© M. S. Shell 2008
1/22
last modified 9/21/2009
How the microscopic world works
ChE210A
In order to understand the molecular origins of thermodynamic equilibrium, it is important to
first understand the fundamental principles by which molecules interact.
Moreover, as science
and technology increasingly elucidate and rely upon nanoscale phenomena, molecular driving
forces are a critical piece of any technical education.
Quantum theory
So, how does the world really work?
What are the most fundamental, basic principles that
form the basis of reality as we know it?
Currently our understanding of reality rests upon two
principle fields in physics: quantum theory and relativity.
Both of these theories have been the
subject of stringent experimental tests over the past century.
Perhaps surprisingly, there are
some incompatibilities between quantum theory and relativity, which are driving the develop
ment of new ideas like string theory.
Luckily differences between the two manifest only under
particular circumstances, as in black holes; moreover, it turns out that relativistic effects are not
usually relevant to most everyday problems and to the kinds of problems studied in chemical
thermodynamics.
So we can safely start our discussion at the level of quantum theory.
Quantum mechanics describes the complete timeevolution of any system.
It is most easily
described by imagining a system of fundamental particles, like electrons or protons.
Each
particle has associated with it a position variable,
g1818
g2869
,g1818
g2870
,…
, etc., where each
g1818
is a vector
g4666g1876,g1877,g1878g4667
.
Then, the complete state of the system at any point in time is specified by a function
Ψg4666g1818
g2869
,g1818
g2870
,…,g1872g4667
called the
wavefunction
.
The wavefunction takes on complex values, of the form
g1853g3397g1854g1861
.
The
physical
significance of the
wavefunction is that
Ψ
g1499
g4666g1818
g2869
,g1818
g2870
,…,g1872g4667Ψg4666g1818
g2869
,g1818
g2870
,…,g1872g4667
gives the
joint probability
that particle one is
at
g1818
g2869
, particle 2 is at
g1818
g2870
, etc, at the time
g1872
.
Here
g1499
denotes the complex conjugate, i.e.,
g1853g3398g1854g1861
.
The quantity
Ψ
g1499
Ψ
therefore is always real and positive,
g1853
g2870
g3397g1854
g2870
.
The central focus of quantum mechanics is the complexvalued
wavefunction
Ψ
,
whose interpretation is that
Ψ
g1499
Ψ
gives the probability the system is at a specific
microscopic state at a given time.
The wavefunction describes the evolution of
probabilities
.
This is very different from Newto
nian mechanics, in which each particle has an exact position at time
g1872
, and not a probability
distribution of positions.
Quantum mechanics says that this distribution is the
most
we can
possibly know about the system; we cannot predict the position of the particles to more
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© M. S. Shell 2008
2/22
last modified 9/21/2009
accuracy.
There is some inherent randomness in nature, and the best that we can do is predict
the probabilities of different possible outcomes.
This may sound a bit strange, because we are
not used to this kind of behavior at the macroscopic scale.
Indeed, for large objects, these
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 Spring '09
 Energy, M. S. Shell

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