The basic equation of relativistic quantum mechanics; stated by Paul Dirac in 1928. It expresses the behaviour of electron waves in a way consistent with special relativity, requiring that electrons have spin ½, and predicting the existence of an antiparticle partner to the electron (the positron).
In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and 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.
Introduction
Since the Dirac equation was originally invented to describe the electron, we will generally speak of "electrons" in this article.
The Dirac equation describes the probability amplitudes for a single electron.
Despite these successes, Dirac's theory is flawed by its neglect of the possibility of creating and destroying particles, one of the basic consequences of relativity.
A similar equation for spin 3/2 particles is called the Rarita-Schwinger equation.
Details
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 ψ(x, t) is a four-component wavefunction.Dirac matrices
A convenient (but not unique) choice of αs is
known as Dirac matrices.
These matricies are often called gamma matrices, and they form a Clifford algebra whose defining property is
where η is the Minkowski metric and I is the Identity matrix.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, ψ(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. 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. 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. 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}≡AB+BA, and δ is the Kronecker delta, which has the value 1 if its two subscripts are equal and 0 otherwise.
These conditions cannot be satisfied if the α's are ordinary numbers, but they can be satisfied if the α's are matrices.
In the introduction, we presented the representation used by Dirac. This representation can be more compactly written as
where 0 and I are the 2×2 zero and identity matrices, respectively, and the σj's (j = 1, 2, 3) are the Pauli matrices.
The Hamiltonian in this equation,
is called the Dirac Hamiltonian.
Quaternion representation
The Dirac equation can be written nicely in quaternion notation. We write it in terms of two quaternion fields representing the left-handed (Ψ) and right-handed (Φ) electrons:
It is important which side the unit quaternions are multiplied on for this to work.
Nature of the wavefunction
Since the wavefunction ψ is acted on by the 4×4 Dirac matrices, it must be a four-component object.
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.
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.
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 ψ0 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 ψ0 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 −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.
Hole theory
The negative E solutions found in the preceding section are problematic, for it was assumed that the particle has a positive energy.
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.
Dirac further reasoned that if the negative-energy eigenstates are incompletely filled, each unoccupied eigenstate – called a hole – would behave like a positively charged particle.
It is not entirely satisfactory to describe the "vacuum" using an infinite sea of negative-energy electrons.
In certain applications of condensed matter physics, however, the underlying concepts of "hole theory" are valid.
Electromagnetic interaction
So far, we have considered an electron that is not in contact with any external fields. 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 φ are the electromagnetic vector and scalar potentials, respectively.
By setting φ = 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
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.
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 ρ is the electric charge density and j is the electric current density defined earlier. It is a relativistically covariant scalar quantity, as we can see by writing it in terms of the current-charge four-vector j = (ρc,j) and the potential four-vector A = (φ/c,A):
where η is the metric of flat spacetime:
η00 = 1,Lagrangian
The classical Lagrangian density of a spin 1/2 fermion with a mass m is given by
whereTo obtain an equation of motion, one can plug this lagrangian into the Euler-Lagrange equation:
Evealutating the two terms separately:
Plugging those back into the Euler-Lagrange equation results in
which is identitcal to the Dirac equation:
Relativistically covariant notation
Let us return to the Dirac equation for the free electron.
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.
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 ψ.
A notation called the "Feynman slash" is sometimes used. Writing
the Dirac equation becomes
and the expression for the action becomes
In this notation electromagnetic interaction can be added simply by promoting the partial derivative to gauge covariant derivative:
Dirac bilinears
There are five different (neutral) Dirac bilinear terms not involving any derivatives:
(S)calar: (scalar, P-even) (P)seudoscalar: (scalar, P-odd) (V)ector: (vector, P-even) (A)xial: (vector, P-odd) (T)ensor: (antisymmetric tensor)A Dirac mass term is an S coupling.
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