentum uncertainty relation by using p=E/c and t=x/c to give
(66)ΔEΔt≥ℏ2.
GUP1 leads to the generalized time-energy uncertainty relation
(67)ΔEΔt≥ℏ2[1+3βc2((ΔE)2+〈E〉2)],
which implies Δt≥Δtmin=β0tP. GUP2 leads to
(68)ΔEΔt≥ℏ2[1-2αc〈E〉+4α2c2((ΔE)2+〈E〉2)],
which implies Δt≥Δtmin=α0tP.
An important application of the time-energy uncertainty is calculating the mean life τ of short-lived particles, by using the full width Γ divided by two as a measure of ΔE [21]; that is, τ=ℏ/Γ, because Γ is easier to determine experimentally than τ. Applying (67) and (68) instead of (66) leads to an extremely small change in the mean life of particles.
In Table 2, the mass m and the full width Γ are from [22]. The mean life was calculated via (66), while ΔτGUP1 and ΔτGUP2 were calculated via (67) and (68), respectively. The rest mass was used as a measure of 〈E〉.
Effect of the modified time-energy uncertainty principle on the mean life of particles.
Particle
Mass m (MeV)
Full width Γ (MeV)
Mean life τ (sec.)
ΔτGUP1/τ
ΔτGUP2/τ
Z
91.19×103
2.49×103
2.64×10-25
1.7×10-34β0
-2.3×10-17α0
η
547.85
1.30×10-3
5.06×10-19
6.1×10-39β0
-8.9×10-20α0
μ
105.66
2.99×10-16
2.197×10-6
2.2×10-40β0
-1.7×10-20α0
The effect of the generalized time-energy uncertainty principle on the mean life is too small to measure experimentally, but it might affect the Planck era cosmology [23]. In [23] the authors investigate the effect of similar relations to (67) and (68) on the values of the main Planck quantities, like tP, and reach the conclusion that they were larger at the Planck era than now by a factor of (10–104) under specific conditions. If true, then the effect of (67) and (68) on the mean life of particles was greater at the early universe and might leave traces in present day cosmology.
6. Conclusions
In this paper, we investigated some implications of the GUP1 and GUP2. We calculated the GUP-corrections to the energy of the quantum harmonic oscillator for all energy levels to first and second order perturbation, and although the corrections are small, current and future experiments can be used to set bounds on the values of the GUP parameters. We also found that the difference between corrections due to GUP1 and GUP2 gets bigger with increasing n; this may provide a way to experimentally determine which GUP is correct.
Then, we investigated the GUP-effect on the spectrum of atomic hydrogen, because spectroscopy provides increasingly more precise measurements for transition frequencies in atoms. We also found that GUP1 and GUP2 have almost the same effect on the spectrum of hydrogen.
After that, we investigated how the GUP-corrected classical Hamiltonian leads to a modified Lorentz force law. We also found that it might be possible to detect the effect of the modified Lorentz force law with current technology, unless the GUP is only applicable near the Planck scale.
Finally, we saw how the GUP leads to a generalized time-energy uncertainty principle and considered its effect on the mean life of some particles, which was too small to measure experimentally. However, its effect in the early universe might be detectable in present day cosmology.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank Dr. Ahmed Farag Ali for his support and for the interesting discussions we had on the GUP. The author would also like to thank the anonymous referees whose useful comments and suggestions made this paper much better.
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