A quantum system can have many states,
differing in (typically microscopic) internal
details, in which the
properties amenable to external observation are substantially the same. When
the externally observable properties are exactly the same, such states are said
to be (exactly) degenerate; but limitations on precision of measurement mean
that even (sufficiently accurate) approximate equality of external observables
can lead to such (approximately degenerate) states being effectively
indistinguishable by external observation. Such degeneracy among possible
states is particularly common when the system is made up
of a large number of identical sub-systems. Such a
system may have some extreme
states that are uniquely (or nearly so)
identified by their gross observable properties, with states closer to such
extremes being subject to less degeneracy.
Our knowledge of the system comprises its measured properties; when there
are many states consistent with those properties, we have no knowledge of which
such state the system is in. A given description of the internal details of the
system many give us a discrete set of states consistent with our measurements;
but, in quantum mechanics, these states can be mixed in arbitrary proportions
(and relative phases) to produce infinitely many possible states with the
measured properties; and a different description of the internal details would
be apt to produce a different set of discrete states, each of which would be
expressible (in terms of the first description's states) by some such
mixture. The official term for such mixtures (when properly formalised)
is superpositions
. The system itself cares not how we describe it
– only how we measure it – so is in no way constrained to be in a
state which fits nicely with our description. However, this very lack of
constraint implies that the system is most probably in one of the states as
disinterested as possible
in how they are described – and there are
plenty of such states. These most probable allowed states can be
represented as superpositions, of the
allowed states of a given description, in specific proportions (with arbitrary
phases) determined by the observed properties of the system and the details of
the chosen description.
It is usual for one of a system's observable properties to be energy; and it
is usual for the system's state with lowest energy to be quite isolated –
there will be little or no degeneracy in this ground state
. While
an isolated
system will generally change in such manner as will reduce
its energy, hence cause it to decay towards its ground state, to do so it must
shed the surplus in one manner or another – if it can do that, it's not
truly isolated
, but our standard meaning of isolated
only
addresses impact from outside as opposed to impact on its
surroundings; the universe is leaving the system alone, but the system is
not obliged to leave the universe alone. However, in the real world, just as
the system tends to lose energy to its surroundings, it surroundings will be
tending to lose energy to theirs – of which the system is a
part. Thus the system will settle on an equilibrium in which it is shedding
energy (by decaying into lower-energy states) as fast as it is being inundated
with energy. This need not be the system's ground state; and the more surplus
energy the system has, typically, the more states available to it with (at least
approximately) that energy.
Thus, in practice, real material systems generally have observed properties
which leave ample ambiguity as to (the internal details of) their state, hence
ample scope for random superposition of possible states. When two such systems
are in a position to exchange energy, determining which will shed more to the
other than it gets back depends on understanding their respective states and the
effect, on each, of gaining and losing energy. The study of such questions is
known as thermodynamics
. One simple case that's fairly
tractable to analyse is a model of a solid in terms of
each molecule vibrating in three directions, modelling each direction's
vibration as an independent simple harmonic
oscillator.
In so far as the internal states of molecules support a meaningful
temperature, one may study how they exchange
energy – both internal and gross (external) kinetic – in
collisions: this we may understand as micro-thermodynamics. On the larger
scale, we can consider how bodies, each comprising many molecules, exchange
energy when they interact: this macro-thermodynamics
is what is usually
meant by thermodynamics.
The first law of thermodynamics says heat is (a form of) energy (also known as work or mass). This is a standard observable of quantum systems; and the formalism for handling quantum superpositions of highly degenerate systems is all predicated on the observables of the sysem; so let's see what rôle heat gets to take in the solution.
Written by Eddy.