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The Relativistic Bohr Model
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:

mv²/r = α/r²
and hence
r = α/(mv²)

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

m = m0/(1−β²)½

where m0 is the rest mass of the electron.

Kinetic energy K is given by

K(β) = m0c²[1/(1−β²)½ −1]

which, to the first approximation, is equal to ½m0v².

The expression for orbital radius r in terms of relative velocity β is

r = α(1−β²)½/(m0c²β²)

Angular momentum pθ is defined as mvr, but it is quantized to lh, where l is an integer. When pθ is expressed in terms of β and set equal to lh only discrete values of β are allowed; i.e.,

pθ = mvr
or, equivalently
pθ = [m0cβ/(1−β²)½][α(1−β²)½/(m0c²β²)]
which reduces to
pθ = α/(cβ)


α/(cβ) = lh
and hence
β = (α/(ch)(1/l) β = γ/l
where γ=α/(ch)

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 Other Characteristics

The quantization of β then provides a quantization of r. Since

r = α(1−β²)½/(m0c²β²)
β = γ/l
it follows that
r = α(1−(γ/l)²)½/(m0c²(γ/l)²)

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.


V(l) = −(m0c²(γ/l)²)/(1−(γ/l)²)½

The quantization of kinetic energy is not complicated either since K(β)=m0c²[1/(1−β²)½ − 1]


K(l) = m0c²[1/(1−(γ/l)²)½ − 1]

And finally,

E(l) = m0c²[(1−(γ/l)²)½ − 1]

The natural unit of energy is the rest mass energy of the electron; i.e.,

E(l)/(m0c²) = [(1−(γ/l)²)½ − 1]

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 2.4685×1015 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
  Measured Bohr Model  Relativistic Model Finite Mass
Relativistic Model
  wavelength wavelengthProportional
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 -1.445467455.644 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 NumbersError Ratios
Upper LowerRelativistic/Bohr Finite Mass
Finite Mass Relativistic/Relativistic
214.599675851 -4.256077796 -0.925299507
320.634714446 -1.805151176 -2.844036697
312.029571515 -1.246529873 -0.614183764
43-0.12887841 0.454905954 -3.52972973
42-1.985905567 5.689922481 -2.86515259
413.998611111 -3.579861111 -0.895276138
540.000456288 -0.086774842 -190.1754386
530.102220037 -0.280525502 -2.744329897
520.223877031 -0.643835616 -2.875844895
511.27260954 -1.081588603 -0.849898236
65-0.520215018 -0.596156241 1.145980451
64-0.709995011 -0.348378513 0.49067741
63-0.090782665 0.26164295 -2.882080513
62-0.318600159 0.919895466 -2.887303852
610.785771704 -0.671757771 -0.854901961
73-0.11094739 0.253842805 -2.287956535
72-0.280012704 0.8729621 -3.11758034
830.325717267 -0.810328638 -2.487828315
820.435205522 -1.273611547 -2.926459986
92-0.34814257 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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