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S1s. The classical limit of quantum mechanics
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Classical mechanics is often seen as the formal limit hbar-->0 of
quantum mechanics. Strictly speaking, this cannot be true since hbar
is a constant of nature, which is often even set to one to have
convenient units. The classical limit really is the limit of large
quantum numbers M (typically of mass, number of particles, or size of
angular momentum), when attention is limited to quantities whose
uncertainties are small compared to their expectations.
In these situations, the effect is similar to taking the limit
hbar --> 0. In these cases the relative uncertainties scale with
sqrt(hbar/M), which becomes small if either hbar is made formally
tiny or if M is large.
Indeed, a quantum system is essentially classical if its relevant
quantities have uncertainties that are small compared to their
expectations.
The relation between classical mechanics is most easily seen if --
as in statistical mechanics -- quantum mechnaics is presented in terms
of mixed states, which correspond to density matrices.
(Almost all quantum mechanics applied to real systems not in
the ground state needs density matrices, since pure states are very
difficult to create and propagate unless a system is in the ground
state. Pure states describe only an idealized version of quantum
reality, which in statistical mechanics appears as the approximation
in the cold limit T-->0.)
Density matrices are intrinsically quantum mechanical.
Nevertheless they exhibit very close analogies to classical densities.
Therefore everyone interested in the relations between classical and
quantum mechanics is well-advised to look at both theories in the
statistical mechanics version, where the analogies are obvious, and
the transition from quantum to classical takes the form of a simple
approximation.
QM in the statistical mechanics version is almost as intuitive as
classical statistical mechanics. The only somewhat nonintuitive part
is in both cases how to interpret probability. (This is already a
severe problem in classical statistical mechanics, as the book by
Laurence Sklar, Physics and Chance, explains in detail.)
A density matrix describes the stochastic behavior of a quantum system
in the same way as a density function describes the stochastic behavior
of a classical system. In both cases, if the system is nice enough that
the stochastic uncertainties (square roots of variances) in the
quantities of interest are much smaller than the quantities themselves,
one can form a deterministic approximation.
This deterministic approximation is given by a classical dynamical
system for the (expectations of the) quantities of interest.
Thus, in a sense, classical variables are simply expectations of
relevant quantum variables with small uncertainty. Then (and only then)
is a deterministic approximation adequate. The small uncertainty
makes these variables approximately predictable in each individual
event, and hence classical.
Classicality therefore develops whenever the uncertainties of the
quantities of interest become small compared to their expectations.
Of course, there is significant interest in quantum systems where this
does not happen, since these are decidedly non-classical, but quantum
theory gets its strange, counterintuitive feature only when one
concentrates on these systems only.
For more details, see, e.g., Sections 7.3-7.5 of
A. Neumaier and D. Westra,
Classical and Quantum Mechanics via Lie algebras
http://de.arxiv.org/abs/0810.1019