Do the constants apply to quantum theory

Quantum Mechanics and Uncertainty Relation

Atoms, electrons and other quantum particles behave fundamentally different from what we know from everyday life. They have both particle and wave properties, their future is absolutely indeterminate and even if you measure in the now, the result is not always precise.

Interference pattern of electrons

The matter we know consists of atoms, electrons and other quantum particles. And yet the laws that apply in the microscopic world of particles and the macroscopic world of bodies could not be more different. The peculiarities of the quantum particles are particularly impressive in the double slit experiment: An electron beam hits a screen with two narrow, parallel slits, penetrates through it and lands on a photo plate mounted behind it. Each impacting particle blackens this plate at the point of impact and leaves a well-localized black point there.

In this respect, electrons do not behave differently than macroscopic particles such as grains of sand and thus demonstrate their particle properties. After a sufficient number of hits, however, an essential difference becomes apparent: While grains of sand form two separate piles behind the double slit because they have flown either through the left or right slit, the electrons on the photo plate create a structured blackening pattern made up of strips of different intensity exists (see figure on the right).

This so-called interference pattern is also known from light or water waves if you let them run through a similar double slit. In fact, the creation of the electron impact pattern can also be described with the help of a wave theory. In contrast to grains of sand, electrons also have wave properties. Incidentally, this also applies to other quantum particles such as protons, neutrons, atoms and even fullerenes - complex molecules made up of 60 carbon atoms. They all generate the characteristic interference pattern in the double slit test. The classical physical approach fails to explain this “particle-wave dualism”.

Only the quantum mechanics of Werner Heisenberg, Erwin Schrödinger and others provided concepts for understanding: quantum particles have to be described by a wave function! In 1929, the German physicist Max Born proposed a probability interpretation of the quantum mechanical wave function. More precisely, the square of this wave function corresponds to the probability with which a particle is currently at this or that location, i.e. would be measured there.

Real coincidence

For the double slit experiment, this means: Before an electron hits the photo plate and its position can therefore be viewed as measured due to the local blackening, only certain probabilities can be specified for the location of this particle - and these can be calculated using the electron wave function. Incidentally, this is generally very extensive in space; In principle, the electrons in a double slit experiment at DESY could even be located in America - the probability of this being negligible, however. On the other hand, it is much more likely that the electrons will move through one of the two slits if their wave function spreads in this direction. If this happens perpendicular to the aperture and symmetrically to the two columns in it, the probability of being in each column is 50 percent. The partial waves from the two columns overlap behind the diaphragm - just like with a water or light wave - and there form an interference pattern through mutual reinforcement of the wave crests and extinction in the wave troughs.

Electrons at the double slit

For the probability of presence, which results from the square of the wave function, this means that the electrons are more likely to hit places where the partial waves are amplified than at those where the partial waves almost cancel each other out. Where exactly an electron actually "materializes" on the photo plate - that is, a real black spot is created on the photo plate - is only decided at the moment of impact and cannot be predicted.

This is an essential difference between macroscopic and microscopic objects. In our macroscopic everyday world, everything can be calculated in advance - even the lottery numbers for next Saturday. You would only have to know in advance all the decisive parameters such as the position of the balls, the speed of the drum and so on. The supposed coincidence is based here only on a lack of information about the initial values, but also on our inability to evaluate the complex dynamics of so many quantities.

In the case of the electrons in the double slit experiment (and generally of all quantum particles), however, even an exact knowledge of the wave function would be of no use: Completely identically prepared particles strike at different points on the photo plate - the wave function only provides the probabilities for this. (Real) chance alone decides where the particles actually land. However, if a large number of such quanta run through the double slit, the distribution of hits on the plate becomes reproducible: with each pass, the characteristic blackening pattern of dark and light stripes is created, which can be described with the help of wave theory.

In overlay

The double-slit experiment already suggests that one has to give up the classic term orbit in the quantum world. While grains of sand run through one of the two gaps, quantum particles seem to penetrate both at the same time - their wave function does exactly that: The interference effects of the wave function after the aperture can also be observed when the electrons are sent individually one after the other through the arrangement. But what happens if you experimentally "look up" directly behind the columns to see which path a particle has actually chosen?

If you observed many particle passes, one half would be found behind gap 1 and the other half behind gap 2. This coincides with the calculated probability of presence, which is 50 percent in each gap (assuming the above-mentioned symmetry of the experiment). However, the blackening pattern on the photo plate would now be different: Instead of the typical stripe pattern, two “electron clusters” appear on the screen - similar to the classic case.

The measurement of the gap through which the respective electrons have passed is responsible for the changed pattern. Namely, it changes the probability wave function of the electrons. Because the particles are set on one of the two possible gap passages. For the wave function this means: Instead of a superimposed state of partial waves through the two gaps, the two states now exist behind the diaphragm, each with a known path, which can no longer interfere. As a result, the interference pattern that resulted from the superposition of the two partial waves also disappears. With a symmetrical measuring arrangement, the particles hit behind gap 1 or gap 2 with a probability of 50 percent.

The measurement of the alternative path influences - unlike in our everyday world - the wave function and thus the probabilities of the quantum particle's location. This applies not only to the location of the particles, but also, for example, to their impulses or angular momentum. As long as the wave function of the quantum particle is not "disturbed" by a measurement, it contains all possible measurement results in a superposition. A measurement shrinks the number of possible values.

Uncertainty relation

Werner Heisenberg

Another peculiarity of the quantum world occurs when certain properties of a particle are measured at the same time. Certain physical quantities - the most prominent pair are location and momentum - cannot be specified exactly at the same time, no matter how precisely one tries to measure. This means, for example: If the position of a particle is known very precisely, its speed is largely indeterminate. Conversely, we hardly know anything about its whereabouts if we know its speed very precisely.

The physicist Werner Heisenberg described this law, which is characteristic of quantum particles, in 1927 with his famous uncertainty relation:

$$ \ Delta x \ cdot \ Delta p \ geq \ frac {h} {4 \ pi} $$

where \ (\ Delta x \) is the positional uncertainty, \ (\ Delta p \) is the momentum uncertainty and \ (h \) is Planck's quantum of action - a natural constant which is indispensable for quantum physics and which is almost characteristic. This connection between position and impulse measurement uncertainty can be used to limit how imprecise the two measurement results will be, and must be, if both quantities are measured at the same time.

The position-momentum-uncertainty relation is one of the basic principles of quantum mechanics. In addition to these two physical quantities, there are also other pairs in the quantum world that fulfill such an uncertainty relation - for example, different angular momentum components. It is important that this type of blurring does not result from measurement inaccuracies or measurement errors alone. It is an inescapable consequence of the mathematical structure of quantum physics, from which it can be derived independently of concrete experiments.

A more in-depth article about Heisenberg's uncertainty principle and energy-time uncertainty can be found here.