# Position Dependent Planck’s Constant in a Frequency-Conserving Schrödinger Equation

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

^{17}[4,5,6]. Recent theoretical work includes the impact of time dependent stochastic fluctuations of Planck’s constant [7], and the changes with Planck’s constant on mixed quantum states [8]. An authoritative review of the status of the variations of fundamental constants is given in [9].

_{h}. The fractional variation of Planck’s constant is proportional to the gravitational potential difference and β

_{h}. The value found in [10] for variations in Planck’s constant was |β

_{h}| < 0.007. This parameter is not zero, and is the largest of the violation parameters extracted in the study. The study did not report on the altitude dependence of Planck’s constant above the earth. A very recent study involving the Galileo satellites found that GR could explain the frequency shift of the onboard hydrogen maser clocks to within a factor of (4.5 ± 3.1) × 10

^{–5}[12], improved over Gravity Probe A in 1976 of ~ 1.4 × 10

^{–4}, these are the α

_{rs}redshift violation values that may be compared to β

_{h}.

^{–}

^{1}, and changes in the width of the barrier going as ħ

^{2}. The electromagnetic experiment voltage oscillations were correctly predicted to be 180 degrees out of phase with the radioactive decay oscillations. This data can be made available for independent analysis by requesting it from the author of [19].

^{2}whose representative field squared divided the derivative terms. There is also the field theory of Modified Gravity (MOG) of Moffat, and the Tensor-Vector-Scalar (TeVeS) gravity of Bekenstein. There are many ways all the constants might be represented as fields, and many ways they might be coupled. Coupling fields together in this way is the accepted approach for the treatment of a constant, but is not the only possible approach, and here, something different will be tried.

## 2. Derivation of the Expectation Value Time Derivative

## 3. Time Evolution Operator under F

^{†}, it is seen that U

^{†}≠ U

^{–1}, the normalization is not conserved noting (27), and from (26) for the non-normalized wavefunctions, the expectation values of A are not constants of the motion even if A commutes with F (and therefore U).

## 4. Result for Expectation Values of Operators Commuting with the Frequency Operator F

## 5. Symmetrized Hermitian Frequency Operator F_{h} and modified Schrödinger Equation

^{†}A might also be tried. It was shown in [39] that such a symmetry results in an expectation value that changes with time in inverse proportion to the wavefunction normalization, while the latter is not conserved noting (15).

_{h}= F

_{h}

^{†}, it is seen that U

_{h}

^{†}= U

_{h}

^{–1}, time evolution is unitary, and the normalization is now again conserved,

## 6. Free Particles under F_{h}

_{h}is introduced. The wavefunction time dependence is still given by (5); however, the spatial wavefunction of a free particle is not of the usual form,

_{o}<< 1, so the ηx in the first term of (46) can be dropped. An oscillating solution will be investigated. The result is a second order homogeneous differential equation with solution,

^{2}<8mωħ

_{o}, where the exponential terms can sum to cos(kx) or sin(kx) depending on the boundary conditions, resulting in quantization of frequency in the usual way, by restriction of the allowed values of k.

_{o}and K

_{o}are the modified Bessel functions of the first and second kind, oscillating functions with a decay envelope. The first term of (49) is the relevant one, as it has no divergences. Noting the square root in the argument containing x, there is not a clearly definable constant wavenumber despite that the particle is “free”. Using I

_{o}(iz

^{1}

^{/2}) = J

_{o}(z

^{1/2}) is found the Bessel function of the first kind. For a particle in a box, the infinite sidewall positions must be located such that L

_{1,2}≥ −ħ

_{o}/η, so that ħ is positive. The wavefunctions are then concentrated on the low Planck’s constant side of the box, decaying to the right of the leftmost sidewall. For quantization, the relation between the frequency and the two of the zeroes of the Bessel function Z [J

_{o}] is,

## 7. Lack of Conservation of Energy, Momentum, and Ehrenfest’s Theorems under F_{h}

_{h}− W

_{h}one sees that,

_{h}is “kinetic frequency” acting together to conserve total frequency as the particle moves. Energy is not conserved now, and in addition, even if the particle is free, the momentum is also not conserved, both changing value with position in the absence of an external potential. Frequency, however, is conserved. Changes in V/ħ from a starting to an ending position is the frequency equivalent of work done on or by the system.

