
Research Article


A Proposal for Spectral Line Profile of Hydrogen Atom Spectrum in the SubNanoMeter Space Time 

Abbas Farmany



ABSTRACT

Based on the minimal length uncertainty, both the spectral lines and natural broadening of spectral lines of hydrogen atom may be corrected in the subnanometer space time. It is interesting to obtain the related spectral line profile of the corrected hydrogen atom spectrum, which is presented in a proposal for the spectral line profile of hydrogen atom spectrum based on the minimal length uncertainty. It is shown that, based on the minimal length uncertainty, the hydrogen atom spectrum is modified.







INTRODUCTION
Both the spectral lines (Brau, 1999); Akhoury
and Yao, 2003; Kempf, 1997; Udem
et al., 1997; Hossenfelder et al., 2003;
Kempf and Managano, 1997) and the natural broadening
of spectral lines (Farmany et al., 2007, 2008)
of hydrogen atom may be corrected based on the minimal length uncertainty relations.
These corrections are based on the minimal length uncertainty analysis. In this
study we obtain a proposal for the spectral line profile of hydrogen atom based
on the minimal length uncertainty. Let us begin with the minimal length uncertainty.
An exiting quantum mechanical implication of the microphysics is a modification
of the uncertainty principle as (Das and Vagenas, 2008,
2009; Benczik et al., 2005;
Stetsko and Tkachuk, 2008):
where, is
the Planck length. Based on the relation (1), we have obtained the corrected
natural broadening of hydrogen atom spectrum as (Farmany
et al., 2007, 2008):
where, and
A_{ji} are the Planck time and Einstein coefficient, respectively. When
atomic emission, absorption or fluorescence spectra are recorded, narrow spectral
lines are obtained. With ordinary spectrometer, the widths of the lines are
determined not by atomic system but by properties of the spectrometer employed
(slit function and spectral band pass). With very high resolution monochromators
or with FabryPerot interferometer, the actual widths obtained are the result
of a variety of line broadening phenomena. These processes give rise to a spectral
distribution or spectral profile of photons which are called S_{v} or
S_{λ}. The quantity S_{v}dv and S_{λ}dλ
can be interpreted as the fraction of photons with frequencies in the interval
v to v+dv or with wavelengths in the interval λ to λ+dλ. The
spectral distribution function is normalized by:
Because line broadening expression is simpler and easier to interpret than in wavelength units, we shall deal most often with the spectral distribution in the terms of frequency,. The distribution S_{v} has the units of time or Hz^{1}. It can be converted to s_{λ} in length units (such as nm^{1}) by the relation: where, λ_{m} is the peak wavelength. In the thermal equilibrium, the forward rate of a microscopic process must be equal to the reverse rate of that same process which is known as the principle of detailed balancing. The detailed balancing principle allows to state that the emitted and observed photons from a continues radiation field in equilibrium have the same spectral distribution, namely S_{v}. Here, we consider the factors that contribute to the distribution and keep in mind that the results apply equally well to absorption and emission as long as equilibrium conditions prevail. Spontaneous emission of photons leads to an exponential time decay of the excited state population. To determine the frequency distribution S_{v} of the emitted radiation, it is necessary to convert a time domain description to a frequency domain description through Fourier transformation. The Fourier transformation of an exponentially damped sine wave is a Lorentzian function. Since, the normalized spectral profile of the natural broadening is a Lorentzian dispersion function as:
where, v_{m} is the frequency at the line center. Note that the Lorentzian
profile is symmetric with respect to the line center. Combing Eq.
5 with 2 we obtains:
Equation 6 is the corrected spectral line profile of natural broadening. Comparing Eq. 6 with the noncorrected spectral line profile of natural broadening we obtain: In wavelength units the half intensity width is:
where, λ_{m} is the wavelength of maximum intensity. From Eq.
2 and 8 we obtain the corrected wavelength formula as:
And comparing Eq. 9 with 8 we can write:
RESULTS
In this study we have obtained a proposal for spectral line profile of hydrogen
atom spectrum based on the minimal length uncertainty. As the main result of
the present study, it is interesting that the modified spectral line profile
of the hydrogen atom spectrum is presented in a proposal for the spectral line
profile of hydrogen atom spectrum based on the minimal length uncertainty. It
is shown that, based on the minimal length uncertainty, the hydrogen atom spectrum
is modified.
ACKNOWLEDGMENTS The financial supports of Azad University of Ilam Is acknowledged.

REFERENCES 
1: Akhoury, R. and Y.P. Yao, 2003. Minimal length uncertainty relation and the hydrogen spectrum. Phys. Lett. B, 572: 3742. CrossRef 
2: Benczik, S., L.N. Chang, D. Minic and T. Takeuchi, 2005. Hydrogenatom spectrum under a minimallength hypothesis. Phys. Rev. A, 72: 012104012108. Direct Link 
3: Brau, F., 1999. Minimal length uncertainty relation and the hydrogen atom. J. Phys. A, 32: 76917696. CrossRef  Direct Link 
4: Das, S. and E.C. Vagenas, 2008. Universality of quantum gravity corrections. Phys. Rev. Lett., 101: 221301221305. CrossRef  Direct Link 
5: Das, S. and E.C. Vagenas, 2009. Phenomenological implications of the generalized uncertainty principle. Can. J. Phys., 87: 233240. CrossRef  Direct Link 
6: Farmany, A., S. Abbasi and A. Naghipour, 2007. Probing the natural broadening of hydrogen atom spectrum based on the minimal length uncertainty. Phys. Lett. B, 650: 3335. CrossRef 
7: Farmany, A., S. Abbasi and A. Naghipour, 2008. Erratum: Probing the natural broadening of hydrogen atom spectrum based on the minimal length uncertainty. Phys. Lett. B, 659: 913914. CrossRef 
8: Hossenfelder, S., M. Bleicher, S. Hofmannb, J. Rupperta, S. Scherera and H. Stocker, 2003. Signatures in the Planck regime. Phys. Lett. B, 575: 8599. CrossRef 
9: Kempf, A., 1997. Nonpointlike particles in harmonic oscillators. J. Phys. A, 30: 20932110. CrossRef  Direct Link 
10: Kempf, A. and G. Managano, 1997. Minimal length uncertainty relation and ultraviolet regularization. Phys. Rev. D, 55: 79097920. CrossRef  Direct Link 
11: Stetsko, M.M. and V.M. Tkachuk, 2008. Orbital magnetic moment of the electron in the hydrogen atom in deformed space with minimal length. Phys. Lett. A, 372: 51265130. CrossRef 
12: Udem, T., A. Huber, B. Gross, J. Reichert, M. Prevedelli, M. Weitz and T.W. Hänsch, 1997. Phasecoherent measurement of the hydrogen 1S2S transition frequency with an optical frequency interval divider chain. Phys. Rev. Lett., 79: 26462649. Direct Link 



