This Quantum World/Serious illnesses/Bohr

BohrEdit

In 1913 Niels Bohr postulated that the angular momentum   of an orbiting atomic electron was quantized: its "allowed" values are integral multiples of  :

  where  

Why quantize angular momentum, rather than any other quantity?

  • Radiation energy of a given frequency is quantized in multiples of Planck's constant.
  • Planck's constant is measured in the same units as angular momentum.

Bohr's postulate explained not only the stability of atoms but also why the emission and absorption of electromagnetic radiation by atoms is discrete. In addition it enabled him to calculate with remarkable accuracy the spectrum of atomic hydrogen — the frequencies at which it is able to emit and absorb light (visible as well as infrared and ultraviolet). The following image shows the visible emission spectrum of atomic hydrogen, which contains four lines of the Balmer series.

 
Visible emission spectrum of atomic hydrogen, containing four lines of the Balmer series.

Apart from his quantization postulate, Bohr's reasoning at this point remained completely classical. Let's assume with Bohr that the electron's orbit is a circle of radius   The speed of the electron is then given by   and the magnitude of its acceleration by   Eliminating   yields   In the cgs system of units, the magnitude of the Coulomb force is simply   where   is the magnitude of the charge of both the electron and the proton. Via Newton's   the last two equations yield   where   is the electron's mass. If we take the proton to be at rest, we obtain   for the electron's kinetic energy.

If the electron's potential energy at infinity is set to 0, then its potential energy   at a distance   from the proton is minus the work required to move it from   to infinity,

 

The total energy of the electron thus is

 

We want to express this in terms of the electron's angular momentum   Remembering that   and hence   and multiplying the numerator   by   and the denominator   by   we obtain

 

Now comes Bohr's break with classical physics: he simply replaced   by  . The "allowed" values for the angular momentum define a series of allowed values for the atom's energy:

 

As a result, the atom can emit or absorb energy only by amounts equal to the absolute values of the differences

 

one Rydberg (Ry) being equal to   This is also the ionization energy   of atomic hydrogen — the energy needed to completely remove the electron from the proton. Bohr's predicted value was found to be in excellent agreement with the measured value.

Using two of the above expressions for the atom's energy and solving for   we obtain   For the ground state   this is the Bohr radius of the hydrogen atom, which equals   The mature theory yields the same figure but interprets it as the most likely distance from the proton at which the electron would be found if its distance from the proton were measured.