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Dirac equation
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The Dirac equation is a relativistic quantum mechanical wave equation
formulated by Paul Dirac in 1928. It provides a description of
elementary spin-½ particles, such as electrons, consistent with
both the principles of quantum mechanics and the theory of special
relativity. It also accounts in a natural way for the nature of particle
spin. The equation demands the existence of antiparticles and actually
predated their experimental discovery, making the discovery of the
positron, the antiparticle of the electron, one of the greatest triumphs
of modern theoretical physics.
Since the Dirac equation was originally invented to describe the
electron, we will generally speak of "electrons" in this article.
Actually, the equation also applies to quarks, which are also elementary
spin-½ particles. A modified Dirac equation can be used to
approximately describe protons and neutrons, which are not elementary
particles (they are made up of quarks). Another modification of the
Dirac equation, called the Majorana equation, is thought to describe
neutrinos.
The Dirac equation is
where m is the rest mass of the electron, c is the speed of light, p is
the momentum operator, is the reduced Planck's constant, x and t are
the space and time coordinates respectively, and Ï?(x, t) is a
four-component wavefunction. (The wavefunction has to be formulated as a
four-component spinor, rather than a simple scalar, due to the demands
of special relativity. The physical meanings of the components are
discussed below.)
The αs are linear operators that act on the wavefunction. Their
most fundamental property is that they must anticommute with each other.
In other words,
αiαj = â?' αjαi,
where , and i and j range from zero to three. The simplest way to obtain
such properties is with 4Ã?4 matrices. There is no set of matrices
of smaller dimension fulfilling the anticommutation requirements. That
four dimensional matrices are necessary turns out to have physical
significance.
A convenient (but not unique) choice of αs is
,
,
known as Dirac matrices. All possible choices are related by similarity
transformations because Dirac spinors are unique representation
theoretically.
The Dirac equation describes the probability amplitudes for a single
electron. This is a single-particle theory, in other words it does not
account for the creation and destruction of the particles. It gives a
good prediction of the magnetic moment of the electron and explains much
of the fine structure observed in atomic spectral lines. It also
explains the spin of the electron. Two of the four solutions of the
equation correspond to the two spin states of the electron. The other
two solutions make the peculiar prediction that there exist an infinite
set of quantum states in which the electron possesses negative energy.
This strange result led Dirac to predict, via a remarkable hypothesis
known as "hole theory", the existence of particles behaving like
positively-charged electrons. This prediction was verified by the
discovery of the positron in 1932.
Despite these successes, the theory is flawed by its neglect of the
possibility of creating and destroying particles, one of the basic
consequences of relativity. This difficulty is resolved by reformulating
it as a quantum field theory. Adding a quantized electromagnetic field
to this theory leads to the theory of quantum electrodynamics (QED).
A similar equation for spin 3/2 particles is called the Rarita-Schwinger
equation.
Contents
1 Derivation of the Dirac equation
1.1 Nature of the wavefunction
1.2 Energy spectrum
1.3 Hole theory
2 Electromagnetic interaction
2.1 Interaction Hamiltonian
3 Relativistically covariant notation
4 See also
5 References
5.1 Selected papers
5.2 Textbooks
[edit]
Derivation of the Dirac equation
The Dirac equation is a relativistic extension of the Schrödinger
equation, which describes the time-evolution of a quantum mechanical
system:
For convenience, we will work in the position basis, in which the state
of the system is represented by a wavefunction, Ï?(x,t). In this
basis, the Schrödinger equation becomes
where the Hamiltonian H now denotes an operator acting on wavefunctions
rather than state vectors.
We have to specify the Hamiltonian so that it appropriately describes
the total energy of the system in question. Let us consider a "free"
electron isolated from all external force fields. For a non-relativistic
model, we adopt a Hamiltonian analogous to the kinetic energy of
classical mechanics (ignoring spin for the moment):
where the p's are the momentum operators in each of the three spatial
directions j=1,2,3. Each momentum operator acts on the wavefunction as a
spatial derivative:
To describe a relativistic system, we have to find a different
Hamiltonian. Assume that the momentum operators retain the above
definition. According to Albert Einstein's famous mass-momentum-energy
relationship, the total energy of a system is given by
This prescribes something like
This is not a satisfactory equation, for it does not treat time and
space on an equal footing, one of the basic tenets of special
relativity. The square of this equation leads to the Klein-Gordon
equation. Dirac reasoned that, since the right side of the equation
contains a first-order derivative in time, the left side should contain
equally simple first-order derivatives in space (i.e., in the momentum
operators). One way for this to happen is if the quantity in the square
root is a perfect square. Suppose that you set
Here, I stands for the identity element. You'll gain the free Dirac
equation:
where the α's are constants to be determined thanks to the
relativistic total energy.
Expanding the square and comparing coefficients on each side, we obtain
the following conditions for the α's:
These last conditions may be written more concisely as
where {...} is the anticommutator, defined as {A,B}â?¡AB+BA, and
δ is the Kronecker delta, which has the value 1 if its two
subscripts are equal and 0 otherwise. See Clifford algebra.