_{h}, this author is unable to identify a simple operator for momentum. In light of (51a), a possible momentum operator is (51b),

_{h}is not generally invariant to the infinitesimal displacements owing to ħ(x), therefore,

_{o}k/m. For the particle in a box with a slight ħ gradient of (49) and full solution, it has not been shown that all eigenstates lead to a real result for (56).

^{2}/m)

^{1/2}. If the particle is then suddenly found at any position other than x = 0 with no source of energy, the particle velocity is imaginary, and the magnitude of the imaginary velocity tells you the extent of the energy non-conservation.

## 8. Average Value of ħ under F_{h}

## 9. Time Dependence of the Expectation Value of ħ under F_{h}

_{h}take up total conserved frequency between them, it is interesting to see if there is a simple quantity taken up by ħ distinctly. That is, what quantity is stored in ħ? Since F

_{h}and ħ do not commute,

^{–}

^{1}becomes the “potential frequency”.

## 10. Indeterminacy of ħ under F_{h}

_{h}and ħ do not commute but are Hermitian,

## 11. Uncertainty under F_{h}

_{x}] = iħ(x), it is found that Δp

_{x}Δx ≥ $|\langle \hslash (x)\rangle |/2$. Note that there is an integration over the spatial domain in the latter being performed, or the average of ħ. For frequency and time, it can be seen from the same arguments applied in normal quantum mechanics that,

## 12. Discussion

_{X}traverse the mild ħ-gradient to Y with fixed total frequency ω, where its local energy E

_{Y}can be measured. With no external potential active in the traversal (or the impact subtracted out if there is one), there will be an energy change ΔE

_{YX}= Δħ

_{YX}ω due to the ħ gradient. If experimenters in X and Y communicate and both agree on the frequency and report the local energies, the differences measured in ħ in X and Y could be confirmed. While this may be difficult to arrange, in principle, it can be tested.

_{c}>> ħ. In the latter, it is not that ħ is actually going to zero, rather, masses and kinetic energies are getting very large, and the classical behavior is recovered. In the model of this paper, it is suggested that should ħ be found to vary spatially anywhere, then somewhere else ħ is minimum. Recall in the result of Section 6 of this paper, that wavefunctions are concentrated in areas of lower ħ—particles would want to collect in those regions, for reasons beyond gravity, and in collecting, also approach the classical limit.

^{2}, analyzed as if conserving total frequency Ω

_{TOT}= ω

_{∞}, not total energy. The Newtonian field for a spherical mass of g = −GM/r

^{2}is integrated from ∞ to r to produce the gravitational potential φ = −GM/r, which is then multiplied by the gravitational photon mass, but without inclusion in the prior integration. This approach produces the result of GR for the gravitational frequency shift to first order. So, with no higher theory of a total frequency-conserving stress tensor, the sum of the kinetic frequency and potential frequency are per (65b), from which (65c) follows. Kinetic frequency is what is measured. Note that ħ(r) has cancelled in (65c), and is precisely the same expression derived when ħ is constant. Equation (65a) is the usual expression from GR for a constant ħ, conserving total energy.

_{TOT}= ħ

_{∞}ω

_{∞}with a position dependent ħ, then (65d,e) results. A functional form of the LPI violation for ħ

_{∞}/h(r) is chosen to resemble (65a), written with the Schwarzschild radius R

_{S}= 2GM/c

^{2}. If total frequency is actually conserved and not total energy, the value of the LPI violation parameter β

_{h}returned will be zero, even if ħ is not constant (at least to first order).

_{h}| < 0.007.

