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IDENTICAL PARTICLES
David J. Jeffery
2012 January 1
1.
INTRODUCTION
Macroscopic objects can never be identical at the microscopic level in practice. Say you
have two macroscopic solid objects that are as identical as the best manufacturing process
can make them. At the micro level the two objects will differ in ways that in number are of
order the number of atoms in the objects. Each pair corresponding atoms (assuming such a
pair is well defined at all) will be in a slightly different thermal state most of the time. In
fact, the bonding arrangement atoms in the two objects is probably almost entirely different
at the micro level.
Microscopic classical particles do not actually exist. But imagine that they do. They
could be also be imagined to be identical. But you could still tell them apart at all times
if you had sufficient information. For example say that you knew all the initial conditions
of the system in which they were embedded. Then, in principle, you could calculate the
future trajectories of all the identical particles and would know which was which at all
times. Alternatively, since they are classical particles, you could imagine observing them as
distinct objects at all times.
Identical quantum mechanical particles are different. First of all, they exist. Fundamental particles like electrons and photons are absolutely identical. They can be distinguished
in space if their wave functions do not overlap or by internal quantum state, most notably
spin, but not otherwise. But otherwise they are absolutely identical. One cannot put marks
on them; they do not have enough degrees of freedom. The two methods for keeping track of
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classical identical particles do not work. When the wave functions of the identical quantum
mechanical particles overlap, they have no distinct trajectories as we will see below. For the
same reason, you cannot observe and thereby keep track of which is which.
What of composite particles like atoms and molecules and for that matter protons
and neutrons which in modern theory are composed of quarks. Composite particles will
be in different internal states due to thermal effects and/or perturbations. However, their
properties (including their behaviors under varying conditions) are identical to a degree
impossible for macroscopic objects. The treatment of composite particles at the level of
SQM is a straightforward generalization of that of fundamental particles as we will see in
§ 4.
We should note that although protons and neutrons are composite particles. There
composite nature only manifests itself in environments (e.g., very kinetic energy environments) thate out of the scope of SQM. So we will usually treat them and speak of them as
fundamental particles—which is OK as long as we know what we mean.
The treatment of identical particles requires another basic axiom in addition to the 5
given in the lecture Elements of Quantum Mechanics. This axiom is the symmetrization
principle—which was alluded to in the Elements of Quantum Mechanics. The symmetrization principle as an axiom cannot be derived, of course. However, there is an argument
based on what is called the exchange degeneracy which shows that symmetrization principle
or something else—we know not what—is needed for quantum mechanics to be consistent.
2.
EXCHANGE DEGENERACY
Consider to identical fundamental particles. We assume that internal degree of freedom
(i.e., spin) are in the same state, and so the only coordinate needed for each is a spatial
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coordinate. Let Ψ(~r1 , ~r 2 ) be their joint wave function, where ~r1 is the coordinate for particle 1
and ~r2 is the coordinate for particle 2. The labels 1 and 2 are formal—i.e., they are allowed
by the formalism—they do not imply that we can distinguish the particles in anyway—and,
in fact, we cannot.
Say we are given wave function Ψ(~r1 , ~r 2 ) and then exchange the particle coordinates
to obtain Ψ(~r2 , ~r 1 ) In a mathematical coordinate sense, Ψ(~r1 , ~r 2 ) and Ψ(~r2 , ~r 1 ) are distinct
functions of the coordinates in general just as functions
f (x, y) = x + y 2
and
f (x, y) = y + x2
(1)
are mathematically distinct functions of the coordinates. However, the two wave functions
are physically the same wave function since exchanging identical particles can make absolutely no physical difference: to be more specific, neither to the appearance of the state at
an instant nor to its evolution. In fact, as we argued in
s and we have just exchanged the formal labels of identical particles.
By valid, we mean that they both satisfy the Schrödinger equation.
The coordinate exchange is usually called particle exchange. The expression particle
exchange makes clear sense for distinct particles. After the exchange, the wave functio If the
particles
Now say that we take linear combination wave function
Ψcom (~r1 , ~r 2 ) = c1 Ψ(~r1 , ~r 2 ) + c2 Ψ(~r2 , ~r 1 ) ,
(2)
where the coefficients are set to preserve normalization. The wave function Ψcom (~r1 , ~r 2 ) is
mathematically distinct in general from Ψ(~r1 , ~r 2 ) and Ψ(~r2 , ~r 1 ) just as
g(x, y) = c1 (x + y 2 ) + c2 (y + x2 )
(3)
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is distinct from f (x, y) = x+y 2 and f (x, y) = y +x2 . We also have to say—as far as we know
so far—that Ψcom (~r1 , ~r 2 ) is physically distinct from Ψ(~r1 , ~r 2 ) and Ψ(~r2 , ~r 1 ). Exchanging the
3.
THE SYMMETRIZATION PRINCIPLE
4.
COMPOSITE PARTICLES
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ACKNOWLEDGMENTS
Support for this work has been provided by the Department of Physics of the University
of Evansville.
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