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Assumptions

The Bohr theory provides a very simplified description of the motion of an electron around an atomic nucleus.  In this spirit, we need to make the following assumptions about this motion:

The electron can be treated as a particle.

The motion of the electron around the nucleus is circular (though in a later section we will relax this assumption to see what happens).

The atomic nucleus is stationary (though this assumption can be relaxed).

The electron does not radiate any electromagnetic radiation in its orbit.

This final assumption is contrary to what classical physics predicts.  In classical physics, any charged particle that undergoes an acceleration should radiate electromagnetic radiation.  Our orbiting electron should therefore be radiating as it goes about its orbit, losing energy in the process.  The electron would then spiral into the nucleus.  However, we know that atoms are stable, so we will hold onto this assumption and see what happens.

Equation of Motion

With these assumptions, we get an orbital motion something like that shown in the figure below.

The motion of the electron around the nucleus can be described by noting that the centripetal acceleration required to maintain the electron in its orbit is provided by the attractive electric force between the positively charged nucleus and the negatively charged electron.  Assume that the nucleus is a single proton, so that the atom we are dealing with is a hydrogen atom.  The equation of the orbit can be written as

where  is the mass of the electron, v is its orbital velocity, e is the magnitude of the charge on a proton (hydrogen nucleus) and the orbiting electron, and we are using SI units to specify the magnitude of the electrostatic force between the electron and nucleus.

Introducing Quantisation

Now we come to the most important assumption of the Bohr theory.  It is with this assumption that we start to inject some “quantum” ideas into the analysis.

The only allowed values of the electron angular momentum in its orbit are those that are equal to an integral multiple of Planck’s constant divided by a factor 2π.

Symbolically this condition becomes

where n is an integer.  An interesting way to interpret this condition is to recall that the de Broglie wavelength of an electron with momentum p is equal to

so that this assumption is equivalent to writing

That is, the circumference of the electron’s orbits must equal an integral number of de Broglie wavelengths.  This sounds reasonable –  if we were considering a wave travelling in a circular motion such that the circumference of the circle were not an integral number of wavelengths, then the wave would undergo destructive interference and could not persist in its motion.  Similarly, we know that an electron can have wavelike properties, so there is sense in requiring that the allowed stable orbits of the electron are those that would not result in destructive interference of the de Broglie wave.

Electron Energy

Finally, we need to consider the energy of the electron in its orbit around the nucleus.  This is simply the sum of the kinetic and potential energies of the electron, given by

The potential energy of the electron is negative because the electric force between the electron and the nucleus is attractive.  Now from the equation of motion given above, we have

so that the total energy is given by

Thus the total energy is equal to half the potential energy.  A similar result also applies to circular gravitational orbits.

In the next section, we will use these formulae to calculate the allowed radii of the electron’s orbits, and the energies of those orbits.

On to Bohr Theory of the Hydrogen Atom