Читать книгу Quantum Physics is not Weird. On the Contrary. - Paul J. van Leeuwen - Страница 32

Figure 4.6: Line spectrum of glowing hydrogen gas on a logarithmic scale.

Source: Wikimedia Commons.

One riddle, in the steadily growing mass of new questions that emerged at the end of the 19^th century, that could not be solved by classical physics concerned the existence of spectral lines ^[18]. A hot glowing gas emits EM radiation in only very specific wavelengths or frequencies, a line spectrum. See figure 4.6. Every element we know has its own unique identifying line spectrum signature. As a side: this is how we know that hydrogen is by far the most common element in our visible universe. Those precise lines presented an intriguing puzzle. How did these discrete separate frequencies of a glowing gas come about? Why only these special wave lengths?

The physicists Rydberg, Lyman, Balmer and Paschen had already discovered along empirically ways that a mathematical relationship existed between the wavelengths of glowing hot hydrogen gas, the Rydberg formula ^[19]:

1/= R(1/n²-1/m²) R=1,097373 x 10⁷ m^-1

In this formula, n and m represent integer positive numbers (1,2,3,4,.. etc.) where m> n. Supplying numbers for n and m will yield specific distinct wave lengths. Choosing for instance n=2 and m=3 yields λ = 656 nanometer (nm), which is orange light and is the first Balmer line (Ba-α). The Rydberg formula proved very impressive in its precise predictions, so all the more provocative that a theoretical derivation from classical physical principles had not been found for this rather simple mathematical relation. Nobody at the start of the 20^th century could present a good Newtonian theory as to why glowing hydrogen did not show a continuous spectrum comparable with that of an incandescent lightbulb or a white-hot glowing poker.

In 1912, under the supervision of Rutherford, Niels Bohr ^[20] (1885-1962) investigated as a post-doctoral the structure of the atom. The Rutherford model with a small positive nucleus with fast orbiting electrons, presented, as already mentioned, such great problems that it was not generally accepted as a real possible model by the physics community.

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).

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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:

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- ΔE_3-1= ΔE_3-2 + ΔE_2-1

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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.