6.4 Bohr’s model of the hydrogen atom  (Page 4/37)

Bohr’s model of the hydrogen atom    , proposed by Niels Bohr in 1913, was the first quantum model that correctly explained the hydrogen emission spectrum. Bohr’s model combines the classical mechanics of planetary motion with the quantum concept of photons. Once Rutherford had established the existence of the atomic nucleus, Bohr’s intuition that the negative electron in the hydrogen atom must revolve around the positive nucleus became a logical consequence of the inverse-square-distance law of electrostatic attraction. Recall that Coulomb’s law describing the attraction between two opposite charges has a similar form to Newton’s universal law of gravitation in the sense that the gravitational force and the electrostatic force are both decreasing as $1\text{/}{r}^2,$ where r is the separation distance between the bodies. In the same way as Earth revolves around the sun, the negative electron in the hydrogen atom can revolve around the positive nucleus. However, an accelerating charge radiates its energy. Classically, if the electron moved around the nucleus in a planetary fashion, it would be undergoing centripetal acceleration, and thus would be radiating energy that would cause it to spiral down into the nucleus. Such a planetary hydrogen atom would not be stable, which is contrary to what we know about ordinary hydrogen atoms that do not disintegrate. Moreover, the classical motion of the electron is not able to explain the discrete emission spectrum of hydrogen.

To circumvent these two difficulties, Bohr proposed the following three postulates of Bohr’s model    :

1. The negative electron moves around the positive nucleus (proton) in a circular orbit. All electron orbits are centered at the nucleus. Not all classically possible orbits are available to an electron bound to the nucleus.
2. The allowed electron orbits satisfy the first quantization condition : In the n th orbit, the angular momentum $L_n$ of the electron can take only discrete values:
   
   $L_n=n\hslash ,\text{where}n=1,2,3,...$
   
   This postulate says that the electron’s angular momentum is quantized. Denoted by $r_n$ and $v_n,$ respectively, the radius of the n th orbit and the electron’s speed in it, the first quantization condition can be expressed explicitly as
   
   $m_e{v}_n{r}_n=n\hslash .$
3. An electron is allowed to make transitions from one orbit where its energy is $E_n$ to another orbit where its energy is $E_m.$ When an atom absorbs a photon, the electron makes a transition to a higher-energy orbit. When an atom emits a photon, the electron transits to a lower-energy orbit. Electron transitions with the simultaneous photon absorption or photon emission take place instantaneously . The allowed electron transitions satisfy the second quantization condition :
   
   $hf=|E_n-E_m|$
   
   where $hf$ is the energy of either an emitted or an absorbed photon with frequency f . The second quantization condition states that an electron’s change in energy in the hydrogen atom is quantized.

These three postulates of the early quantum theory of the hydrogen atom allow us to derive not only the Rydberg formula, but also the value of the Rydberg constant and other important properties of the hydrogen atom such as its energy levels, its ionization energy, and the sizes of electron orbits. Note that in Bohr’s model, along with two nonclassical quantization postulates, we also have the classical description of the electron as a particle that is subjected to the Coulomb force, and its motion must obey Newton’s laws of motion. The hydrogen atom, as an isolated system, must obey the laws of conservation of energy and momentum in the way we know from classical physics. Having this theoretical framework in mind, we are ready to proceed with our analysis.