San José State University |
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of a Hydrogen-like Atom, Original and Relativistic |
The Bohr model of the atom was replaced by the Quantum Mechanics model based upon the Schroedinger equation in the 1920's. The great success of the Bohr model had been in explaining the spectra of hydrogen-like (single electron around a positive nucleus) atoms. The Schroedinger equation replicated this explanation in a more sophisticated manner and the Bohr analysis was considered obsolete. But the Schroedinger equation approach works only for a very limited number of models. Beyond this limited set the Schroedinger equation approach gives no insights, whereas the Bohr model does provide insights into diverse cases. In particular the Schroedinger equation approach cannot be applied to the case which takes into account the relativistic effects. On the other hand, the Bohr analysis can.
The details of the application of the Bohr analysis to a hydrogen like atom in the
relativistic regime is given elsewhere. The significant result is
that the angular momentum is quantized in units of Planck's constant divided by 2π,
h.
The rudiments of the analysis are given here. The potential energy of an electron, V(r), is given by −α/r, where r is the radius of an elecrons orbit and α is a constant equal to the force constant for electrostatic attraction times the square of the charge of an electron. The attractive force is given by −α/r².
In a circular orbit the balance of the attractive force and the centrifugal force requires that:
where v is the orbital velocity of the electron and m is its mass.
For relativistic effects the relevant variable is the velocity relative to the speed of light, β=v/c. The inertial mass m of the electron is given by
where m_{0} is the rest mass of the electron.
Kinetic energy K is given by
which, to the first approximation, is equal to ½m_{0}v².
The expression for orbital radius r in terms of relative velocity β is
Angular momentum p_{θ} is defined as mvr, but it is quantized
to l
Thus
The value of γ for the hydrogen atom is 1/137.036, the so-called fine structure constant. This means the correction for relativistic effects is relatively minor even for the extreme case of l=1. For larger values of the principal quantum number l the correction for relativitic effects is even smaller.
The model would equally well apply to an electron in orbit about a nucleus with a positive charge of Z (a hydrogen-like atom), in which case the value of γ would be Z/137.036. For large values of Z, say 100, this could result in significant correction for relativistic effects. And, of course, for Z≥137 there would be the awkward problem of physical impossiblity of the model results. Fortunately it appears that values of Z are limited to about 110.
The quantization of β then provides a quantization of r. Since
Since r and β are quantized so are potential energy, kinetic energy and total energy.
The quantization of potential energy is simple since V(r)=−α/r.
Thus
The quantization of kinetic energy is not complicated either since K(β)=m_{0}c²[1/(1−β²)^{½} − 1]
Thus
And finally,
The natural unit of energy is the rest mass energy of the electron; i.e.,
For γ=1/137.036 and l=1 the RHS is equal to −2.664×10^{−5}. For l=2 the RHS is −6.66×10^{−6} so a transition from l=2 to l=1 would release an amount of energy equal to 1.997×10^{−5} times the rest mass energy of the electron, which is 0.511 Mev. Thus the energy of the photon released in such a transition should be 10.21 ev or 1.6356×10^{−18} joules. Dividing this energy by Planck's constant gives the frequency of the photon produced by the transition as being 2.4685×10^{15} cycles per second and a wavelength of 1.21513×10^{−7} meters or 121.513 nanometers.
The measured wavelength of the radiation corresponding to this transition is 121.566 nm. Thus the computed value deviaties from the measured value by only 0.0436 of 1%.
For the transitions from l=3 to l=2 and from l=3 to l=1 the wavelength of the radiation are 656.185 nm and 102.527 nm, respectively. The measured values are 656.272 nm and 102.583 nm. The errors are 0.0132 of 1% and 0.0546 of 1%, respectively.
The conventional non-relativistic Bohr model which gives the energy level as 13.605ev/l² gives a photon energy of 10.20 ev which corresponds to a wavelength of 121.651 nm for a transition from the l=2 level to the l=1 level. Compared to the measured wavelength of 121.566 nm this is an error of only 0.01234 of 1%, but the relativistic version of the model has an error of about 0.05 of 1%. For the l=3 to l=2 transition the conventional Bohr model predicts a wavelength of 656.813 nm, which is an error of 0.0824 of 1% compared to the measured value. In this case the relativistic version of the model has notably less error. For the l=3 to l=2 transition the predicted wavelength of the conventional Bohr model is 102.627 nm, an error of 0.0429 of 1%. The low relative error of the Bohr Model for the l=2 to the l=1 level transition is the exceptional case, as the table below shows.
Both versions of the model need to be corrected for the difference between the orbit radius r and the distance between the d. In the literature this is referred to as the correction for the finite mass of the nucleus. The orbit is about the center of mass of the two particles. This makes r=(1836/1837)d. The effective value of γ is then (1/137.036)(1836/1837)=1/137.1106. This changes the predicted wavelength for the l=2 to l=1 transition in the relativistic model to 121.645 nm. The error for this value relative to the measured value is 0.065 of 1%. Although this is a larger error, the correction needs to be made. It happens in this case that the error due to the limitations of the model happened to cancel out some of the error due to not correcting for the difference between orbital radius and distance between the particles.
