A Simple Remark Leading to a Basic Precision Estimate for Non-Relativistic (NR) Real Values of Quantum Mechanics Operators

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1. Introduction

It is customary in almost all text books on Quantum Mechanics that all physical quantities, as ${m}_{e}$ electron mass, e electric charge, and h are not given with an infinite precision leaving the physicist to ask the following question.

Will a small uncertainty on the fundamental nature constant h or $\hslash =\frac{h}{2\text{\pi}}$ ,

(leaving outside electric charge e and electron mass ${m}_{e}$ ) blur the expected results that one finds using h, such as energy levels for most electron systems constrained to a Hamiltonian? Just because this basic uncertainty ( $\delta \hslash $ ) exists! The purpose of this letter is to show how a small variation $\delta \hslash $ , that one introduces in the NR Hamiltonian ${H}_{0}$ can produce an effect on numerical quantum observable such as energy levels. This work puts forward a way to calculate analytically the uncertainty of a quantum datum, such as ${I}_{H}$ the ionization energy of hydrogen, when one pays attention to the slight modification of the Hamiltonian ${H}_{0}$ , inserting $\hslash \pm \delta \hslash $ rather than h in the

kinetic energy ${T}_{c}=-\frac{{\hslash}^{2}}{2{m}_{e}}{\Delta}_{r}$ .

2. Non-Relativistic Hamiltonian

This atomic unit system serves to simplify numerical coefficients to establish the Hamiltonian ${H}_{0}$ and to solve the Schrödinger equation and find the wave function. In the end, the numerical results are obtained restoring the numerical values of ${m}_{e}$ , e electric charge and h. The simple equations written below hint at a way to show how to estimate the error on of quantum observables, such as impulse p or energy E when making the assumption that the Planck constant h is known within the experimental precision:

${H}_{0}=-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}+V\left(r\right)=E$ (1)

${H}_{\delta \hslash}=-\frac{{\left(\hslash \pm \delta \hslash \right)}^{2}}{2m}{\Delta}_{r}+V\left(r\right)=E\pm \Delta E$ (2)

subtracting (3) from (2)

${H}_{\delta \hslash}-{H}_{0}=\pm \Delta E$ (3)

Taking the positive sign for the correction term: $\left(\hslash +\delta \hslash \right)$

then:

${H}_{\delta \hslash}-{H}_{0}=+\Delta E$ (4)

Neglecting terms $O\left(\delta {\hslash}^{2}\right)$ in the subtraction ${H}_{\delta \hslash}-{H}_{0}=\Delta E$ . We arrive simply to:

$\Delta E=\frac{\delta \hslash}{\hslash}.\frac{{\hslash}^{2}}{m}{\Delta}_{r}$ (5)

$\frac{{\hslash}^{2}}{2m}{\Delta}_{r}=\frac{\Delta E}{2}.\frac{\hslash}{\delta \hslash}$ (6)

There is the kinetic energy term:

${T}_{c}=\frac{{p}^{2}}{2m}=-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}$ (7)

Let'us use the virial theorem:

$2\langle {T}_{c}\rangle +\langle U\rangle =0$ (8)

Replacing:

$\langle {T}_{c}\rangle =-\langle \frac{U}{2}\rangle =\frac{\Delta E}{2}\mathrm{.}\frac{\hslash}{\delta \hslash}$ (9)

3. Relativity Correction

At this point I will show that the relativity term that appears below (that is H) compared to ${H}_{0}$ is indeed negligible, for what are correction terms order of $\delta {\hslash}^{2}$ and greater. Using the relativity energy correction term [1] to the ${H}_{0}$ Hamiltonian:

$H={m}_{e}{c}^{2}\left(\sqrt{1+\frac{{p}^{2}}{{m}_{e}^{2}{c}^{2}}}-1\right)-\frac{{e}^{2}Z}{r}$ (10)

The quantity $\frac{{p}^{2}}{{m}_{e}^{2}{c}^{2}}$ is dimensionless so that the hamiltonian H is indeed an energy. (Mechanics of the atom pp 202-204)

$H={H}_{0}+{H}_{1}$ (11)

Considering $\alpha =\frac{{p}^{2}}{{m}_{e}^{2}{c}^{2}}$ , a small quantity, and proceeding to a Taylor expansion order of ${\alpha}^{2}$ for the square root term, the correction ${H}_{1}$ is given by:

${H}_{1}=-\frac{{p}^{4}}{8{m}_{e}^{3}{c}^{2}}$ (12)

${H}_{1}={E}_{1}$ with:

${E}_{1}=-\frac{{p}^{4}}{8{m}_{e}^{3}{c}^{2}}$ (13)

${p}^{2}=-{\hslash}^{2}{\Delta}_{r}$ (14)

${E}_{1}=-\frac{1}{2{m}_{e}{c}^{2}}\frac{{\hslash}^{4}}{4{m}_{e}^{2}}{\left({\Delta}_{r}\right)}^{2}$ (15)

Replacing the square impulse operator in ${H}_{1}$ lead to a ${p}^{4}$ term that is :

$-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}+V\left(r\right)-\frac{{p}^{4}}{8{m}_{e}^{3}{c}^{2}}=E+{E}_{1}$ (16)

${T}_{c}=\frac{{p}^{2}}{2m}=-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}$ (17)

As before, one adds the relativity term ${H}_{1}$ to both Equations (1) and (2) to give:

$-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}+V\left(r\right)-\frac{{p}^{4}}{8{m}_{e}^{3}{c}^{2}}=E+{E}_{1}$ (18)

$-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}+V\left(r\right)-\frac{1}{8{m}_{e}{c}^{2}}\frac{{\left(\hslash {\Delta}_{r}\right)}^{2}}{4{m}_{e}^{2}}=E+{E}_{1}$ (19)

