Ground state energy density of the quantum harmonic oscillator

The total energy of the ground state of the quantum harmonic oscillator is obtained with minimal assumptions. The vacuum energy density of the universe is derived and a cutoff frequency is obtained for the upper bound of the quantum harmonic oscillator.


Introduction
In this paper we find an expression for the ground state energy of the quantum harmonic oscillator (QHO) to obtain the vacuum energy density of the universe. We use the addition of velocities of Einstein's Special Relativity theory to derive a contraction of the Planck time. Using Heisenberg's Uncertainty Principle we obtain an expression for the total energy of the QHO ground state. We also obtain a cutoff frequency for the QHO. This paper is a parallel version of an earlier e-publication by the author [1] which was based on Carmeli's Cosmological Special Relativity theory [2].

Addition of velocities and the contracted Planck time
Consider an inertial reference frame K 1 which moves at a velocity v 1 along the x axis of inertial frame K. If an object is measured to have a velocity v 2 along the x 1 axis relative to the origin of frame K 1 then the velocity v 1+2 of the object relative to the origin of frame K is given by the addition of velocities of Einstein's Special Relativity theory [3], where c is the speed of light in vacuum.
Substituting these variables into (1) gives where approximations are made due to δv ′ ≪ c. Simplifying (2) yields Assume velocity v is very close to the speed of light c such that it is given by For this value of v the second term in the first factor on the right hand side of (3) is given by where we make the approximation again since δv ′ ≪ c. Substituting (5) into (3) and simplifying we obtain Now, define the acceleration constant a 0 = cH 0 , where H 0 is Hubble's constant, and use it to define the expansion velocity parameter where the Planck time[4] T P = hG/c 5 , whereh is the reduced Planck constant and G is Newton's constant. Eq. (7) is just Hubble's law [5] given by V = H 0 L P where L P = cT P is the Planck length. It is a fundamental principle of Special Relativity that the laws of physics are invariant for all observers in inertial reference frames, from which it follows that the physical constants are the same for all inertial observers. Sinceh, G, c and H 0 are physical constants, hence a 0 and therefore V are both constants in any inertial reference frame. Then, substituting from (7) for the velocity in (6) such that δv ′ = V = hG/c 3 H 0 , we obtain the expression for the contracted expansion velocity What this means is that while the observer in K 1 detects that the object has an expansion velocity V, the observer in K will say that the object actually has an expansion velocity of V C relative to the origin of K 1 so that the expansion velocity of the object in K is v 2K = c − V + V C . In order to obtain a time value from the contracted velocity, divide a 0 into (8) to get the time interval We call T P C the contracted Planck time since it has the form

Quantum harmonic oscillator ground state energy
For a simple linear quantum harmonic oscillator (QHO) in one dimension the energy levels for the plane wave modes k and frequencies ω k are given by [7,8] E (k) n = n + 1 2 hω k , where the quantum number n = 0, 1, 2, ... is the state number. The ground state energy for the frequency ω k is given for n = 0, For three linear QHO's oriented along orthogonal axes, and for two polarizations, the ground state energy of the electromagnetic field is given by To get the total energy density ρ vac for all modes k in the ground state |0 we must sum over all oscillator mode frequencies between zero and a finite cut-off frequency ω max to obtain the expected value where E is the total energy of the ground state and V is the volume of the universe. Performing the integration in (14), the relationship between ρ vac and ω max is given by We can obtain an expression for the density by defining the energy E in terms of the contracted Planck time T P C according to Heisenberg's Uncertanty Principle [9] of the form ∆E ∆t ≥h.
Taking ∆t = T P C from (9) and using (16) with E = ∆E, we assume for the ground state the total energy For a universe with Hubble radius R H = c/H 0 the volume is given by Substituting for E and V from (17) and (18) into (14) we get the vacuum energy density [14] ρ where the last right hand side expression is identical in form to the critical energy density ρ c of standard cosmology. We note that this expression for ρ vac was obtained through a general application of the Hubble law, Special Relativity theory and the Heisenberg Uncertainty Principle. Solving (15) for ω max using (19) we obtain Substituting values for the parameters, the value of the cut-off frequency (20) is given by The oscillator ground state energy at the cutoff frequency is which is 0.067% of the upper limit for the electron neutrino rest mass energy of 26 eV of the Kamiokande II experiment [10], also just 0.69% of the upper limit for the electron antineutrino rest mass energy of 2.5 eV of the Troitsk neutrino mass experiment [11] and between 4.3% to 8.6% of the upper limit of the summed neutrino rest mass energies m ν c 2 < (0.2 − 0.4) eV from [12] and the Planck experiment [13]. The value of the vacuum energy density ρ vac ≈ 8.73 × 10 −9 erg cm −3 ≈ 9.71 × 10 −30 gm c 2 cm −3 or equivalent to about 5.8 Hydrogen (HI) atoms per cubic meter. The cosmological constant [14,15,16] is expressed by where κ = 8πG/c 4 is Einstein's constant. Substituting for the parameters we get a value Λ ≈ 1.81 × 10 −56 cm −2 , for a flat universe with no matter. This compares with the High-Z Supernova Search Team experiment [17] where Λ ≈ 1.07×10 −56 cm −2 for a flat universe with H 0 ≈ 65.2 km s −1 Mpc −1 , with vacuum density parameter Ω Λ ≈ 0.72 and mass density parameter Ω M ≈ 0.28.

Conclusion
From a general application of the Hubble law and Special Relativity we obtained a time interval T P C which is 60 orders of magnitude smaller than the Planck time T P . By the Heisenberg Uncertainty Principle we associated this time with energy and derived the vacuum energy density which, from (15) and (20) , can be put in the form Since the ground state energy cutoff (22) is approximately 4% to 8% of the present day upper limits of the sum of neutrino rest mass energies, and as the neutrino mass upper limits appear to be diminishing by an order of magnitude with more sensitive experiments, it is probably not too far-fetched to speculate [18,19] that ǫ max ≈ m ν c 2 .