The Rise of Quantum Mechanics

The first decades of the 20th century were clearly witnessing a revolution in the way physicists thought about the structure and behaviour of matter. Not least was the realization that the behaviour of systems with very low energy, such as individual atoms, was not described satisfactorily by the ‘classical’ mechanics embodied in Newton’s Laws of Motion. The first indication of this came when attempts were made to understand the nature of the electromagnetic radiation discussed in the last chapter when emitted from a heated body due to oscillations of its component electrons. If the temperature is sufficiently high then we can actually see the radiation as visible light as, for example, emitted by the filament in an electric light bulb. The light emitted in this case is essentially ‘white’ and, there is a spread in the frequencies emitted. With a bulb, radiant heat is also generated so that infrared rays of lower frequency than visible light are also present. The intensity of the radiation at the different frequencies is characteristic of the temperature of a heated body. For example, at a lower temperature, as with a domestic radiator, no visible light is present at all but there is still a characteristic spread of lower frequencies in the infrared region. An object at higher temperatures will emit visible light and, at extremely high temperatures, most of the radiation will be at frequencies characteristic of ultraviolet light and x-rays. There is a simple relationship (Wien’s Law) between the frequency around which most of the radiation is emitted and the absolute temperature (T) , namely

frequency = ( l0¹¹ T ) oscillations per second.

Classical ideas were unable to account for the nature of these temperature dependent frequency distributions. In particular classical theory predicted that, whatever the temperature, the radiation would be increasingly intense the higher its frequency. This meant that even a cool body would be emitting light, ultraviolet rays and x-rays in dramatic contradiction to observation. The solution to this problem was first intimated by Max Planck in 1900, who suggested that the least energy that an oscillating system could have, other than no energy at all, was proportional to its frequency. This meant that it would be impossible to set very high-frequency oscillations in train because they would require much higher energy than would be available due to thermal motion. This would be particularly so when the emitting body was at a low temperature. The constant of proportionality, which is now called Planck’s constant, is one of the key fundamental constants of physics. Planck’s proposal thus has the form

  Lowest energy of an oscillator = Planck’s constant × frequency = hv

where h is the symbol universally used for the constant and has the value h = 6.626×10^-34 J s. The Greek letter v is generally used as the symbol for frequency. Planck went further than this, however, and proposed that the only allowed energies an oscillator could have-its energy levels were integer multiples of this lowest energy (i.e. nhv where n = 0, 1, 2 . . .). So, whilst classically an oscillator’s energy can creep up continuously. From zero (when it is at rest) to any desired value, with the Planck hypothesis the energy of an oscillator is quantized; only discrete values are allowed determined by the quantum number n. Suffice it to say here that this hypothesis accounted perfectly for the nature of the electromagnetic radiation emitted by a hot body, but it was completely contrary to the current classical theories and was the beginning of a major revolution in physics. The next key development was connected with the photoelectric effect. This is the effect in which electrons (referred to in this context as photoelectrons) are emitted from a metallic surface when electromagnetic radiation, in the form of, for example, light or ultraviolet radiation, falls on to it. This can be understood simply as energy being carried by the oscillating electric and magnetic fields of the radiation being transferred to atomic electrons and knocking them out of the atoms. When this process was carefully studied three curious features emerged.

1.Contrary to natural expectations, the energy of the individual emitted electrons did not depend on the intensity of the radiation, although the more intense it was the more electrons were emitted.

2. The energy of the electrons depended directly on the frequency of the radiation.

3. No electrons were emitted if the frequency fell below a certain value.

These features could not be understood at all in terms of classical ideas which, for example, would suggest that the more intense the radiation, the more energetic its oscillations and, therefore, the more energetic the photoelectrons. It raised the question as to why weak blue light produces more energetic photoelectrons than strong red light. The solution to this dilemma was suggested by Einstein who, influenced by Planck’s quantum ideas, suggested that in the photoelectric process radiation behaves like a stream of energy packages or particles having an energy proportional to the frequency of the radiation. These particles are called photons or quanta and the energy of each photon is given by

photon energy = Planck’s constant × frequency = hv.

In the photoelectric effect the photon gives up its energy to an electron. The more intense the radiation the more photons there are; hence more photoelectrons are emitted. The energy of a photoelectron is simply the energy of the photon, which depends only on the frequency, less the energy needed to escape from the metal. Clearly, below a certain frequency, there will not be sufficient energy to escape. Thus the three features of the photoelectric effect mentioned earlier are all neatly explained and Einstein was awarded the Nobel prize for this work not for relativity. This explanation does, of course, introduce a completely new idea into physics, namely that electromagnetic radiation which in many circumstances behaves as a wave-witness the  can also behave as a stream of particles. This is the first instance of what is called wave particle duality and is a key feature of the quantum revolution. The particle nature of electromagnetic radiation was subsequently directly confirmed by Compton in 1923, who showed that electrons scatter x-rays in just the same way as one billiard ball scatters another. This scattering could not be understood at all treating the x-rays as waves. These ideas then stimulated Niels Bohr in 1913 to suggest a simple form for the structure of atoms. He proposed that electrons moved around the atomic nucleus in circular orbits, rather like the planetary system, constrained by the electric attractive force due to the positively charged nucleus. Classically, such electrons would emit electromagnetic radiation since they are continually accelerating-their direction of motion is changing all the time. They would thus continually lose energy and eventually spiral into the nucleus. To avoid this problem Bohr proposed that only certain orbits were allowed for an atomic electron and that, whilst in these orbits, no radiation was emitted. He specified these orbits by hypothesizing that in them the angular momentum  of the electron is an integer multiple of h / 2π which is universally symbolized by h. In mathematical terms the angular momentum is only allowed to have the values nh where n = 1, 2, 3, . . .. This, in turn, means that the orbiting electron can only have certain energies determined by the value of the quantum number n. Bohr then proposed that radiation is only emitted when an electron jumps from a higher energy orbit to one of lower energy and that its frequency, U, is given by simply equating the energy released to hv, here using the idea of Einstein and treating the emitted radiation as a photon. This means that emitted radiation can only have certain frequencies, determined by the values of the quantum number n before and after the jump. Further, the energy of an electron can never be zero since the lowest value allowed for n is 1 (not 0) which means that an atomic electron is always in orbit and, therefore, does not fall into the nucleus. It was a great triumph for Bohr that observed frequencies of the electromagnetic radiation emitted (the atomic spectrum) by hydrogen, which has only one electron, when bombarded by electrons in a discharge tube, agreed very closely with this simple theory. Such a spectrum is observed as a series of discrete lines, corresponding to the different allowed frequencies, when refracted through, for example, a glass prism. It was also very satisfactory that the radii of electron orbits calculated on this basis were consistent with the known sizes of atoms. Bohr’s theory, succesful as it was, could not be carried through to account for the more detailed properties of atoms. It was a hybrid theory relying on both classical and quantum ideas. For example the concept of an electron in orbit is classical and, as mentioned above, it should be continually emitting electromagnetic radiation. Yet, according to Bohr, radiation is only emitted when an electron jumps from one orbit to another and then it had to be treated as a photon rather than a classical wave motion. For this and other reasons the theory had to be developed further into a full quantum theory. This took place during the following decade and was based on the implications of wave-particle duality already emphasized.