Entropy and quantum mechanics

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Thus, in classical mechanics the

The non-unique value of the entropy comes about because
there is no limit to the precision to which the state of a classical system can be
specified. In other words, the cell size can be made arbitrarily small, which
corresponds to specifying the particle coordinates and momenta to arbitrary
accuracy. However, in quantum mechanics the uncertainty principle sets a
definite
limit to how accurately the particle coordinates and momenta can be specified.
In general,

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Consider a simple quantum mechanical system consisting of non-interacting
spinless particles of mass confined in a cubic box of dimension .
The energy levels of the th particle are given by

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Clearly, as the energy approaches the ground-state energy,
the number of accessible states becomes far less than
the usual classical estimate . This is true for all quantum mechanical systems.
In general, the number of microstates
varies roughly like

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At low temperatures, great care must be taken to ensure that equilibrium thermodynamical arguments are applicable, since the rate of attaining equilibrium may be very slow. Another difficulty arises when dealing with a system in which the atoms possess nuclear spins. Typically, when such a system is brought to a very low temperature the entropy associated with the degrees of freedom not involving nuclear spins becomes negligible. Nevertheless, the number of microstates corresponding to the possible nuclear spin orientations may be very large. Indeed, it may be just as large as the number of states at room temperature. The reason for this is that nuclear magnetic moments are extremely small, and, therefore, have extremely weak mutual interactions. Thus, it only takes a tiny amount of heat energy in the system to completely randomize the spin orientations. Typically, a temperature as small as degrees kelvin above absolute zero is sufficient to randomize the spins.

Suppose that the system consists of atoms
of spin . Each spin can have two possible orientations. If there is enough
residual heat energy in the system to randomize the spins then each orientation
is equally likely. If follows that there are
accessible spin
states. The entropy associated with these states is
. Below some critical temperature, , the interaction between the
nuclear spins becomes significant, and the system settles down in
some unique quantum mechanical ground-state (*e.g.*, with all spins aligned).
In this situation,
,
in accordance with the third law of thermodynamics. However, for temperatures
which are small, but not small enough to ``freeze out'' the nuclear spin degrees
of freedom, the entropy approaches a limiting value which depends only
on the kinds of atomic nuclei in the system. This limiting value is independent
of the spatial arrangement of the atoms, or the interactions between them.
Thus, for most practical purposes the third law of thermodynamics can be written

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