In 1897, J.J. Thomson measured the ratio of charge to mass of the electron and also measured the velocity of electrons in a cathode-ray tube (something like a TV tube). In 1909, Millikan was able to show that charge existed only in quantized amounts. All charge was some integral multiple of the electron charge, which Millikan found to be equal to 1.6´ 10^-19 C. With Thomson’s measurement, this fixed the mass of the electron as 9.1´ 10^-31 Kg. Since matter was neutral and since the electron’s mass could not account for the density of matter, there had to be more massive positive charges that existed. However, the size and distribution of the positive and negative charges was unknown. In 1911, Rutherford scattered helium nuclei (radioactively emitted, positively charged particles called alpha particles) off a thin gold foil. He found that a significant number of the alpha particles were back-scattered through 180 degrees. This could happen only if there were very small positively charged particles in the gold foil. This led to a model of the atom in which there was a positively charged nucleus which had a radius on the order of 10^-15 m and electrons which orbited the nucleus at a radius of about 10^-10 m. Thus, most of "matter" was made up of nothing at all! However there were problems associated with this model.

There were two basic problems associated with the planetary model of the atom. The first of these was the stability of atoms. An accelerating electron emits radiation. Simple estimates indicate that an electron orbiting a nucleus should radiate away all its energy and fall into the nucleus in a time on the order of 10^-9 s. If this were the case, there would be no atoms at all as we know them. The second problem related to the spectra of atoms. Individual atoms were known to give off radiation at specific frequencies. In other words, the spectra consisted of a distinct set of lines. Any theory of atoms would have to explain why this occurred. Moreover the frequencies of the lines obeyed an empirical formula, f=C_nm(1/n²-1/m²), where C_nm is a constant and n and m are positive integers.

To address these problems, Niels Bohr, in 1913, proposed a theory of the hydrogen atom. There wasn’t much of a physical basis for the model, but it seemed to work. Bohr assumed that the electron in the hydrogen atom could take on only quantized energy amounts. It is a little easier to restate the assumption in terms of a quantity called angular momentum. Simply stated, Bohr’s first postulate is "The electron orbits the proton in circular orbits having quantized angular momentum L=mvr=nÑ (it seems that h-bar appears as N-tilde when viewed on the web), where n=1,2,3,4,…" As a consequence of this postulate and Newton’s Second Law (see course pack notes on Bohr Atom) it also follows that the speed is quantized, v=ac/n (a=k_ee²/Ñ c> 1/137 is known as the fine structure constant), the radius is quantized, r=n² a₀ (a₀=Ñ /amc > 5.29´ 10^-11 m is known as the Bohr radius) and energy is quantized, E= -E_R/n² (E_R=(1/2)a²mc²=13.6 eV is known as the Rydberg energy). Thus the electron can take on only the set of values shown below:

As we shall see, radius and speed are not quantized in the correct quantum theory, but angular momentum and energy are quantized.

To explain the stability and spectrum of the hydrogen atom, Bohr used his second postulate, "The electron radiates or absorbs energy only when changing orbits. If it goes from an orbit with n=n₁ to a lower orbit with n=n₂, the frequency of radiation emitted is f=E_R(1/n₂²-1/n₁²)/h." In this way the hydrogen atom is stable since the electron in the n=1 orbit does not have a lower orbit to go to and it does not radiate in the n=1 state, only when changing orbits. Moreover, the frequencies predicted by the Bohr formula were in excellent agreement with the experimentally observed frequencies. To get the frequency of a transition simply subtract the two energies and divide by h. For example, in going from n=3 to n=2, the change in energy is –1.51 eV – (-3.4 eV)=1.89 eV. The corresponding frequency is 1.89 eV/4.14´ 10^-15 eV s =4.57´ 10¹⁴ Hz, which corresponds to a wavelength of c/f=657 nm (visible light in the red).

Thus Bohr was able to quantitatively explain the structure of atoms and give their size, even if his theory was a bit ad hoc.