Quantum Physics is not Weird. On the Contrary.. Paul J. van Leeuwen

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      Bohr now took Planck's quantum concept - plus Einstein's photon - and applied it in his quantum atom model to explain why the electron did not crash into the nucleus and discovered thus the origin of the Rydberg formula for all the known spectra of hydrogen. In his proposal the electrons could exist only in specific distinct orbits where each orbit had a certain fixed electron energy level: quantized orbits so to speak. On a transition from a higher energy orbit to a lower one, an electron would emit a photon with an amount of energy exactly equal to the energy difference between the two quantized orbits.

      Figure 4.7: Bohr's atomic model with permitted electron orbits.

      Source: Wikimedia Commons.

      According to Bohr, the energy difference between two permitted orbits, ΔE, corresponded to the frequency of the photon according to Planck's formula: ΔE=h.f. This scenario resulted indeed in the Rydberg formula. Bohr's proposal for the explanation of the hydrogen emission spectrum - 1913 - evoked great interest but was considered as reaching too far outside established physics at that time. It did not help that Bohr was not able explain why only these special orbits were allowed by nature.

      In his quantum model, Bohr proposed boldly that the transition from an electron from one orbit to another had to be instantaneous - that is, without intermediate time and location - because otherwise EM radiation energy would already be radiating during transition. It may be clear that his idea was utterly inconsistent with classical physics and certainly not in accordance with the common ideas about the behavior of the material universe. His idea was therefore difficult to accept immediately as a valid scientific model. However, in his idea we can already discern the first echo of something that later would become the quantum collapse and non-locality. It was admitted anyhow as remarkable in Bohr's proposal that the relationship between wavelength and energy levels 1,2,3,4,5 and 6 accurately delivered the Rydberg formula.

      Figure 4.8: The Bohr electron orbits of the hydrogen atom with their energy jumps as the source of the spectral lines.

      Source: Wikimedia Commons.

      In Bohr's model, see figure 4.8, the electrons could only jump between the orbits corresponding to specific permitted energies. He assigned to each orbit a number n, which is a positive integer starting from 1, the lowest energy orbit. There are of course an unlimited number of possible orbits but let us limit them here to the first 6.

      With these 6 orbits every possible jump between the orbits n = 1 to 6 is allowed. There exist 5+4+3+2+1 = 15 possible ways to jump from a higher to a lower energy orbit. This corresponds with 15 possible energy differences ΔE_n-m. An electron can jump:

       from m = (2,3,4,5,6) to the n = 1 orbit, 5 possibilities,

       from m = (3,4,5,6) to the n = (1,2) orbits, 4 additional possibilities,

       from m = (4,5,6) to the n = (1,2,3) orbits, 3 additional possibilities,

       etc.

      A 4-1 jump will then produce a photon with a wavelength of 97 nm, see figure 4.8. The jumps to the lowest orbit n = 1 correspond exactly with the Lyman series, the jumps to the next lowest orbit n = 2, with the Balmer series and the n = 3 jumps with the Paschen series.

      The following exercise is only for those who would like to do a little bit of mathematics, otherwise just go straight to the conclusion:

      You can verify the Bohr energies yourself with the wavelengths given in figure 4.8 with a slightly modified version of the Planck formula: ΔE = (h.c)/ λ. The Planck constant h = 6.626 × 10⁻³⁴ J.s (Joule seconds), the speed of light c = 3 x 10⁸ m/s and 1 nm = 10^-9 m. (nm stands for nanometer).

      .

      The value ΔE_2-1 stands for energy difference between orbit 2 and orbit 1. Figure 4.8 gives a wavelength = 122 nm for that. With that you can calculate the transition energy ΔE = (6.626 × 10⁻³⁴ x 3 x 10⁸)/(122 x 10^-9) = 0.163 x 10^-17 J. Now calculate the transition energy ΔE_3-2 from orbit 3 to orbit 2 with = 656 nm. Finally calculate the direct transition energy from orbit 3 to orbit 1 ΔE_3-1 with = 103 nm. Then the following should be true:

      .

      - ΔE_3-1= ΔE_3-2 + ΔE_2-1

      .

      In the same way you can demonstrate:

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      - ΔE_4-1= ΔE_4-2 + ΔE_2-1= ΔE_4-3 + ΔE_{3-1 =} ΔE_4-3 + ΔE_3-2 + ΔE_2-1

      The apparent conclusion would indeed be that only certain fixed energy levels for electrons, orbiting the hydrogen nucleus, were allowed. But the fundamental reason for that restriction of nature remained at that time a mystery for Bohr and his contemporaries. It was a compelling fact that the measured wavelengths of the spectral lines exactly matched the differences between a limited set of energy levels, but that match was not convincing enough for scientific acceptance. More evidence, together with an acceptable explanation for those permitted energy levels, was needed. A surprising but acceptable explanation was proposed in 1923. This was the result of the daring and inspired insight of a French prince. Before going into that explanation, we will first pay some attention to standing waves in strings.

      Harmonic electron waves

      Figure 4.9: Seven harmonically vibrating modes of a string. From tonic to 6^th harmonic.

      Bron: Wikimedia Commons.

      A vibrating string has only very specific ways in which a standing wave can arise and continue. The condition is that an integer number of half wavelengths fits exactly on the length of the string. Which means ½λ, 1λ, 1½λ, 2λ, 2½λ, etc. You get the lowest vibration - the tonic - if exactly ½λ fits on the string, with 1λ fitting you get the 1st harmonic, with 1½λ fitting on the string you get the 2nd harmonic, and so on. This means that there will always be a lowest vibration for a given string, to evoke a lower tone is impossible. The possible vibrations are clearly 'quantized' and have a lowest limit value. This phenomenon inspired a prince to a rather daring idea.

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