Comparison of Spectral Wavelengths Computed from Three Models of the Hydrogen Atom with the Measured Wavelengths |
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Measured | Bohr Model | Relativistic Model | Finite Mass Relativistic Model | ||||||
wavelength | wavelength | Proportional Deviation | wavelength | Proportional Deviation | wavelength | Proportional Deviation | |||
Quantum Numbers | (nm) | (nm) | (0f 1%) | (nm) | (0f 1%) | (nm) | (0f 1%) | ||
Upper | Lower | ||||||||
2 | 1 | 121.566 | 121.551 | -0.01234 | 121.497 | -0.05676 | 121.62985 | 0.05252 | |
3 | 2 | 656.28 | 655.987 | -0.04465 | 656.094 | -0.02834 | 656.80898 | 0.0806 | |
3 | 1 | 102.583 | 102.549 | -0.03314 | 102.514 | -0.06726 | 102.62538 | 0.04131 | |
4 | 3 | 1875.01 | 1878.509 | 0.18661 | 1874.559 | -0.02405 | 1876.60175 | 0.08489 | |
4 | 2 | 486.133 | 486.202 | 0.01419 | 485.996 | -0.02818 | 486.52548 | 0.08074 | |
4 | 1 | 97.254 | 97.24 | -0.0144 | 97.198 | -0.05758 | 97.30413 | 0.05155 | |
5 | 4 | 4050 | 3999.407 | -1.24921 | 4049.977 | -0.00057 | 4054.39034 | 0.1084 | |
5 | 3 | 1281.81 | 1278.161 | -0.28468 | 1281.437 | -0.0291 | 1282.83361 | 0.07986 | |
5 | 2 | 434.04 | 433.502 | -0.12556 | 433.925 | -0.02811 | 434.39788 | 0.08084 | |
5 | 1 | 94.976 | 94.932 | -0.04633 | 94.92 | -0.05896 | 95.02359 | 0.05011 | |
6 | 5 | 7400 | 7293.036 | -1.44546 | 7455.644 | 0.75195 | 7463.76748 | 0.86172 | |
6 | 4 | 2630 | 2637.907 | 0.30065 | 2624.386 | -0.21346 | 2627.24537 | -0.10474 | |
6 | 3 | 1093.8 | 1097.182 | 0.3092 | 1093.493 | -0.02807 | 1094.68486 | 0.0809 | |
6 | 2 | 410.174 | 410.535 | 0.08801 | 410.059 | -0.02804 | 410.50607 | 0.08096 | |
6 | 1 | 93.782 | 93.712 | -0.07464 | 93.727 | -0.05865 | 93.82902 | 0.05014 | |
7 | 3 | 1004.98 | 1007.981 | 0.29861 | 1004.647 | -0.03313 | 1005.74182 | 0.0758 | |
7 | 2 | 397.002 | 397.377 | 0.09446 | 396.897 | -0.02645 | 397.32937 | 0.08246 | |
8 | 3 | 954.62 | 953.705 | -0.09585 | 954.322 | -0.03122 | 955.36149 | 0.07767 | |
8 | 2 | 388.9049 | 388.657 | -0.06374 | 388.797 | -0.02774 | 389.22063 | 0.08118 | |
9 | 2 | 383.5384 | 383.844 | 0.07968 | 383.432 | -0.02774 | 383.84991 | 0.08122 |
A comparison of the accuracies of the three models can be made most easily by viewing their relative errors. These are shown below.
Quantum Numbers | Error Ratios | |||
Upper | Lower | Relativistic/Bohr | Finite Mass Relativistic/Bohr | Finite Mass Relativistic/Relativistic |
2 | 1 | 4.599675851 | -4.256077796 | -0.925299507 |
3 | 2 | 0.634714446 | -1.805151176 | -2.844036697 |
3 | 1 | 2.029571515 | -1.246529873 | -0.614183764 |
4 | 3 | -0.12887841 | 0.454905954 | -3.52972973 |
4 | 2 | -1.985905567 | 5.689922481 | -2.86515259 |
4 | 1 | 3.998611111 | -3.579861111 | -0.895276138 |
5 | 4 | 0.000456288 | -0.086774842 | -190.1754386 |
5 | 3 | 0.102220037 | -0.280525502 | -2.744329897 |
5 | 2 | 0.223877031 | -0.643835616 | -2.875844895 |
5 | 1 | 1.27260954 | -1.081588603 | -0.849898236 |
6 | 5 | -0.520215018 | -0.596156241 | 1.145980451 |
6 | 4 | -0.709995011 | -0.348378513 | 0.49067741 |
6 | 3 | -0.090782665 | 0.26164295 | -2.882080513 |
6 | 2 | -0.318600159 | 0.919895466 | -2.887303852 |
6 | 1 | 0.785771704 | -0.671757771 | -0.854901961 |
7 | 3 | -0.11094739 | 0.253842805 | -2.287956535 |
7 | 2 | -0.280012704 | 0.8729621 | -3.11758034 |
8 | 3 | 0.325717267 | -0.810328638 | -2.487828315 |
8 | 2 | 0.435205522 | -1.273611547 | -2.926459986 |
9 | 2 | -0.34814257 | 1.019327309 | -2.927901947 |
The table shows that generally the relativistic version is more accurate than the non-relativistic version there are a number of cases, notably the l=2 to l=1 transition, where the non-relativistic version is more accurate. Likewise the relativistic version corrected for the finiteness of the nuclear mass is more accurate than the non-relativistic version. In a comparison between the relativistic version corrected for the finite mass to the relativistic version without correction it appears the accuracy of the version with the correction is lower than the one without the correction. The positive ratios indicate the cases where the two models being compared are in error in the same direction. The negative ratios are where they are in error in opposite directions.
(To be continued.)
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