Unformatted text preview: VI. IDENTICAL PARTICLES
Griffiths Chapter 5, Shankar Chapter 10 1 A. Definition
Two particles are said to be identical if they are exact replicas of
each other in every respect.
There should be no experiment that can detect any intrinsic
difference between them.
This definition of identical particles is the same in classical and
quantum physics.
The implications are very different.
2 B. The Classical Case
Imagine a pool table with four holes labelled 1, 2, 3 and 4.
Place two identical billiard balls,
label them 1 and 2, near holes 1 and
2, respectively. 3 4 Let’s shoot these two balls down the
table at the same time.
Since they are identical, the resulting
configuration should appear exactly the
same if we interchanged the two of
them.
3 1 1 2 2 B. The Classical Case 3 4 Let’s shoot the billiard balls towards
the diagonally facing pockets and have
two physicists predict what will happen.
Physicist 1 predicts that ball 1 will end
up in pocket 4, and ball 2 in pocket 3.
Physicist 2 predicts that ball 1 will end
up in pocket 3 and ball 2 in pocket 4. 1 2 1 2 Upon shooting the balls simultaneously, suppose we find that
physicist 2 is right: ball 1 lands in pocket 3 and ball 2 in pocket 4.
Now if a judge walks in only at the end of the experiment, who
will he say is right?
4 B. The Classical Case Now if a judge walks in only at the end of the experiment, who
will he say is right?
3 4 He only sees the final configuration,
which is identical upon exchanging the
balls, and he doesn’t know which ball is
which.
He will conclude that both physicists
are correct.
What is it that we know, that this judge
doesn’t, that allows us to see that
physicist 2 is right?
We have seen the trajectories of the balls.
5 1 2 B. The Classical Case In classical physics, we can distinguish between identical particles by
following their nonidentical trajectories.
We can, in principle, measure these trajectories without disturbing them.
Consequently, two configurations related by exchanging identical particles
are physically nonequivalent.
In quantum physics, we have no trajectories that we can measure without
disturbing the system.
To be true to our postulates and the empirical evidence that inspired them, we
can’t allow ourselves to distinguish configurations based on any notions of
different trajectories.
Consequently, two configurations related by exchanging identical particles
must be treated as one and the same configuration, described by the same
quantum wave function.
6 C. The Quantum Implications
1. Symmetric and Antisymmetric States Suppose we have two distinguishable particles 1 and 2, and a
position measurement shows particle 1 to be at x = a and
particle 2 to be at x=b. We could write this composite state as
ψ (x1 = a, x2 = b) or just ab
position of
p article #1 position of
p article #2 See board VI.C.1.a Since the particles are distinguishable, the exchange state ba>
is a different state.
7 C. The Quantum Implications Now let’s repeat this experiment but with two identical particles.
Suppose we again find one particle at a and the other at b.
Which is the correct state, ab> or ba>?
Answer: Neither!
The are two possibilities for the correct state:
ab + ba symmetric state vector or ab − ba See board VI.C.1.b antisymmetric state vector In either case, it doesn’t matter which particle is at a and which particle is
at b (aside from a minus sign for the antisymmetric case).
8 C. The Quantum Implications 2. Bosons and Fermions Suppose an electron, for example, could exist in either the
symmetric or the antisymmetric state.
Then, by postulate #2, it could also exist in a superposition of
those states.
But this linear combination of symmetric and antisymmetric
states is itself neither symmetric nor antisymmetric. (Check this
yourself!)
So, according to our thinking, this is not a possible state.
Therefore, an electron must be either symmetric or
antisymmetric permanently.
9 C. The Quantum Implications It happens that electrons are antisymmetric.
So are protons and neutrons. These are all examples of FERMIONS.
Photons turn out to be symmetric. Photons, gravitons and pions are
all examples of BOSONS.
Existing in symmetric or antisymmetric states under exchange of
identical particles is then, like spin, an intrinsic property of each type
of quantum particle.
In fact, it turns out (using relativistic quantum mechanics) that all half
integer spin particles are Fermions (they exist in antisymmetric
states) and all integer spin particles are Bosons (they exist in
symmetric states).
10 C. The Quantum Implications How can we be sure that nature behaves like we are predicting here, with
particles being Fermions or Bosons?
Experimentally, we find that Fermions obey the Pauli Exclusion Principle,
whereas Bosons do not.
This means that two identical Fermions cannot exist in the same quantum state.
See board VI.C.2 As a result, Fermions and Bosons obey different statistics when doing
statistical mechanics.
Fermions obey FermiDirac Statistics and Bosons obey BoseEinstein
Statistics. See Grifﬁths section 5.3: Solids
One can therefore do experiments to show that electrons are Fermions, for
example, and that nature does behave with respect to identical particles just as
we think.
...or one can use quantum field theory...
11 C. The Quantum Implications 3. Other Consequences Griffiths goes into a lot of depth about many of the implications
of the existence of Fermions and Bosons for quantum identical
particles.
Exchange Forces
Atoms and the Periodic Table
Solids and Band Structure
Quantum Statistical Mechanics and the Blackbody
Spectrum
Note that none of his discussion deals with entangled states. See
his footnote 2 in section 5.1.1.
12 ...
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 Spring '08
 RESCH
 Photon, Quantum Field Theory, Identical particles

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