Heisenberg’s uncertainty principle (1927) [1]
.. (so named after the physicist who formulated its basic principles, Werner Heisenberg (1901-1976)), is another fundamental feature of quantum mechanics which postulates that at a microscopic level there are related features of the universe, such as the location and velocity of a particle, which cannot be known with complete precision. If you know one integer (location) with some clarity, you cannot determine the other (velocity) with any degree of precision. The more you know about one, the less you can know about the other. In other words, electrons (and everything else, for that matter) cannot be described as simultaneously being at such and such a location and having such and such a velocity: the more you squeeze a particle into a smaller space, the more it tends to pick up momentum and try to escape. Heisenberg's Uncertainty Principle dictates that the uncertainty of the position of the particle times the uncertainty in its speed is always greater than a quantity called Planck's constant h, divided by twice the particle's mass.[2]
In order to determine the position and velocity of an electron, the obvious way to do so is by shining light on it. Some of the waves of light will be scattered by the particles and this will indicate its position. However, you will not be able to determine the position of the particle more accurately than the distance between the wave crests of light[3]. If you bounce light off an object using high frequency (short wavelength) light you can locate an electron with greater precision, but since the photons are very energetic they will give the electron a kick sharply disturbing its velocity – rather like trying to detect the speeds and directions of a frenzied group of school children whose momentary positions you may know with great accuracy[4]. If on the other hand you use low frequency (long wavelength) light you minimise the impact of the electron’s motion, since the low frequency photons have comparatively low energy, but we sacrifice precision in determining the electron’s position. Another aspect of the uncertainty principle endows so-called virtual particles with a licence to borrow some energy for a small amount of time, but the more that is borrowed, the more quickly it must be repaid. [5]
Such uncertain aspects of the microscopic world become even more severe as the distance and time scales on which they are considered become ever smaller. Particles and fields undulate and jump between all possible values consistent with quantum uncertainty. The microscopic realm is thus “a roiling frenzy awash in a violent sea of quantum fluctuations”. The notion of a smooth spatial geometry, the central principle of general relativity, is destroyed by the violent fluctuations of the quantum world on short distance scales. On ultramicroscopic scales, the uncertainty principle, a fundamental feature of quantum mechanics, is in direct conflict with the central feature of general relativity – the smooth geometrical model of space and time[6].
In other words, the notions that we tend to take for granted in the ordinary everyday world that objects have definite positions and speeds and that they have definite energies at different times simply do not apply in the quantum world. Hawking describes Heisenberg’s uncertainty principle as a fundamental, inescapable property of the world[7]. Stephanie Wehner and Esther Hanggi of the National University of Singapore’s Centre for Quantum Technology have gone even further, suggesting that if you take away the uncertainty principle, you end up with a perpetual motion machine in violation of the second law of thermodynamics[8]. And, as Carlos Calle points out, the uncertainty principle doesn’t apply just to electrons and atoms. It applies to everything. However, because it involves Planck’s constant, which is very small, you don’t notice its effects when you are watching baseballs, cars, or planets move.[9]
Matrix mechanics
Another important contribution made by Heisenberg concerned the measurement of the mechanics of the orbits of the electron in the atom. The important discovery that Heisenberg made is that "quantities that can be measured, like position or speed, can’t be represented by ordinary numbers. They must be represented by these arrays, these matrices, and must follow their special rules" which elevate the order in which you do the calculations as a factor of significant importance, such that when you do things in a different order the final configuration changes.
As Carlos Calle explains, in Bohr’s model the electrons orbit the nucleus in very specific orbits that he called stationary orbits. In these orbits, the electrons are safe, and they don’t radiate any energy. However, the electrons are allowed to move between orbits, and when they jump to a lower orbit, they give off energy. These energies are quantised, with the electrons jumping from one orbit to another as they absorb or eject a photon with the right energy. Bohr used Planck’s theory to calculate the energies of the allowed stationary orbits for the atom, and when he compared his calculations with experimental data, his theory agreed exactly[10].
