Download Quantum mechanical description of identical particles

IDENTICAL PARTICLES
Identical particles, or indistinguishable particles, are particles that cannot be
distinguished from one another, even in principle. Species of identical particles
include elementary particles such as electrons, as well as composite microscopic
particles such as atoms and molecules.
There are two main categories of identical particles: bosons, which can share quantum
states, and fermions, which are forbidden from sharing quantum states (this property
of fermions is known as the Pauli Exclusion Principle.) Examples of bosons are
photons, gluons, phonons, and helium-4 atoms. Examples of fermions are electrons,
neutrinos, quarks, protons and neutrons, and helium-3 atoms.
The fact that particles can be identical has important consequences in statistical
mechanics. Calculations in statistical mechanics rely on probabilistic arguments,
which are sensitive to whether or not the objects being studied are identical. As a
result, identical particles exhibit markedly different statistical behavior from
distinguishable particles. For example, the indistinguishability of particles has been
proposed as a solution to Gibbs' mixing paradox.
Quantum mechanical description of identical particles
We will now make the above discussion concrete, using the formalism developed in
the article on the mathematical formulation of quantum mechanics.
For simplicity, consider a system composed of two identical particles. As the particles
possess equivalent physical properties, their state vectors occupy mathematically
identical Hilbert spaces. If we denote the Hilbert space of a single particle as H, then
the Hilbert space of the combined system is formed by the tensor product
.
Let n denote a complete set of (discrete) quantum numbers for specifying singleparticle states (for example, for the particle in a box problem we can take n to be the
quantized wave vector of the wavefunction.) Suppose that one particle is in the state
n1, and another is in the state n2. What is the quantum state of the system? We might
guess that it is
which is simply the canonical way of constructing a basis for a tensor product space
from the individual spaces. However, this expression implies that we can identify the
particle with n1 as "particle 1" and the particle with n2 as "particle 2", which conflicts
with the ideas about indistinguishability discussed earlier.
Actually, it is an empirical fact that identical particles occupy special types of multiparticle states, called symmetric states and antisymmetric states. Symmetric states
have the form
Antisymmetric states have the form
Note that if n1 and n2 are the same, our equation for the antisymmetric state gives
zero, which cannot be a state vector as it cannot be normalized. In other words, in an
antisymmetric state the particles cannot occupy the same single-particle states. This is
known as the Pauli Exclusion Principle, and it is the fundamental reason behind the
chemical properties of atoms and the stability of matter.
Exchange symmetry
The importance of symmetric and antisymmetric states is ultimately based on
empirical evidence. It appears to be a fact of Nature that identical particles do not
occupy states of a mixed symmetry, such as
There is actually an exception to this rule, which we will discuss later. On the other
hand, we can show that the symmetric and antisymmetric states are in a sense special,
by examining a particular symmetry of the multiple-particle states known as exchange
symmetry.
Let us define a linear operator P, called the exchange operator. When it acts on a
tensor product of two state vectors, it exchanges the values of the state vectors:
P is both Hermitian and unitary. Because it is unitary, we can regard it as a symmetry
operator. We can describe this symmetry as the symmetry under the exchange of
labels attached to the particles (i.e., to the single-particle Hilbert spaces).
Clearly, P² = 1 (the identity operator), so the eigenvalues of P are +1 and −1. The
corresponding eigenvectors are the symmetric and antisymmetric states:
In other words, symmetric and antisymmetric states are essentially unchanged under
the exchange of particle labels: they are only multiplied by a factor of +1 or −1, rather
than being "rotated" somewhere else in the Hilbert space. This indicates that the
particle labels have no physical meaning, in agreement with our earlier discussion on
indistinguishability.
We have mentioned that P is Hermitian. As a result, it can be regarded as an
observable of the system, which means that we can, in principle, perform a
measurement to find out if a state is symmetric or antisymmetric. Furthermore, the
equivalence of the particles indicates that the Hamiltonian can be written in a
symmetrical form, such as
It is possible to show that such Hamiltonians satisfy the commutation relation
According to the Heisenberg equation, this means that the value of P is a constant of
motion. If the quantum state is initially symmetric (antisymmetric), it will remain
symmetric (antisymmetric) as the system evolves. Mathematically, this says that the
state vector is confined to one of the two eigenspaces of P, and is not allowed to range
over the entire Hilbert space. Thus, we might as well treat that eigenspace as the
actual Hilbert space of the system. This is the idea behind the definition of Fock
space.