These conditions cannot be satisfied if the α's are ordinary
numbers, but they can be satisfied if the α's are matrices. The
matrices must be Hermitian, so that the Hamiltonian is Hermitian. The
smallest matrices that work are 4Ã?4 matrices, but there is more
than one possible choice, or representation, of matrices. Although the
choice of representation does not affect the properties of the Dirac
equation, it does affect the physical meaning of the individual
components of the wavefunction.
In the introduction, we presented the representation used by Dirac. This
representation can be more compactly written as
where 0 and I are the 2Ã?2 zero and identity matrices, respectively,
and the Ï?j's (j=1,2,3) are the Pauli matrices.
The Hamiltonian in this equation,
is called the Dirac Hamiltonian.
[edit]
Nature of the wavefunction
Since the wavefunction Ï? is acted on by the 4Ã?4 Dirac
matrices, it must be a four-component object. We will see, in the next
section, that the wavefunction contains two sets of degrees of freedom,
one associated with positive energies and the other with negative
energies, with each set containing two degrees of freedom that describe
the probability amplitudes for the spin to be pointing "up" or "down"
along a specified direction.
We may explicitly write the wavefunction as a column matrix:
The dual wavefunction can be written as a row matrix:
where the * superscript denotes complex conjugation. By comparison, the
dual of a scalar (one-component) wavefunction is just its complex
conjugate.
As in ordinary single-particle quantum mechanics, the "absolute square"
of the wavefunction gives the probability density of the particle at
each position x and time t. In this case, the "absolute square" is the
scalar product of the wavefunction with its dual:
The conservation of probability gives the normalization condition
By applying Dirac's equation, we can examine the local flow of
probability:
The probability current J is given by
Multiplying J by the electron charge e yields the electric current
density j carried by the electron.
The values of the wavefunction components depend on the coordinate
system. Dirac showed how Ï? transforms under general changes of
coordinate system, including rotations in three-dimensional space as
well as Lorentz transformations between relativistic frames of
reference. It turns out that Ï? does not transform like a vector
under rotations and is in fact a type of object known as a spinor.
[edit]
Energy spectrum
It is instructive to find the energy eigenstates of the Dirac
Hamiltonian. To do this, we must solve the time-independent
Schrödinger equation,
where Ï?0 is the time-independent part of the energy eigenfunction:
Let us look for a plane-wave solution. For convenience, we align the z
axis with the direction in which the particle is moving, so that
where w is a constant four-component spinor and p is the momentum of the
particle, as we can verify by applying the momentum operator to this
wavefunction. In the Dirac representation, the equation for Ï?0
reduces to the eigenvalue equation:
For each value of p, there are two eigenspaces, both two-dimensional.
One eigenspace contains positive eigenvalues, and the other negative
eigenvalues, of the form:
The positive eigenspace is spanned by the eigenstates:
and the negative eigenspace by the eigenstates:
where
The first spanning eigenstate in each eigenspace has spin pointing in
the +z direction ("spin up"), and the second eigenstate has spin
pointing in the â?'z direction ("spin down").
In the non-relativistic limit, the ε spinor component reduces to
the kinetic energy of the particle, which is negligible compared to pc:
In this limit, therefore, we can interpret the four wavefunction
components as the respective amplitudes of (i) spin-up with positive
energy, (ii) spin-down with positive energy, (iii) spin-up with negative
energy, and (iv) spin-down with negative energy. This description is not
accurate in the relativistic regime, where the non-zero spinor
components have similar sizes.
[edit]
Hole theory
The negative E solutions found in the preceding section are problematic,
for relativistic mechanics tells us that the energy of a particle at
rest (p = 0) should be E = mc² rather than E = â?'mc².
Mathematically speaking, however, there seems to be no reason for us to
reject the negative-energy solutions. Since they exist, we cannot simply
ignore them, for once we include the interaction between the electron
and the electromagnetic field, any electron placed in a positive-energy
eigenstate would decay into negative-energy eigenstates of successively
lower energy by emitting excess energy in the form of photons. Real
electrons obviously do not behave in this way.
To cope with this problem, Dirac introduced the hypothesis, known as
hole theory, that the vacuum is the many-body quantum state in which all
the negative-energy electron eigenstates are occupied. This description
of the vacuum as a "sea" of electrons is called the Dirac sea. Since the
Pauli exclusion principle forbids electrons from occupying the same
state, any additional electron would be forced to occupy a
positive-energy eigenstate, and positive-energy electrons would be
forbidden from decaying into negative-energy eigenstates.
Dirac further reasoned that if the negative-energy eigenstates are
incompletely filled, each unoccupied eigenstate â?" called a hole
â?" would behave like a positively charged particle. The hole
possesses a positive energy, since energy is required to create a
particleâ?"hole pair from the vacuum. Dirac initially thought that
the hole was a proton, but Hermann Weyl pointed out that the hole should
behave as if it had the same mass as an electron, whereas the proton is
over a thousand times heavier. The hole was eventually identified as the
positron, experimentally discovered by Carl Anderson in 1932.