_{h}| = 0.007 and the mass and radius of the Earth, (66) results in very small fractional changes near the surface of the Earth relative to infinity, on the order of one part in 10

^{12}. The form (66) does not persist beneath the Earth surface due to volume filling matter. The same order of magnitude for the fractional change is found in the ratio of ħ at the maximum and minimum radii of the Earth’s orbit around the sun. These variations are four orders of magnitude lower than the very best terrestrial laboratory measurement capability, achieving on the order of 10

^{–8}relative uncertainty using the superconducting Watt balance [41]. Therefore, the authors of [10] may have used the GPS data to attempt to measure changes four orders of magnitude smaller than the capability of the very best earthbound metrology, if Equation (66) is operative.

_{h}would have to be 7 to 9 orders larger to account for the difference. At the surface of the sun using |β

_{h}| = 0.007 the fractional change in ħ is 1 part in 10

^{8}, getting closer to the relative uncertainty of the best terrestrial measurement. Thus, (66) may not be the correct description, in light of all the data from the two experiments.

_{h}and look specifically for a systematic change in ħ with altitude may be worthwhile. An independent analysis of the data of the diode experiment in reference [19], along with analyses of the theory of the measurement are both needed. Repeats of all of the experiments by independent investigators with higher precision equipment would be critical.

## 13. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Classical Field Equation of Motion Compared to Frequency-Conserving Schrödinger Equation

#### Appendix A.2. Energy Squared Lagrangian

^{1/2}supporting the field φ is from [25],

^{2}, then multiplying (A3a) by φ/ψ and adding to (A3b) after division by ψ in the latter, one finds for the equation of motion for the combined fields,

^{2}in the large bracketed term of (A10). In Equations (A8) and (A9) occurrences of ∇ψ or ∇

^{2}ψ that are either second order and/or multiplied by x

_{o}are assumed to be negligible, and only the first terms of (A8) and (A9) remain. This condition implies a combination of an early epoch and/or second order spatial changes. Then, substituting the resulting equations (A11), (A12), (A8) and (A9) into (A10),

_{m}is very much larger than the frequency of the non-relativistic field, so that over much less than one cycle of the latter, the term B in (A14) would average to zero. That still leaves the problematic term A.

#### Appendix A.3. Frequency Squared Lagrangian

^{2}= (βψ)

^{2}and the latter is absorbed into L, but appears in the denominator of the mass term,

^{2}in the denominator of the mass term, and that the fields are uncoupled if m = 0. Multiplying (A16a) by ψ and (A16b) by φ/2 and adding them produces,

#### Appendix A.4. Discussion

#### Appendix A.5. Conclusions

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**Figure 1.**Plot of the overall form Equation (49), demonstrating that the wavefunction amplitude increases when Planck’s constant is lower. Planck’s constant increases with increasing x position. The argument of the Bessel function is z = 50(1+x)

^{1/2}.

**Table 1.**The case for a photon (or clock) changing position in gravity radially that would register a detectable change in frequency deviating from GR is when total energy is conserved, and Planck’s constant is position dependent. It is concluded that a variable Planck’s constant may show an apparent consistency with the Einstein Equivalence Principle, to first order, for total conserved frequency, in experiments with clocks and light.

Conserved Quantity | ħ Dependence | ω(r) |
---|---|---|

E_{TOT} = ħ_{∞}ω_{∞} | constant | ω_{GR}(r) |

E_{TOT} = ħ_{∞}ω_{∞} | position dependent | ω_{GR}(r) ħ_{∞}/ħ(r) |

Ω_{TOT} = ω_{∞} | constant | ω_{GR}(r) |

Ω_{TOT} = ω_{∞} | position dependent | ω_{GR}(r) |

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Dannenberg, R.
Position Dependent Planck’s Constant in a Frequency-Conserving Schrödinger Equation. *Symmetry* **2020**, *12*, 490.
https://doi.org/10.3390/sym12040490

**AMA Style**

Dannenberg R.
Position Dependent Planck’s Constant in a Frequency-Conserving Schrödinger Equation. *Symmetry*. 2020; 12(4):490.
https://doi.org/10.3390/sym12040490

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2020. "Position Dependent Planck’s Constant in a Frequency-Conserving Schrödinger Equation" *Symmetry* 12, no. 4: 490.
https://doi.org/10.3390/sym12040490