How Bohr Put The Quantum In Mechanics

In this article I`ll shortly go over the derivation of the quantised energy levels of electrons in atoms using only A-level knowledge.

By the time Bohr came up with his model of electrons orbiting a nucleus at set distances, a few things were already known about atoms:

- The nucleus is tiny compared to the whole atom.
- The nucleus contains most of the mass of an atom, and is therefore considered stationary (in the same way that we consider the sun to be stationary as the earth revolves around it)

Let`s look at the atom from a classical point of view first:

The nucleus contains a number Z of protons, each carrying a charge of +e, elementary charge.
An electron in orbit around the nucleus has charge -e.

We know that the electrostatic force of attraction between the two is then :

F = Ze² / 4*pi*eps*r²

To make things a bit easier to read, let`s make k = 4*pi*eps so that we just have

F = Ze² / kr² .

If an electron is to have a stable orbit, this has to equate centripetal force

F = Ze² / kr² = m_ev² / r

Hence kinetic energy 1/2 m_ev² can be written as KE = Ze² / 2kr

The total energy of the electron is then

E = KE + PE = Ze² / 2kr - Ze² / kr² = -Ze² / 2kr where PE is just the electrostatic interaction.

Now according to this equation, energy is more negative, i.e. the atom is more stable, for small r.
Hence classically, the electron would prefer to collapse into the nucleus rather than stay at a set distance.

This is where the cool stuff comes in:
Bohr, in a fit of randomness/genius/magic, decided to impose a arbitrary rule on his model. He decided that angular momentum, L, should be quantised, meaning it is only allowed to have a set of specific values.
There was no scientific basis for that assumption, and there still isn`t, apart from the fact that it works extremely well to explain what we see from measurements.

The quantisation is given by L = mvr = nh / 2pi where h is planck`s constant and n is the integer responsible for the quantisation. So we see angular momentum can only come in multiples of h / 2pi.

Now comes a bit more math:

we have mvr = nh / 2pi.
we know that mv is momentum p, and that E = p² /2m_e .

This leads to E_n = -Ze² / 2kr_n identical as classically, except that r can now only take set values, and thus E can only have set values.
r_n here is given by r_n = n²a₀ / h where we call a₀ the bohr radius, which is constant at 0.053 nm.

Now that the math is over with, let`s see what we got out of that:
- r can only take set values, that increase as a square series (1,4,9,16,...)
- There is a minimum energy E₀ associated to an electron in the smallest orbit, the ground state.
- If we supply more energy (i.e. make energy less negative) we can increase the radius of the electron orbit in set amounts, until we supply enough E to completely detach the electron from the atom, like a rocket reaching escape velocity.

This idea of quantization that Bohr came up with out of the blue laid the foundations on which Schrodinger, Heisenberg, Dirac and many others build our understanding of the extremely tiny.