The heart of quantum mechanics – Squaring the circle

On our journey to the heart of quantum physics, next, we compare the wave picture with the particle picture of light. In the wave picture, we can calculate the interference pattern which corresponds to the intensity distribution of the light as square of the amplitude of the electromagnetic wave.

In comparison to that, in case of individual photons, the intensity I(x) on the screen is proportional to the probability density ρ(x) multiplied by the number N of photons. We combine the equation I(x) is proportional to N ρ(x) with the equation I(x) is equal to the square of the amplitude. In combination, we obtain an unfamiliar, new expression: The square root of the probability density ρ(x), multiplied by the number N of photons.

What can be concluded from this ‘squaring of the circle’? Let us take a look at an individual photon, i.e. set N=1. For the individual photon, the probability density ρ(x) can then be calculated in a similar way as intensity distribution of a wave: it is the square of a rotating wheel with a rotation frequency, an amplitude, and a phase.

Thus, we have found a fundamental, new description of light: A probability wave, symbolised by a rotating wheel. The radius corresponds to the square root of the probability density. The vector may be either positive or negative, owing to the phase. We can, however, not directly observe the phase. We can only indirectly recognise the phase based on the impact on the interference pattern if the path difference Δ between various different partial waves leads to phase differences.

This “square root probability” with an invisible phase is termed, in quantum mechanics, as wave function Ψ. Mathematically speaking, the observation probability P(x), the probability density ρ(x) and the wave function Ψ(x) relate to one another as follows. This equation yields the probability of detecting the photon in the range from x to (x+ δx).

The rotating wheel symbolises an oscillation in the quantum dimension that cannot be directly observed, and thus forms the heart of quantum physics.