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 The Bohr Model of the Hydrogen Atom

Niels Bohr's model of the hydrogen atom was one of the first great successes of quantum theory. Its predictions of the wavelengths of the hydrogen spectrum are within a few tenths of one percent or better of the actual values. It failed however in the explanation of the spectra of helium and the higher elements. Nevertheless it is still a valuable theory for providing insights into the mechanism of electron transition phenomena on the subatomic level.

The great success of the Bohr model had been in explaining the spectra of hydrogen-like atoms;i.e., ions with a single electron around a positive nucleus. The Schroedinger equation replicated this explanation in a more sophisticated manner and the Bohr analysis was considered obsolete. But the Schroedinger equation approach can be solved for only 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. See Relativistic Bohr Model.

The details of the application of the Bohr analysis to a hydrogen like atom in the nonrelativistic regime are given here. The significant initial assumption is that the angular momentum is quantized in units of Planck's constant divided by 2π, h. The angular momentum of the electron is mvr, where r is the radius of the elecron's orbit, v is its orbital velocity and m is its mass. Thus

#### mvr = nh

where n is a positive integer which is called the quantum number of the electron.

The potential energy of an electron, V(r), is given by −κ/r, where κ 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 −κ/r².

In a circular orbit the balance of the attractive force and the centrifugal force requires that:

#### mv²/r = κ/r²and hence v² = κ/(rm)

But from the quantization of angular momentum

#### v = nh/(mr) and hence v² = n²h²/(m²r²)

Equating the two expressions for v² gives

#### κ/(rm) = n²h²/(m²r²) which reduces to r = n²h²/(mκ)

This is the quantization condition for the orbit radius. The quantization of the other characteristics of the state of the electron follow from that for r.

Orbital velocity is given by

#### v = nh/(mr) = κ/(nh)

Kinetic energy K is given by

#### K = ½mv² = ½mκ²/(n²h²)

The potential energy V is then

#### V = −κ/r = −mκ²/(n²h²)

From the expressions for K and V it is seen that

#### K = −½V and total energy T is T = K + V = −½mκ²/(n²h²)

If total energy decreases by ΔT half of the decrease goes into increased kinetic energy and the other half goes into an emitted photon. Thus the energy γ is given by

#### γ = ½(mκ²/h²)(1/nF²−1/nI²)

where nI is the initial quantum number for the electron and nF is its final quantum number. The photon energy U is converted into wavelength λ via the relationship

#### λ = hc/U

where c is the speed of light.

Comparison of Spectral Wavelengths
Computed from Bohr Model
of the Hydrogen Atom with
the Measured Wavelengths
Measured Bohr ModelProportional
Deviation
UpperLower wavelength
(nanometers)
wavelength
(nanometers)
Of 1%
2 1 121.566 121.551 -0.01234
3 2 656.28 655.987 -0.04465
3 1 102.583 102.549 -0.03314
4 3 1875.01 1878.509 0.18661
4 2 486.133 486.202 0.01419
4 1 97.254 97.24 -0.0144
5 4 4050 3999.407 -1.24921
5 3 1281.81 1278.161 -0.28468
5 2 434.04 433.502 -0.12556
5 1 94.976 94.932 -0.04633
6 5 7400 7293.036 -1.44546
6 4 2630 2637.907 0.30065
6 3 1093.8 1097.182 0.3092
6 2 410.174 410.535 0.08801
6 1 93.782 93.712 -0.07464
7 3 1004.98 1007.981 0.29861
7 2 397.002 397.377 0.09446
8 3 954.62 953.705 -0.09585
8 2 388.9049 388.657 -0.06374
9 2 383.5384 383.844 0.07968

As can be seen in the table the errors are typically a small fraction of one percent.