# Development of Quantum Physics VIII – Tiny Jumping Planets, Everywhere

The previous article introduced the following problem: Atoms can absorb and emit radiation of certain energies only. There’s a quite simple formula for the hydrogen atom which is able to describe the absorption and emission lines in its spectrum. But how do atoms have to be built in order to produce the observed spectra? In our case: How does a hydrogen atom look like? We already found out that it has to have a nucleus, in which almost all of the atom’s mass is concentrated, and also a comparatively huge electron shell.

After much effort, Neils Bohr was able to construct a model of the hydrogen atom which seemed to fit the experimental data. It’s this very model which I’d like to discuss in this article.

Niels Bohr (around 1922)
(Source: www.nobelprize.org)

(I’ll spare the mathematical backgrounds of Bohr’s model here, although they are quite easy to understand. But I’ll write a seperate article about them for the interested reader.)

You can find all the other articles of this series here.

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Bohr’s atom model is also known as the “planetary model”. There’s a simple reason for this: In Bohr’s model, the electron circles around the atom’s nucleus like a tiny planet orbiting a central star. (For the sake of simplicity, we will only deal with the hydrogen atom – so our atom consists of just one electron and a nucleus.) If we treated our system of particles really thoroughly, we would have to describe a situation in which the electron and the nucleus circle a common barycenter, but we can make our lives a lot easier if we assume that the atomic nucleus sits in the center of the electron’s circular orbit motionlessly. In doing so, we don’t make a big error in describing the situation properly, because more than 99.9 % of the atom’s mass is concentrated in the nucleus. Our simplification is legitimate and sufficiently exact.

Well, the radial force, which always points towards the central nucleus (centripetal force) and keeps the electron on its curved trajectory, is the Coulomb force. This is just the technical term for the electromagnetic attraction between the negatively charged electron and the positive nucleus (= one proton).

From the relation “centripetal force equals Coulomb force” alone, we can derive an expression for the radius of the electron’s orbit. By now, every radius is allowed. (The distance between electron and nucleus can be, for example, 5 (arbitrary) units of length or 38 units or 26.1 units or every other value imaginable.)

Now, let’s introduce some quantum physics – suddenly it gets a bit trickier!

Classical particles (as electrons) can be described in terms of their matter wave, as I explained before. Let’s see what will happen if we apply this knowledge to the planetary model!

The electron, which have been considered as a particle so far, from now on will be treated as kind of a wave which forms around the atomic nucleus. But the wave has to obey certain conditions in order not to fall into the nucleus or leave the atom. (We are interested in these particular cases only, because the hydrogen atom is obviously stable over time!) The condition goes as follows: “A stationary (= time independent) state requires standing waves.” Equally, we could say that the circumference of the electron’s trajectory has to be an integer multiple of the matter wave’s wavelength.
However, how can we find out the electron’s wavelength? – This, too, has been illustrated before. We have to use Louis de Broglie’s formula for calculating a “particle’s” wavelength:
$\lambda_D=\frac{h}{m\cdot v}$. ($h$ is the Planck constant, $m$ is the mass of the particle, and $v$ is its speed. It all describes the de Broglie wavelength which is denoted as $\lambda_D$.)

Suddenly, there aren’t infinitely many possible radii for the electron path anymore, but instead discrete ones only! (And it’s all because of the postulation that only standing matter waves result in stable atoms!)

A possible standing (or stationary) wave around the nucleus.
(Source: uni-ka.the-jens.de)

Imagine you drawing a wave around the nucleus: The (“horizontal”) distance between the peeks or troughs always has to be of the same size. If you reach the starting point again after making a single revolution (= if you’re able to “close” the wave after a revolution), you drew a standing wave. Now, the distance between two peeks (or troughs, respectively) equals the wavelength of the standing wave. You can of course shorten (or maybe even lengthen) it by a certain amount and get other stationary waves. But the important thing here is that there are wavelengths which don’t result in standing waves and therefore cannot form a stable atom.

Another possible standing wave.
(Source: http://web.physik.rwth-aachen.de)

Thus, the electron is only allowed to be in certain areas around the nucleus. There is, moreover, a smallest possible radius for the electron paths: It’s value is $5.2917\cdot 10^{-11}~m$ ($\approx 0.5~\AA$).  (A little side effect: This smallest radius can also be obtained by taking Heisenberg’s Uncertainty Principle and including the electron’s energy. It’s always pleasant to have a consistent theory, isn’t it? 😉 )

In a nutshell, as soon as we describe electrons in terms of their matter waves and assume that the stable orbits correspond to standing (or stationary) waves, we see that the “allowed” orbital paths of the electrons become quantized.

Equipped with these insights, we are now able to understand the absorption and emission spectra of the hydrogen atom.
If light (i. e. light quanta of the energy $E=h\cdot\nu$) hits the atom, the atom’s electron can absorb the incoming energy – but only if the law of conservation of energy is satisfied! Since the radii of the electron’s possible circular orbits are quantized and discrete, the electron isn’t able to soak up arbitrary energies, it only can absorb energies which put it on other possible orbits directly. The conservation law of energy – combined with the Bohr model – implies that atoms can absorb light of certain frequencies only! (This goes for energy as well, because energy and frequency are proportional via $E=h\cdot\nu$.) Photons can only be absorbed by an atom if the photon energy corresponds directly to the energy which has to be spent to raise the electron to another orbital path.

Once raised, the electrons tend to drop back to the innermost orbit. Like everything else in nature, electrons “want” to be in a state of minimum energy. This lowest-energy state equates to the innermost orbit, and therefore electrons in higher orbits fall down again almost immediately after being raised. At it, the conservation of energy certainly has to be fulfilled! This results in the electron emitting the same amount of energy which was absorbed by it earlier. The emission of the energy happens in the form of a photon ($E=h\cdot\nu$).

Now that we have an idea of the energy household of an atom, the absorption and emission spectra of the hydrogen atom actually seem to make sense, don’t they?
In the following picture you can see spectra of hydrogen. The characteristic lines in the spectra of other atomic sorts look different, which is due to the differing distribution of the atomic energy levels (= “electron orbits”). Hence, the electrons of other atoms have to absorb and emit other amounts of energy.

Absorption and emission spectrum of hydrogen.
(Source: www.astronomyknowhow.com/hydrogen-alpha.htm)

Is the Bohr model the whole truth?

I’m afraid I have to disappoint you – there are several problems with this model, which means that it cannot fully correspond to reality. Let me point out some of its complications here:

• According to classical electrodynamics, the orbiting electron has to emit electromagnetic radiation, because that’s just what moving charges do. If it emitted radiation, it would lose energy and fall into the nucleus (due to the attracting Coulomb forces). But the Bohr model just assumes that this very thing doesn’t happen. In this respect, it contradicts classical electrodynamics.
• Atoms with more than a single electrons aren’t described by the Bohr model.
• The Bohr model ignores the theory of relativity, although the electron already is moving at one per cent of the speed of light.
• There are other split-ups of the hydrogen’s energy levels which we discovered experimentally (the anomalous Zeeman effect and the Lamb shift, for expample).

Nevertheless, we have to grant Bohr’s model that it is able to accurately describe many experimental results. There’s of course some sort of aesthetic appeal to it due to the analogy between the atom model and our macroscopic solar system.

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