In this Essay I summarize what I think were the four key postulates that enabled Bohr to understand the energy levels of the hydrogen atom and the radiation emitted by hydrogen. These postulates remain valid.
1. Stationary states
Bohr postulated, first, that atoms can exist in various states of motion, each state of motion being distinct and characterized by a fixed energy. Such a state is called a stationary state. (Note, of course, that the electrons themselves are not stationary in a stationary state.) A corollary of this idea of stationary states is that energy is a quantized variable in atoms, limited to certain discrete values.
2. Quantum jumps
In 1913, when Niels Bohr took on the task of unraveling atomic structure, scientists knew that a quantum principle governed radiation. This led Bohr to abandon all hope of a classical description of the processes of emission and absorption of light. According to classical radiation theory, an electron in an orbit around a positively charged nucleus should radiate continuously; the radiated frequency should change steadily to create spectral bands, not spectral lines; and the atom should collapse finally to the size of the nucleus. None of these things happened. Bohr postulated that an atom undergoes sudden transitions (quantum jumps) from one stationary state to another. He replaced the idea of smooth steady radiation by an idea of occasional tiny explosions, in each of which the electrons suddenly alter their state of motion and a bundle of light energy is emitted. (Bohr followed Planck’s thinking, not Einstein’s. He assumed that although matter emits energy in lumps, radiation remains continuous. He used Planck’s formula E = hf, in the same sense that Planck had used it. In 1913 he had not yet accepted the reality of light “corpuscles”—later to be called photons. His thinking, albeit it erroneous from a modern perspective, had no effect on his analysis. He was studying matter, not radiation.)
In a letter to Bohr commenting on a pre-publication copy of Bohr’s paper, Ernest Rutherford wrote, “There appears to me to be one grave difficulty in your hypothesis, which I have no doubt you fully realise, namely, how does an electron decide what frequency it is going to vibrate at when it passes from one stationary state to the other? It seems to me that you would have to assume that the electron knows beforehand where it is going to stop.” Rutherford’s question, although a very good one (it had to wait many years for an answer), betrayed one mode of classical thinking that Bohr was willing to abandon. Classically, in order to emit radiation of frequency f, an electron must vibrate at this same frequency f. According to the idea of stationary states connected by quantum jumps, there can be no such simple connection between radiated frequency and the electron’s frequency of motion. An electron may vibrate at one frequency in its initial state, a different frequency in its final state. The frequency of the radiated light, although related to these frequencies, is equal to neither of them. Fortunately, Bohr let neither the unpredictability nor the nonvisualizability of the quantum jump deter him from advancing the idea. Quantum mechanics subsequently made possible calculations of the probability of transitions (part of the problem that worried Rutherford). These transitions, or quantum jumps, remain as nonvisualizable as they were to Bohr, and any particular one nearly as unpredictable.
3. Submicroscopic energy conservation
As mentioned in Essay Q11, Bohr, in developing his theory of atomic structure, held onto as much of classical physics as seemed to work in the atomic domain. In particular he assumed that the law of energy conservation remained rigorously valid for electrons and the emitted bundles of radiation.1 This assumption made possible a simple explanation of the Ritz combination principle, which has been advanced by Walther Ritz in 1908. Ritz pointed out that often the sum of two frequencies radiated by a particular kind of atom is equal to a third frequency emitted by that atom. Consider three stationary states of an atom with energies E1, E2, and E3, as shown in the figure below.
If the atom undergoes a quantum jump from the third to the second state, energy conservation requires that the emitted photon (using modern terminology) carry away exactly the energy difference between these two states:
hfa = E3 – E2 .
For a quantum jump from the second to the first state, energy conservation imposes a similar condition,
hfb = E2 – E1 .
Conservation of energy for the leapfrog transition from the third to the first state can be similarly expressed,
hfc = E3 – E1 .
It is obvious from these equations that the frequencies of the light for these three possible transitions are simply related by the equation,
fc = fa + fb .
This is consistent with the Ritz combination principle.
The additive rule does not apply to wavelengths, and energy conservation explains why. Bohr was not the first to assume that energy is conserved in atomic systems, but he was the first to make good use of the assumption. The significance of the Ritz combination principle as a simple manifestation of energy conservation could be appreciated only after Bohr had broken loose from the idea that radiated frequency is equal to electron vibration frequency.
4. The correspondence principle
The three ideas outlined above—stationary states, quantum jumps, and submicroscopic energy conservation—are evidently intertwined. Together they provide a simple description (not, to be sure, a classically visualizable description) of the processes of emission and absorption of radiation by atoms, and they relate atomic spectra to atomic mechanics. Bohr’s fourth key idea provided an essential bridge between the classical and quantum worlds. His bridge is the correspondence principle, the idea that quantum mechanics must have a “classical limit.” This idea of the classical limit was already a part of relativity theory, where it took a simple form. When particles move slowly compared with the speed of light, the relativistic description of their motion reduces to the classical description. To put it simply, the new theory (be it relativity or quantum mechanics) must agree with the old where the old is known to be correct.
In his theory of the hydrogen atom, Bohr used the correspondence principle in this way. He postulated that classical mechanics should approximately correctly describe an atomic transition when that transition takes place between two stationary states whose fractional difference in energy and in other properties is very small. Through the correspondence principle he could get hold of the mysterious quantum jump. The new and unknown was tied to the old and familiar. In practice this means that an electron moving in a highly excited energy state far removed from the nucleus does have a mechanical frequency of motion nearly the same as the frequency of radiation that it emits as it jumps down sequentially through closely spaced energy states.
1 Years later, Bohr at least temporarily entertained the idea that energy might not be conserved in the nuclear process of beta decay because electrons shot out of particular radioactive nuclei with a range of energies, all less than the energy difference between the two nuclear states. Pauli’s postulated neutrino (1930) and Fermi’s theory of beta decay (1934) saved the day for energy conservation. The electron’s energy plus the neutrino’s energy add up to exactly the energy lost by the nucleus.