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 sub-nano-meter 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.
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DOI: 10.3923/jas.2010.784.785
URL: https://scialert.net/abstract/?doi=jas.2010.784.785
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):
(1) |
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):
(2) |
where, and Aji 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 Fabry-Perot 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 Sv or Sλ. The quantity Svdv and Sλdλ 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:
(3) |
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 Sv has the units of time or Hz-1. It can be converted to sλ in length units (such as nm-1) by the relation:
(4) |
where, λm 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 Sv. 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 Sv 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:
(5) |
where, vm 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:
(6) |
Equation 6 is the corrected spectral line profile of natural broadening. Comparing Eq. 6 with the non-corrected spectral line profile of natural broadening we obtain:
(7) |
In wavelength units the half intensity width is:
(8) |
where, λm is the wavelength of maximum intensity. From Eq. 2 and 8 we obtain the corrected wavelength formula as:
(9) |
And comparing Eq. 9 with 8 we can write:
(10) |
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.
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