The same is perfomed as in Equations (2) and (3). That is:

$-\frac{{\hslash}^{2}}{2m}{\Delta}_{r}+V\left(r\right)-\frac{1}{2{m}_{e}{c}^{2}}\frac{{\hslash}^{4}{\Delta}_{r}^{2}}{4{m}_{e}^{2}}=E+{E}_{1}$ (20)

$-\frac{{\left(\hslash \pm \delta \hslash \right)}^{2}}{2m}{\Delta}_{r}+V\left(r\right)-\frac{1}{2{m}_{e}{c}^{2}}\frac{{\left(\hslash \pm \delta \hslash \right)}^{4}{\Delta}_{r}^{2}}{4{m}_{e}^{2}}$ (21)

$=E\pm \Delta E+{E}_{1}\pm \Delta {E}_{1}$ (22)

Using symbolic Mathematica, it is easy to develop the product ${\left(\hslash \pm \delta \hslash \right)}^{4}$ and to subtract Equation (22) from Equation (21). It is shown that leaving out all orders $\ge O\left(\delta {\hslash}^{2}\right)$ in the subtraction there is only one term order $O\left(\delta \hslash \right)$ coming from the relativity correction:

$\Delta {E}_{1}=-\frac{{\Delta}_{r}^{2}{\hslash}^{3}\delta \hslash}{2{c}^{2}{m}_{e}^{3}}$ (23)

This corrected term
$\Delta {E}_{T}$ can be evaluated with the NR
$\Delta E$ (or T_{c}) term as:

$\Delta {E}_{T}=\Delta E+\Delta {E}_{1}=\Delta E\left(1-\frac{\Delta E}{2{m}_{e}{c}^{2}}\right)$ (24)

Numerical Effect on Physical Constants

To check this on the value of the ionization potential I_{H} of hydrogen:
${I}_{H}=13.606\text{\hspace{0.17em}}\text{eV}$ we can use the ionization energy of hydrogen I_{H} to give a value to
$\langle \frac{U}{2}\rangle $ neglecting the irrelevant sign:

$\Delta E={I}_{H}\cdot \frac{\delta \hslash}{\hslash}$ (25)

$\Delta {I}_{H}={I}_{H}\cdot \frac{\delta \hslash}{\hslash}$ (26)

It is enough to define ${m}_{e}{c}^{2}=511\text{\hspace{0.17em}}\text{keV}$ for the rest mass of the electron. The result is:

$\Delta {E}_{T}={I}_{H}\left(1-\frac{{I}_{H}}{2{m}_{e}{c}^{2}}\right)\frac{\delta \hslash}{\hslash}$ (27)

The quantity $-\left(\frac{{I}_{H}}{2{m}_{e}{c}^{2}}\right)$ is numerically:

${E}_{cor}=1.81138\times {10}^{-4}$ (28)

That uctuating energy is on the order of the uncertainty:

$\frac{\delta \hslash}{\hslash}={10}^{-8}$ (29)

$\Delta {I}_{H}=13.606\times {10}^{-8}\text{\hspace{0.17em}}\text{eV}$ (30)

$\Delta {E}_{T}=13.6058\times {10}^{-8}\text{\hspace{0.17em}}\text{eV}$ (31)

Equivalent energies in quantum mechanics are found in [2] Mécanique Quantique and can be linked to an equivalent temperature T_{k}:

${T}_{k}=1.36\times 11605\times {10}^{-7}\approx 0.00157\text{\hspace{0.17em}}\text{K}$ (32)

$1\text{\hspace{0.17em}}\text{eV}\approx 11605\text{\hspace{0.17em}}\text{K}$ (33)

4. Conclusions and Suggestions

I can define the temperature T_{T} with the corrected value
$\Delta {E}_{T}=13.6058\times {10}^{-8}\text{\hspace{0.17em}}\text{eV}$ . It will give an uncertainty on
$\delta {T}_{k}=\pm 2.321\text{\hspace{0.17em}}\text{nK}$ . The relativity correction gives an uncertainty 10^{−3} compared to the fluctuation caused by the basic uncertainty

$\frac{\delta \hslash}{\hslash}={10}^{-8}$ . I can conclude that an uncertainty such as $\delta \hslash $ on the Planck

constant can produce an important (in se) fluctuation for energy in atomic systems enough to be experimentally seen. Elsewhere an ultra cold temperature obtained by ultra cold atoms could detect this effect, and can be used to determine an ultra precise h Planck constant. Recalling that ultra cooled atoms can be cooled to a temperature
$T=1\text{\hspace{0.17em}}\mu \text{K}$ well under the temperature T_{k} [3] .

${T}_{k}=157\text{\hspace{0.17em}}\mu \text{K}$ (34)

obtained with the newly uncertainty on Planck constant used for S.I. units system [4] :

$\frac{\delta \hslash}{\hslash}={10}^{-8}$ (35)

Acknowledgements

The author expresses his deep thanks to Pr. Ronald Mc Caroll, for the the improvement of the manuscript, and thanks to Dr Régis Courtin for his critical reading of this paper and correcting the english grammar.

References

[1] Born, M. (1967) Mechanics of the Atom. Ungar, New York.

[2] Cohen-Tannnoudji, C., Diu, B. and Laloe, F. (1973) Mécanique Quantique. Hermann, Paris.

[3] Cohen-Tannoudji, C. (1990) Atomic Motion in Laser Light. In: Dalibard, J., et al., Eds., Les Houches Summer School, Session LIII, Elsevier, New York.

[4] Quinn, T.J. (1995) Base Units of the Système International d’Unités, Their Accuracy, Dissemination and International Traceability. Metrologia, 31, 515-527.

https://doi.org/10.1088/0026-1394/31/6/011