Fermions

The set of the fermions consists of all particles that carry an odd half-integer multiple of Dirac’s constant (aitch-bar) as their inherent spin. In the fundamental particles the set includes the leptons, the quarks, and the baryons. If a large number of the same kind of fermion comes together in a system (such as a gas), then the overall state function describing that system is antisymmetric with respect to any interchange of particles; that is, if particle A occupies state-1 and particle B occupies state-2 (ψ = ψ(A, B), then ψ(B, A) = -ψ(A, B). That fact means that only one particle can occupy a single given state, other particles in the system being barred from that state by Pauli’s exclusion principle. Thus, large collections of fermions conform to the requirements of Fermi-Dirac statistics.

The state function of any particle in the set conforms to the Dirac equation for particles of spin-1/2 and to the Rarita-Schwinger equation for particles of spin-3/2. Those equations appear respectively as

(Eq’n 1)

and

(Eq’n 2)

In the Dirac equation the state function (ø)
takes the form of a four-component column matrix whose elements represent the
eigenstates of a particle (for example, electron with spin up, electron with
spin down, positron with spin up, and positron with spin down). In the
Rarita-Schwinger equation ψ_{q}
takes the form of a vector-valued spinor that has more components than does the
Dirac spinor.

The complete state function must include a description of the direction in which the particle’s spin axis points, even if only implicitly. The Dirac equation and the Rarita-Schwinger equation constrain the form that the state function can take, in accordance with the principle of least action density by way of the Euler-Lagrange equations. In the absence of an interaction with other particles, the direction in which a particle’s spin axis points does not affect the action that the particle plays out. So once we have solved either of those equations, we can include the spin orientation by multiplying the state function by some factor that behaves as a constant with respect to the differentiation operators in the equations.

Just as there is no absolute velocity, so there is no absolute orientation. The Universe has no state of absolute rest and it has no North Pole. Thus we are free to orient our coordinate frame as we wish: we can, for example, simply draw a straight line from the center of our sun to the center of another star and say that it defines the z-direction of our Cartesian grid. By convention, physicists use the z-axis of the grid as the primary reference in the quantum theory of angular momentum.

A spin-1/2 particle has two eigenstates, which we identify by their eigenvalues: +1/2 (spin up relative to the positive z-direction) and -1/2 (spin down). A spin-3/2 particle has four eigenstates, whose eigenvalues are +3/2, +1/2, -1/2, and -3/2.

Three spin operators exist to extract the relevant measurements of spin from the state function and Reality mimics what they do. Those three exist, one for each axis of the coordinate system, because we cannot measure all three components of a spin vector at once. In mathematical terms, the spin operators don’t commute with each other; that is, we have the commutators ([A,B] = (AB-BA)) as

(Eq’ns 3)

The operators in those commutators take the form

(Eq’ns 4)

for spin-1/2 particles (the Dirac equation) and they take the form

(Eq’ns 5)

for spin-3/2 particles (the Rarita-Schwinger equation).

In order to calculate the expectation value for one of the components of a fermion’s spin vector, we must start by noting that we must include something resembling a column vector, a spinor, as a factor in the state function; that is, we must have

(Eq’n 6)

in which alpha, beta, gamma, and delta represent what look like direction
cosines, but are actually the partial probabilities, of the spin vector. We
interpret the partial probabilities, which are complex numbers, by noting that
α*α
yields the probability of finding the particle in the state s_{x}=+S/2
(the asterisk on the first alpha refers to the complex conjugate of alpha). In
setting up that spinor we have tacitly assumed that we can somehow determine the
values of the partial probabilities, perhaps, for example, by putting the
fermion into a magnetic field and measuring how it interacts with radio waves.
We then calculate the expectation value by the usual method, exploiting Born’s
theorem;

(Eq’n 7)

for the z-component of the spin of a spin-1/2 particle. In that equation I’ve written the spinor as the Hermitian conjugate of a matrix (the complex conjugate of the transposed matrix) in which the asterisks signify the complex conjugates of alpha, beta, gamma, and delta. Because the spinors and the spin matrix do not depend explicitly upon the spatial coordinates, they and their matrix multiplication commute with the psi-functions and the integration, so I pulled them out to the left in going from the first to the second line. Then I used the fact that the remaining integral equals one by definition and carried out the matrix multiplication.

The magnitude of the spin vector itself does not equal one unit of aitch-bar, oddly enough. Angular momentum does weird things in the quantum theory. We calculate the magnitude of the spin vector from

(Eq’n 8)

in which s=1/2 or s=3/2 in this case. Thus

(Eq’ns 9)

That statement tells us that in order to have one Dirac unit of spin parallel to the z-axis the particle’s spin axis must tilt into the x-y plane in an indeterminate way.

When we discuss large numbers of fermions, we don’t refer explicitly to their spin states; rather, we describe the collection of particles with Fermi-Dirac statistics, which describe the distribution of the fermions among the various energy states available to them. Fermi-Dirac statistics, like Bose-Einstein statistics, differ from the classical Maxwell-Boltzmann statistics (reflected in the Maxwell distribution) in the way in which the spins alter the count of microstates that constitute a given macrostate.

In the classical case we treat the particles as distinguishable objects (in essence each particle carries a unique license-plate number), so interchanging two particles between the two states that they occupy (moving particle A from state-1 to state-2 and moving particle B from state-2 to state-1) creates a different microstate, which must be counted toward the total in the given macrostate. Quantum particles of a given species are perfectly indistinguishable from one another, so interchanging two such particles in a collection does not create a new microstate. Thus fermions have fewer microstates in a given macrostate than classical particles do.

Because fermions are indistinguishable, we need only
concern ourselves with the number of particles occupying each single-particle
state, n_{r}, noting that, because of the Pauli exclusion principle, the
options are only zero and one. In a gas consisting of N particles summing over
all of the available states gives us

(Eq’n 10)

and

(Eq’n 11)

in which ε_{r}
represents the per-particle energy of the r-th state, R labels the possible
quantum macrostates of the gas, and E_{R} represents the total energy
contained by the R-th state. The Boltzmann factor associated with the gas,

(Eq’n 12)

describes the relative probability of finding the gas in a particular R-state at the absolute temperature T. With that factor we can calculate the average number of particles in state-s as

(Eq’n 13)

in which Z represents the partition function associated with the gas.

The nature of the exponential allows us to draw the factor describing the state-s away from the rest of the exponential and write the above equation as

(Eq’n 14)

in which Z_{s}(N-n_{s}) represents the partition function
associated with the second sum in both the numerator and the denominator on the
first line. We thus have, taking proper account of Pauli’s exclusion principle,
the average occupation of the state-s as

(Eq’n 15)

Because N represents an extremely large number, the natural logarithm of the partition function will change by a minuscule amount when we subtract relatively small numbers from N, so we can write

(Eq’n 16)

To a good approximation the value of
α_{s}
is independent of s, so α_{s}=α
and we have

(Eq’n 17)

That fact transforms Equation 15 into

(Eq’n 18)

Because of the way in which we define it, α=-μ/kT, represents the chemical potential of the gas. Thus we have the fundamental formula describing the Fermi-Dirac distribution. As required, it produces values between zero and one.

Now we have the basic information that applies to fermions.

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