When his turn came to explain the mechanics of the atom, Heisenberg first conceptualised the Bohr atom as containing very large orbits. As he attempted to devise a code for connecting the quantum numbers and energy states in the atom with the experimentally determined frequencies and intensities (brightness) of the light spectra, the realisation dawned that the quantities involved such as the electron’s position or speed can’t be represented by plain numbers. Whilst in the classical world p x q always equals q x p that is not necessarily so in the quantum world. So instead Heisenberg utilised an array of numbers that followed some simple but very specific rules – as it turned out, the rules of matrix algebra - to explain such things as the electron’s linear momentum(p) and displacement from equilibrium(q). Heisenberg solved the equation of the aspect of the atom's motion by classical means, and then calculated the energy of the particle in its quantum state.
In so doing, Heisenberg devised a formula for including all possible states within the atom, and was able to show that the energy states were quantised and time independent as in the Bohr atom. The theory came to be known as matrix mechanics: a purely mathematical formulation whereby the energy levels of atoms were described purely in terms of numbers, difficult to use and impossible to visualise in picture form and where the order in which the calculations are made is critical. In the process the internal mechanics of the atom appeared as yet another manifestation of the quantised graininess of the quantum world determined by Planck’s constant.[11] Thanks to Heisenberg and Schrödinger, we now have not just one atomic theory but two: matrix mechanics and wave mechanics.
Some short time later, the English physicist Paul Dirac came up with a third to the effect that these two versions were in fact equivalent, and special instances a more general formulation which he came up with on his own linking quantum theory with classical physics, the essence of which was Heisenberg's idea that the order in which you perform the calculations was of the utmost importance.[12] Dirac recognised that this mathematical form had the same structure as aspects of the classical dynamics of particle motion. From this thought, he was able to develop a general quantum theory incorporating links with aspects of classical physics that was structured on the same basis as Heisenberg's matrix mechanics. [13].
Heisenberg's brilliant insights notwithstanding, it is interesting to note that he almost failed his Ph D oral examination by his inability to answer questions of a fairly elementary nature (to those who understand these things) about the resolving power of optical instruments, and it was only after the intervention by his supervisor, a famous theoretical physicist, that the examiner was persuaded to pass Heisenberg and then only with the lowest possible pass mark. Years later, the same problem came back to haunt him when he was in the process of formulating his uncertainty principle. He used an example which again failed to take into account the resolving power of the microscope until the problem was pointed out to him by Niels Bohr. Despite his blind spot on such a ‘basic’ issue, Heisenberg is recognised as the leading theoretical formulators of one of the fundamental principles underpinning quantum mechanics[14] - so a blind spot here and there is no necessary bar to greatness if your theory can otherwise withstand experimental analysis and stand on its own feet in the eyes of your peers!
[1] See Greene (2000) 112-131, esp at 113, 118 and 129; 424.
[2] Step[hen Hawking's posthumous mémoire Brief Answers to the Big Questions, John Murray, London, 2018, 93; JP McEvoy, Introducing Quantum Theory: A Graphic Guide, Icon Books Ltd. Kindle Edition, locations 1549-1568
[3] Hawking, 54; Greene (2000), 113.
[4] Greene (2000), 118.
[5] Virtual particles are considered on the page Feynman's sum over paths - a more detailed analysis
[6] Greene (2000), 129
[7] Hawking (1988), 55. It has been suggested in some quarters that the word 'uncertainty' is perhaps a mis-interpretation of the German original, and that a better word would be 'indeterminacy' or better still 'indeterminability': Bill Bryson, A Short History of Nearly Everything, Broadway Books, 2003, 144, citing Michael Frayn.
[8] Jessica Griggs, “To be quantum is to be uncertain”, New Scientist, 23 June 2012, 8.
[9] Carlos I Calle, Einstein For Dummies (Kindle Locations 4329-4330). Wiley, 2005. Kindle Edition.
[10] Ibid, Locations 4212-4213; 4219-4226, 4244-4266, 4249-4252.
[11] These paragraphs comprise an edited summary of Calle, op cit, locations 4242-4213, 4220-4228 and J P McEvoy, J.P, Introducing Quantum Theory: A Graphic Guide, Icon Books Ltd. Kindle Edition, locations 1186-1285.
[12] Ibid, Locations 1473-1478; 4269-4270; 4275-4276.
[13] https://en.wikipedia.org/wiki/Paul_Dirac
[14] Tony Hey and Patrick Walters, The New Quantum Universe, Cambridge University Press, Cambridge, 2004, 23-24.