----------------------------------------------------
S1o. Does quantum mechanics apply to single systems?
----------------------------------------------------

(See also the entry ''Entropy and knowledge'' elsewhere in this FAQ.)

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 only apply - as is often 
claimed - 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 
     L. Sklar, 
     Physics and Chance, 
     Cambridge Univ. Press, Cambridge 1993.
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 7 of the book  
    Arnold Neumaier and Dennis Westra,
    Classical and Quantum Mechanics via Lie algebras,
    Cambridge University Press, to appear (2009?).
    http://www.mat.univie.ac.at/~neum/papers/physpapers.html#QML
    arXiv:0810.1019 
and in my recent slides
    A. Neumaier, 
    Classical and quantum field aspects of light,
    http://www.mat.univie.ac.at/~neum/papers/physpapers.html#lightslides
and
    A. Neumaier, 
    Optical models for quantum mechanics,
    http://www.mat.univie.ac.at/~neum/papers/physpapers.html#optslides
and explored in more detail in my German 
   Ein Theoretische Physik FAQ
   http://www.mat.univie.ac.at/~neum/physik-faq.txt
under the name ''consistent experiment interpretation''

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.