The distribution begins as a Dirac delta function, indicating that all the particles are located at the origin at time t = 0, and for increasing times they become flatter and flatter until the distribution becomes uniform in the asymptotic time limit.

Examples include states whose wavefunctions are Dirac delta functions or infinite plane waves.

for α > − 1 / 2 and where δ is the Dirac delta function.

Augustin-Louis Cauchy exploited such a density function in 1827 with an infinitesimal scale parameter, defining what would now be called a Dirac delta function.

* The Dirac delta function in mathematics

where δ is the Dirac delta function.

An " impulse " in a continuous time filter means a Dirac delta function ; in a discrete time filter the Kronecker delta function would apply.

One common practice ( not discussed above ) is to handle that divergence via Dirac delta and Dirac comb functions.

where is the representation of the wavefunction in Dirac notation, and is the Kronecker delta function.

However, one can measure the position ( alone ) of a moving free particle, creating an eigenstate of position with a wavefunction that is very large ( a Dirac delta ) at a particular position x, and zero everywhere else.

where stands for the Dirac delta function.

where the Dirac delta function denotes a unit source concentrated at the point

Schematic representation of the Dirac delta function by a line surmounted by an arrow.

The Dirac delta function, or function, is ( informally ) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.

From a purely mathematical viewpoint, the Dirac delta is not strictly a function, because any extended-real function that is equal to zero everywhere but a single point must have total integral zero.

While from this perspective the Dirac delta can usually be manipulated as though it were a function, formally it must be defined as a distribution that is also a measure.

In many applications, the Dirac delta is regarded as a kind of limit ( a weak limit ) of a sequence of functions having a tall spike at the origin.

Rigorous treatment of the Dirac delta requires measure theory or the theory of distributions.

The Dirac delta is used to model a tall narrow spike function ( an impulse ), and other similar abstractions such as a point charge, point mass or electron point.

Further developments included generalization of the Fourier integral, " beginning with Plancherel's pathbreaking L < sup > 2 </ sup >- theory ( 1910 ), continuing with Wiener's and Bochner's works ( around 1930 ) and culminating with the amalgamation into L. Schwartz's theory of distributions ( 1945 )...", and leading to the formal development of the Dirac delta function.

The Dirac delta function as such was introduced as a " convenient notation " by Paul Dirac in his influential 1930 book Principles of Quantum Mechanics.

< u > Center-right column :</ u > Original function is discretized ( multiplied by a Dirac comb ) ( top ).

The Fourier transform of a periodic function, s < sub > P </ sub >( t ), with period P, becomes a Dirac comb function, modulated by a sequence of complex coefficients:

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Consequently, a common practice is to model " sampling " as a multiplication by the Dirac comb function, which of course is only " possible " in a purely mathematical sense .</ ref >

The DTFT of a periodic sequence, s < sub > N </ sub >, with period N, becomes another Dirac comb function, modulated by the coefficients of a Fourier series.

In the classical limit, i. e. at large values of or at small density of states — when wave functions of particles practically do not overlap — both the Bose – Einstein or Fermi – Dirac distribution become the Boltzmann distribution.

The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner.

Although Richard Feynman noted that a simplistic interpretation of the relativistic Dirac equation runs into problems with electron orbitals at Z > 1 / α = 137, suggesting that neutral atoms cannot exist beyond untriseptium, and that a periodic table of elements based on electron orbitals therefore breaks down at this point, a more rigorous analysis calculates the limit to be Z ≈ 173, but also that this limit would not actually spell the end of the periodic table.

The high-pass filter with lower band edge B < sub > H </ sub > is just a transparent filter minus a sinc filter, which makes it clear that the Dirac delta function is the limit of a narrow-in-time sinc filter:

We can also say that the measure is a single atom at x ; however, treating the Dirac measure as an atomic measure is not correct when we consider the sequential definition of Dirac delta, as the limit of a delta sequence.

In the limit of approaching zero, the Wigner semicircle distribution becomes a Dirac delta function.

However, this can be handled by taking the limit, and leads to a Dirac delta function in the integrand.

where is the Dirac delta function and the limit must be understood in the space of Schwartz distributions, then ' is a mollifier.

Reduction of the Dirac equation for an electron in a magnetic field to its non-relativistic limit yields the Schrödinger equation with a correction term which takes account of the interaction of the electron's intrinsic magnetic moment with the magnetic field giving the correct energy.

The Pauli theory may be seen as the low energy limit of the Dirac theory in the following manner.

* Domenico Giulini, André Großardt, " The Schrödinger-Newton equation as non-relativistic limit of self-gravitating Klein-Gordon and Dirac fields " arXiv: 1206. 4250

The Dirac delta is not a function in the traditional sense as no function defined on the real numbers has these properties.

This is the classical result in the sense that it does not take into account that electrons obey Fermi – Dirac statistics.

In order to make any sense out of this, one must assume that the " bare vacuum " must have an infinite positive charge density which is exactly cancelled by the Dirac sea.

: This is necessary to ensure that the transform captures a possible jump in g ( x ) at x = 0, as is needed to make sense of the Laplace transform of the Dirac delta function.

In the sense of distributions, there is no difference between multiplier operators and convolution operators ; every multiplier T can also be expressed in the form for some distribution K, known as the convolution kernel of T. In this view, translation by an amount x < sub > 0 </ sub > is convolution with a Dirac delta function δ (· − x < sub > 0 </ sub >), differentiation is convolution with δ '.

A consequence of this apparent paradox is that the electric field of a point-charge can only be described in a limiting sense by a carefully constructed Dirac delta function.

The attempted interpretation of the Dirac equation as a single-particle equation could not be maintained long, however, and finally it was shown that several of its undesirable properties ( such as negative-energy states ) could be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this case for the quantized " Dirac field " or the " electron field ", with the " negative-energy solutions " pointing to the existence of anti-particles.

The neutral kaon, a boson, is also a Dirac particle in a sense.