Quantum Theory - A Very Short Introduction by John Polkinghorne

Author:John Polkinghorne
Language: eng
Format: mobi, pdf
Published: 2010-10-28T17:19:45.711000+00:00

Chapter 4 Further developments

The hectic period of fundamental quantum discovery in the mid1920s was followed by a long developmental period in which the implications of the new theory were explored and exploited. We must now take note of some of the insights provided by these further developments.

Tunnelling

Uncertainty relations of the Heisenberg type do not only apply to positions and momenta. They also apply to time and energy. Although energy is, broadly speaking, a conserved quantity in quantum theory – just as it is in classical theory – this is only so up to the point of the relevant uncertainty. In other words, there is the possibility in quantum mechanics of ‘borrowing’ some extra energy, provided it is paid back with appropriate promptness. This somewhat picturesque form of argument (which can be made more precise, and more convincing, by detailed calculations) enables some things to happen quantum mechanically that would be energetically forbidden in classical physics. The earliest example of a process of this kind to be recognized related to the possibility of tunnelling through a potential barrier.

The prototypical situation is sketched in figure 7, where the square ‘hill’ represents a region, entry into which requires the

7. Tunnelling

payment of an energy tariff (called potential energy) equal to the height of the hill. A moving particle will carry with it the energy of its motion, which the physicists call kinetic energy. In classical physics the situation is clear-cut. A particle whose kinetic energy is greater than the potential energy tariff will sail across, traversing the barrier at appropriately reduced speed (just as a car slows down as it surmounts a hill), but then speeding up again on the other side as its full kinetic energy is restored. If the kinetic energy is less than the potential barrier, the particle cannot get across the ‘hill’ and it must simply bounce back.

Quantum mechanically, the situation is different because of the peculiar possibility of borrowing energy against time. This can enable a particle whose kinetic energy is classically insufficient to surmount the hill nevertheless sometimes to get across the barrier provided it reaches the other side quickly enough to pay back energy within the necessary time limit. It is as if the particle had tunnelled through the hill. Replacing such picturesque story-telling by precise calculations leads to the conclusion that a particle whose kinetic energy is not too far below the height of the barrier will have a certain probability of getting across and a certain probability of bouncing back.

There are radioactive nuclei that behave as if they contain certain constituents, called α-particles, that are trapped within the nucleus by a potential barrier generated by the nuclear forces. These particles, if they could only get through this barrier, would have enough energy to escape altogether on the other side. Nuclei of this type do, in fact, exhibit the phenomenon of α-decay and it was an early triumph of the application of quantum ideas at the nuclear level to use tunnelling calculations to give a quantitative account of the properties of such α-emissions.

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