Does quantum mechanics apply to single systems?

It is clear phenomenologically that statistical mechanics (and hence quantum mechanics) applies to single systems like a particular cup of tea, irrespective of what the discussions about the foundations of physics say (see many other entries in this FAQ). Thus statistical mechanics and quantum mechanics do not - as is often claimed - only apply to large ensembles of independently and identically prepared systems; when the system is large enough (i.e., macroscopic), a _single_ system is enough. (For smaller single systems, see the entry How do atoms and molecules look like? in the present FAQ.)

In classical statistical mechanics, the traditional bridge between the ensemble view and thermodynamics (which clearly applies to single systems) is the ergodic hypothesis. But there is not enough time in the universe to explore more than an extremely tiny region of the about 10^25-dimensional phase space of the cup of tea to explain the success of the thermodynamical description by ergodicity.

In quantum mechanics, the situation is even worse - usually it is not even attempted here to bridge the gap.

The best treatment I know of the foundational problems involved in classical statistical mechanics is in the book

but it does not present a solution. Other sources are not better in this respect.

My own solution is the thermal interpretation of physics, discussed to some extent in Chapter 10 of the online book

and in my recent slides The key idea is that mathematical expectation has two different interpretations in physics, one as average over a large number of cases, and the other as a means of defining observables. That the two interpretations have the same mathematical properties is the reason they have been confused in the past. The thermal interpretation separates them neatly and thus gets rid of most of the confusing aspects of the foundations of physics.

(See also the entry Entropy and knowledge from this FAQ.)

Arnold Neumaier (Arnold.Neumaier@univie.ac.at)
A theoretical physics FAQ Thermal Interpretation FAQ