# What does space mean in quantum mechanics

## Quantum equilibrium

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Boltzmann's idea of the justification of the statistical hypothesis and Gibbs' statistical approach are independent of the specific form on which the physical theory is based. It doesn't really matter whether the world consists of point particles, of fields or of filamentary particles (strings) or who knows what. As long as there are physical laws for these quantities, nothing stands in the way of Boltzmann's view of typical motion. One can of course be put off by the problem of irreversibility, but one shouldn't. One should only be aware of this problem.

**Figure 9:** The migration of 3 particles in physical space (here in one dimension) corresponds to a path in the 3-dimensional configuration space

Newtonian mechanics is being replaced by quantum mechanics, which means that quantum mechanics is more generally valid than Newtonian mechanics and has to explain it (see e.g. 1997). This view of quantum mechanics is not yet shared by all physicists, in particular it was not the view of Niels Bohr, one of the founders of quantum mechanics. But once you have this view, the motion of the point particles in Newtonian mechanics must result from quantum mechanics. The previous textbook quantum mechanics cannot do this, and there is still no agreement on what a final quantum mechanics that does this will look like (see 1998). An obvious theory is Bohm's mechanics, which is named after its author David Bohm (1917-1992). It is also often referred to as "pilot wave theory" because in it an "abstract wave" generates the movement of particles. Bohmian mechanics is also a law for the motion of particles, Bohmian particles, not Newtonian, which means that the law for motion is not Newtonian.

The feature of this theory is that it is no longer formulated in physical 3-dimensional space, but immediately in phase space, although the phase space of Bohm's mechanics in this case is not 6*N*-dimensional, but only 3*N*-dimensional, i.e. the movement of the particles is already uniquely determined if you know all the locations. The 3rd*N*-dimensional space, in which the particle locations are removed, is traditionally also called configuration space (see Figure 9)

In Bohm's mechanics, the phase space is the configuration space. The law for the movement of the particles is given by the Schrödinger wave function ( also -Function called) formulated, which represents an "abstract wave" on the configuration space. It obeys a wave equation (analogous to Newton's gravitational potential, which satisfies the potential equation), the famous Schrödinger equation.

**Figure 10:** A plane wave falls on a double slit. Spherical waves originate from the gaps and overlap (interfere).

The -Function has wave properties, it can be canceled out (destructive interference) or intensified (constructive interference). If you only look at a single particle, then that is -Function a wave in 3-dimensional space, in which one can visualize the wave image and the particle trajectories of the particle, i.e. the flow (the totality of all trajectories of the particle in the configuration space, which for a single particle is exactly the same as the physical space).

I do not want to write down the law of motion of the particles in detail (see e.g. 1987, 1992 and 1996). Suffice it to say that the orbits accumulate in the wave crests and valleys (in the high-dimensional configuration space), and that there, where the -Function disappears, there are no lanes.

Figure 11 shows the possible paths of a Bohm's particle in the so-called two-slit experiment: a particle is sent to two microscopic slits. The one guiding the particle Function behaves like any wave according to the two columns: An interference pattern is generated according to Huygens' principle (Figure 10).

The places without particle trajectories show where the interference canceled the wave.

**Figure 11:** The possible orbits of a Bohm particle in the two-slit experiment. The endpoints of the orbits are the famous interference pattern of the two-slit experiment. In each `` run '' of the experiment in which a particle is sent on its way, a random point blackening is obtained on a photo plate placed at the right end (see Figure 12), which is the end point of a randomly selected path of the possible paths shown is.

The typical wave propagation in all directions ensures that Bohm's orbits are ideally unstable or, more modernly, chaotic: what you had to laboriously understand on Galton's board, Bohm's mechanics provides almost for free. This is shown very clearly in Figure 11: Directly behind the gap, the particle trajectories fan out as far as possible and separate (compare Figure 3).

A frequently asked question is whether one can `` see '' the tracks in the two-slit experiment. What is meant here is whether the path of a particle can be made visible through one of the two gaps (e.g. in a cloud chamber) without changing anything in the impact pattern on the photo plate (Figure 12). In Bohm's mechanics this is impossible, because every interaction with the particle - and looking at the particle trajectory is an interaction - requires a description on the configuration space of the comprehensive system, which consists of particles and `` observing device ''. The new wave function is then one on this configuration space, which means that the possibility of interference is destroyed.

**Figure 12:** In the course of time, the random points of impact of the particles, which pass through the double slit one after the other, form an interference pattern, as occurs with wave interference. The relative frequencies for the points of impact are determined by the absolute square of the wave function.

As luck would have it, quantum mechanics has been viewed as a statistical theory since the invention of the Schrödinger equation, albeit without any connection with the Boltzmann and Gibbs ideas. The physicist Max Born (1882-1970) postulated that indicates the probability distribution of the locations of the particles. In fact, this postulate was based on the so-called quantum flux equation, which was found by Schrödinger (1887-1961) and is to be seen quite analogously to Liouville's theorem. This means that the weighting of each point in the configuration space with results in a volume measure that is distinguished from Bohm's dynamics itself, namely, is "stationary" (in a somewhat more general than the usual sense, which is irrelevant for our discussion). With this we have a physical example of a content of sets distinguished by physics, which, unlike in classical mechanics and Liouville's theorem, is no longer given by the ordinary volume.

Born's postulate is completely analogous to Gibbs' approach. One will wonder why this connection was not immediately recognized. One reason for this is that in Newton's statistical mechanics of our world, Gibbs’s hypothesis is mostly wrong: There is no thermal equilibrium. Born's hypothesis, on the other hand, seemed to be correct without exception: `` quantum equilibrium '' always prevails. That should have led to the conviction that Boltzmann's program is now feasible.

**Figure 13:** Our universe is in quantum equilibrium.

It was not until the physicist John S. Bell (1928-1990) that the self-evident was briefly and naturally noted (1987, before Bell spoke of the orbit of an atomic particle, the - Particles in a cloud chamber):

Suppose, for example, this world contains an actual ensemble of similar experimental set-ups. In the same way as for the -particle track it follows from the theory that the ,, typical`` world will approximately realize quantum mechanical distributions over such approximately independent components. The role of the hypothetical ensemble is precisely to permit definition of the word ,, typical ''.

The elaboration of these few lines is then quite time-consuming, but no more than that (1992). Conceptually, everything was already said by Bell: The configuration of the Bohmian particles in our universe is related to the -Volume typical, being natural represents the wave function of our universe. It is so that in this case the physical law of motion of Bohm's particles determines the initial configuration, namely as a typical configuration. In contrast to Figure 8, the initial condition can now be selected blindly following Bohm's law without `` external help '', which is to be expressed in Figure 13.

It has by no means been said here that our universe is typical from a quantum mechanical point of view! Our universe is still special, only this specialty has been shifted from the particle coordinates to the wave function. It is the wave function of the universe that is responsible for the thermal imbalance of our universe. The particles of the Bohmian universe, however, are "in equilibrium" with the wave function that guides them.

**thanksgiving**

I would like to thank Reinhard Lang for the correction and for many instructive discussions about the basics of this article.

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*D. Duerr*

Wed May 6 16:27:57 MET DST 1998

Wed May 6 16:27:57 MET DST 1998

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