It is not entirely satisfactory to describe the "vacuum" using an
infinite sea of negative-energy electrons. We must postulate that the
negative-energy electrons do not contribute to the total energy and
momentum of the vacuum, which would otherwise be infinite, and that the
negative-energy electrons do not produce an electric field, although
they can be affected by an external field. These difficulties led
physicists to abandon hole theory in favour of Dirac field theory, which
bypasses the problem of negative energy states by treating positrons as
true particles.
In certain applications of condensed matter physics, however, the
underlying concepts of "hole theory" are valid. The sea of conduction
electrons in an electrical conductor, called a Fermi sea, contains
electrons with energies up to the chemical potential of the system. An
unfilled state in the Fermi sea behaves like a positively-charged
electron, though it is referred to as a "hole" rather than a "positron".
The negative charge of the Fermi sea is balanced by the
positively-charged ionic lattice of the material.
[edit]
Electromagnetic interaction
So far, we have considered an electron that is not in contact with any
external fields. Proceeding by analogy with the Hamiltonian of a charged
particle in classical electrodynamics, we can modify the Dirac
Hamiltonian to include the effect of an electromagnetic field. The
revised Hamiltonian is (in SI units):
where e is the electric charge of the electron (in this convention, e is
negative), and A and Ï? are the electromagnetic vector and scalar
potentials, respectively.
By setting Ï? = 0 and working in the non-relativistic limit, Dirac
solved for the top two components in the positive-energy wavefunctions
(which, as discussed earlier, are the dominant components in the
non-relativistic limit), obtaining
where B = Ã?A is the magnetic field acting on the particle. This is
precisely the Pauli equation for a non-relativistic spin-½
particle, with magnetic moment (i.e., a spin g-factor of 2). The actual
magnetic moment of the electron is larger than this, though only by
about 0.12%. The shortfall is due to quantum fluctuations in the
electromagnetic field, which have been neglected. See vertex function.
For several years after the discovery of the Dirac equation, most
physicists believed that it also described the proton and the neutron,
which are both spin-½ particles. However, beginning with the
experiments of Stern and Frisch in 1933, the magnetic moments of these
particles were found to disagree significantly with the predictions of
the Dirac equation. The proton has a magnetic moment 2.79 times larger
than predicted (with the proton mass inserted for m in the above
formulas), i.e., a g-factor of 5.58. The neutron, which is electrically
neutral, has a g-factor of â?'3.83. These "anomalous magnetic
moments" were the first experimental indication that the proton and
neutron are not elementary particles. They are in fact composed of
smaller particles called quarks. Incidentally, quarks are spin-½
particles, which are exactly described by the Dirac equation !
[edit]
Interaction Hamiltonian
It is noteworthy that the Hamiltonian can be written as the sum of two
terms:
where Hfree is the Dirac Hamiltonian for a free electron and Hint is the
Hamiltonian of the electromagnetic interaction. The latter may be
written as
It has the expected value
where Ï? is the electric charge density and j is the electric
current density defined earlier. The integrand in the final expression
is the interaction energy density. It is a relativistically covariant
scalar quantity, as we can see by writing it in terms of the
current-charge four-vector j = (Ï?c,j) and the potential four-vector
A = (Ï?/c,A):
where η is the metric of flat spacetime:
η00 = 1
[edit]
Relativistically covariant notation
Let us return to the Dirac equation for the free electron. It is often
useful to write the equation in a relativistically covariant form, in
which the derivatives with time and space are treated on the same
footing.
To do this, first recall that the momentum operator p acts like a
spatial derivative:
Multiplying each side of the Dirac equation by α0 (recalling that
α0²=I) and plugging in the above definition of p, we obtain
Now, define four gamma matrices:
These matrices possess the property that
where η once again stands for the metric of flat spacetime. These
relations define a Clifford algebra called the Dirac algebra.
The Dirac equation may now be written, using the position-time
four-vector x = (ct,x), as
With this notation, the Dirac equation can be generated by extremising
the action
where
is called the Dirac adjoint of Ï?. This is the basis for the use of
the Dirac equation in quantum field theory.
A notation called the "Feynman slash" is sometimes used. Writing
the Dirac equation becomes
and the expression for the action becomes
[edit]
See also
Breit equation
Klein-Gordon equation
Quantum electrodynamics
[edit]
References
[edit]
Selected papers
P.A.M. Dirac, Proc. R. Soc. A117 610 (1928)
P.A.M. Dirac, Proc. R. Soc. A126 360 (1930)
C.D. Anderson, Phys. Rev. 43, 491 (1933)
R. Frisch and O. Stern, Z. Phys. 85 4 (1933)
[edit]
Textbooks
Dirac, P.A.M., Principles of Quantum Mechanics, 4th edition (Clarendon,
1982)
Shankar, R., Principles of Quantum Mechanics, 2nd edition (Plenum, 1994)
Bjorken, J D & Drell, S, Relativistic Quantum mechanics
Thaller, B., The Dirac Equation, Texts and Monographs in Physics
(Springer, 1992)
Retrieved from "http://en.wikipedia.org/wiki/Dirac_equation";
Categories: Quantum mechanics | Quantum field theory | Equations
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