The End of Time: The Next Revolution in Physics

It is a basic Copenhagen tenet that the probabilistic statements reflect a fundamental property of nature, not simply our ignorance. It is not that before the measurement the particle does have a definite momentum and we simply do not know it. Instead, all momenta in the superposition are present as potentialities, and measurement forces one of them to be actualized. This is justified by a simple and persuasive fact. If we do not perform measurement but instead allow ψ to evolve, and only later make some measurement, then the things observed later (like the two-slit fringes) are impossible to explain unless all states were present initially and throughout the subsequent evolution. Outcomes in quantum mechanics are determined by chance at the most fundamental level. This is the scenario of the dice-playing God that so disturbed Einstein.

If anything, the second cardinal fact disturbed him even more. There seems to be a thoroughgoing indefiniteness of nature even more radical than the probabilistic uncertainties. As we have seen, one and the same state can be regarded as a superposition of either momentum or position eigenstates. It is the way this mathematics translates into physics that is startling. The experimentalist has complete freedom to choose what is to be measured: position or momentum. Both are present simultaneously as potentialities in the wave function. The experimentalist merely has to choose between set-ups designed to measure position or momentum. Once the choice is made, outcomes can then be predicted – and one outcome is actualized when the measurement is made. In fact, the indefiniteness is even greater since other quantities, or observables as they are called, such as energy and angular momentum, are also present as potentialities in ψ.

Only one experiment can be made – for position or momentum, say, but not both. Every measurement ‘collapses’ the wave function. After the collapse, the wave function, which could have been used to predict outcomes of alternative measurements, has been changed irrevocably: there is no going back to the experiment we opted not to perform. It is a very singular business. Whatever observable we decide to measure, we get a definite result. But the observable that is made definite depends on our whim. The many people who, like Einstein, believe in a real and definite world find this immensely disconcerting. What is out there in the world seems to depend on mere thoughts that come into our mind. Most commentators believe that this radical indefiniteness – the possibility to actualize either position or momentum but not both – is the most characteristic difference between classical and quantum physics. In classical physics, position and momentum are equally real, and they are also perfectly definite.

The fact that in quantum mechanics one can choose to measure one but not both of two quantities was called complementarity by Bohr. Pairs of quantities for which it holds are said to be complementary.

HEINSENBERG’S UNCERTAINTY PRINCIPLE

Heisenberg’s famous uncertainty relation gives quantitative expression to complementarity for position and momentum. De Broglie’s relation λ=h/mv=h/p determines the wavelength of a particle of momentum p=mv, where m is its mass and v its velocity. Now, it follows from Fourier’s work on the superposing of waves that a wave packet restricted to a small spatial region contains many waves in a broad spread of wavelengths. To narrow down the spatial positions q, it is necessary to broaden the range of momenta p. Conversely, to get a nearly definite p, we must accept a wide range of positions q.

Mathematically, we can in fact construct wave packets in which the positions are restricted to a small range, from q to q + Δq, and the momenta to a correspondingly small range, from p to p + Δp. Any attempt to make Δq smaller necessarily makes Δp larger, and vice versa. Heisenberg’s great insight – his uncertainty relation – was the physics counterpart of this mathematics. There is always a minimum uncertainty: the product ΔqΔp is always greater than or, at best, equal to Planck’s constant h divided by 4π. If you try to pin down the position, the momentum becomes more uncertain, and vice versa. This is the uncertainty relation. Moreover, a wave packet of minimum dimensions will in general spread: the uncertainty in the position will increase. This is what in quantum mechanics is known as the ‘spreading of wave packets’.