Dirac equation

SearchAdvancedHelp

(0.002 seconds)

1—10 of 27 matching pages

1: 33.22 Particle Scattering and Atomic and Molecular Spectra

… ►

§33.22(iv) Klein–Gordon and Dirac Equations

… ►The motion of a relativistic electron in a Coulomb field, which arises in the theory of the electronic structure of heavy elements (Johnson (2007)), is described by a Dirac equation. …

2: 18.39 Applications in the Physical Sciences

… ►Bound state solutions to the relativistic Dirac Equation, for this same problem of a single electron attracted by a nucleus with $Z$ protons, involve Laguerre polynomials of fractional index. …

3: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►of the Dirac delta distribution. Equation (1.18.19) is often called the completeness relation. … ►Applying the representation (1.17.13), now symmetrized as in (1.17.14), as $\frac{1}{x}\delta \left(x-y\right)=\frac{1}{\sqrt{xy}}\delta \left(x-y\right)$, … ►These latter results also correspond to use of the $\delta \left(x-y\right)$ as defined in (1.17.12_1) and (1.17.12_2). … ►Thus, and this is a case where $q(x)$ is not continuous, if $q(x)=-\alpha \delta \left(x-a\right)$, $\alpha >0$, there will be an $L^2$ eigenfunction localized in the vicinity of $x=a$, with a negative eigenvalue, thus disjoint from the continuous spectrum on $[0,\mathrm{\infty })$. …

4: 1.17 Integral and Series Representations of the Dirac Delta

… ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

5: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). …

6: 1.16 Distributions

… ►

§1.16(iii) Dirac Delta Distribution

… ►The Dirac delta distribution is singular. … ►As distributions, the last equation reads … ►

1.16.40 $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\delta \left(t\right){\mathrm{e}}^{\mathrm{i}xt}dt=1;$$

ⓘ

Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.16.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §1.16(viii), §1.16 and Ch.1

… ►Since the quantity on the extreme right of (1.16.41) is equal to $\sqrt{2\pi }⟨\delta ,\varphi ⟩$, as distributions, the result in this equation can be stated as …

7: Bibliography T

… ►

Uniform asymptotic approximation of Fermi-Dirac integrals. J. Comput. Appl. Math. 31 (3), pp. 383–387.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii).

Encodings:

BibTeX

… ►

Rotating black holes: Separable wave equations for gravitational and electromagnetic perturbations. Phys. Rev. Lett. 29 (16), pp. 1114–1118.

ⓘ

External Links:

Document

Referenced by:

§31.17(ii).

Encodings:

BibTeX

… ►

High Speed Numerical Integration of Fermi Dirac Integrals. Master’s Thesis, Naval Postgraduate School, Monterey, CA.

ⓘ

External Links:

FullText

Referenced by:

§25.21(vii).

Encodings:

BibTeX

… ►

Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

ISBN 0198533187, MathReview Entry

Referenced by:

§1.18(x).

Encodings:

BibTeX

… ►

Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

ⓘ

External Links:

Document

Referenced by:

§25.21(vii).

Encodings:

BibTeX

…

8: 2.6 Distributional Methods

… ►To each function in this equation, we shall assign a tempered distribution (i. … ►The Dirac delta distribution in (2.6.17) is given by … ►To derive the asymptotic expansion of ${𝐼}^{\mu }f(x)$, we recall equations (2.6.17) and (2.6.20). …These equations again hold only in the sense of distributions. … ►For rigorous derivations of these results and also order estimates for ${\delta }_n(x)$, see Wong (1979) and Wong (1989, Chapter 6).

9: Software Index

… ► ►►►►

	Open Source													With Book						Commercial
…
25.21(vii) Fermi–Dirac, Bose–Einstein					✓									✓			✓					✓
…
28 Mathieu Functions and Hill’s Equation
…

…

10: Bibliography L

… ►

Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

Encodings:

BibTeX

… ►

Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

Encodings:

BibTeX

… ►

Integral and series representations of the Dirac delta function. Commun. Pure Appl. Anal. 7 (2), pp. 229–247.

ⓘ

External Links:

ISSN 1534-0392, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(iv), (1.18.57).

Encodings:

BibTeX

►

Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.

ⓘ

External Links:

ISSN 0022-0396, Document, MathReview (John Albert), ZentralBlatt

Referenced by:

§22.19(iii).

Encodings:

BibTeX

… ►

The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

ⓘ

Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

…