Dirac equation

SearchAdvancedHelp

(0.001 seconds)

11—20 of 27 matching pages

11: 1.4 Calculus of One Variable

… ►

1.4.23_1 $$\alpha (d)-\alpha (c)={\int }_c^{d}w(x)dx,$$ $[c,d]\subset I$.

ⓘ

Symbols:

$[a,b]$: closed interval, $dx$: differential of $x$ , $\int$: integral, $\subset$: is contained in and $w(x)$: weight function

Referenced by:

§1.4(v), Erratum (V1.2.0) Section 1.4

Permalink:

http://dlmf.nist.gov/1.4.E23_1

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►

1.4.23_2 $${\int }_a^{b}f(x)d\alpha (x)={\int }_a^{b}f(x)w(x)dx,$$ $f$ integrable with respect to $d\alpha$.

ⓘ

Symbols:

$dx$: differential of $x$ , $\int$: integral and $w(x)$: weight function

Referenced by:

§1.4(v)

Permalink:

http://dlmf.nist.gov/1.4.E23_2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

… ►If, for example, $\alpha (x)=H\left(x-x_n\right)$, the Heaviside unit step-function (1.16.14), then the corresponding measure $d\alpha (x)$ is $\delta \left(x-x_n\right)dx$, where $\delta \left(x-x_n\right)$ is the Dirac $\delta$-function of §1.17, such that, for $f(x)$ a continuous function on $(a,b)$, ${\int }_a^{b}f(x)d\alpha (x)=f(x_n)$ for $x_n\in (a,b)$ and $0$ otherwise. Delta distributions and Dirac $\delta$-functions are discussed in §§1.16(iii), 1.16(iv) and 1.17. … ►Let $d\alpha (x)=w(x)dx+{\sum }_{n=1}^N{w}_n\delta \left(x-x_n\right)dx$, $x_n\in (a,b)$, $n=1,\mathrm{\dots }N$. …

12: 33.14 Definitions and Basic Properties

… ►

§33.14(i) Coulomb Wave Equation

… ►The function $s(ϵ,\mathrm{\ell };r)$ has the following properties: ►

33.14.13 $${\int }_0^{\mathrm{\infty }}s({ϵ}_1,\mathrm{\ell };r)s({ϵ}_2,\mathrm{\ell };r)dr=\delta \left({ϵ}_1-{ϵ}_2\right),$$ ${ϵ}_1,{ϵ}_2>0$,

ⓘ

Symbols:

$s(ϵ,\mathrm{\ell };r)$: regular Coulomb function, $\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and $ϵ$: real parameter

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.0.24):

The constraint ${ϵ}_1,{ϵ}_2>0$ was added.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

►where the right-hand side is the Dirac delta (§1.17). … ►

33.14.15 $${\int }_0^{\mathrm{\infty }}{\varphi }_{m,\mathrm{\ell }}(r){\varphi }_{n,\mathrm{\ell }}(r)dr={\delta }_{m,n}.$$

ⓘ

Symbols:

${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\ell }$: nonnegative integer, $r$: real variable and ${\varphi }_{n,\mathrm{\ell }}(r)$: function

Proof sketch:

Derivable from (33.14.14) by combining it with Dunkl (2003, Theorem 2.2).

Referenced by:

§13.10(vi), §13.23(v), §33.14(iv), Erratum (V1.0.24) for Equation (33.14.15), Erratum (V1.0.24) for Subsection 33.14(iv)

Permalink:

http://dlmf.nist.gov/33.14.E15

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.24):

This equation, originally written as a definite integral ${\int }_0^{\mathrm{\infty }}{\varphi }_{n,\mathrm{\ell }}^2(r)dr=1$, was rewritten more generally as an orthonormality relation.

See also:

Annotations for §33.14(iv), §33.14 and Ch.33

…

13: Bibliography C

… ►

Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

ⓘ

External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

Encodings:

BibTeX

… ►

Elementary Differential Equations. Clarendon Press, Oxford.

ⓘ

External Links:

ISBN 0-19-853139-7, MathReview, ZentralBlatt

Referenced by:

§16.12.

Encodings:

BibTeX

… ►

The fourth Painlevé equation and associated special polynomials. J. Math. Phys. 44 (11), pp. 5350–5374.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§32.8(iv).

Encodings:

BibTeX

… ►

Numerical evaluation of the Fermi-Dirac integrals. The Astrophysical Journal Supplement Series 71, pp. 677–699.

ⓘ

Notes:

Double-precision Fortran, maximum precision 12D.

External Links:

ISSN 0067-0049, Document, Code

Referenced by:

§25.12(iii), §25.18(i), 3rd item, 3rd item.

Encodings:

BibTeX

… ►

Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London.

ⓘ

External Links:

MathReview (M. Zlámal)

Referenced by:

§1.18(ix), §1.18(x).

Encodings:

BibTeX

…

14: 20.13 Physical Applications

… ►The functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, provide periodic solutions of the partial differential equation …with $\kappa =-\mathrm{i}\pi /4$. … ►is also a solution of (20.13.2), and it approaches a Dirac delta (§1.17) at $t=0$. …Theta-function solutions to the heat diffusion equation with simple boundary conditions are discussed in Lawden (1989, pp. 1–3), and with more general boundary conditions in Körner (1989, pp. 274–281). … ►This allows analytic time propagation of quantum wave-packets in a box, or on a ring, as closed-form solutions of the time-dependent Schrödinger equation.

15: Bibliography G

… ►

On the computation of generalized Fermi-Dirac and Bose-Einstein integrals. Comput. Phys. Comm. 74 (2), pp. 233–238.

ⓘ

External Links:

ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§25.12(iii), §25.18(i), 5th item.

Encodings:

BibTeX

… ►

The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

ⓘ

External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

Encodings:

BibTeX

… ►

Algorithm 745: Computation of the complete and incomplete Fermi-Dirac integral. ACM Trans. Math. Software 21 (3), pp. 221–232.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 20D.

External Links:

ISSN 0098-3500, Document, GAMS (module 745; package TOMS), ZentralBlatt

Referenced by:

6th item.

Encodings:

BibTeX

… ►

Generalized Fermi-Dirac functions and derivatives: Properties and evaluation. Comput. Phys. Comm. 136 (3), pp. 294–309.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 20D

External Links:

Document, Code, ZentralBlatt

Referenced by:

7th item.

Encodings:

BibTeX

… ►

Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

ⓘ

External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

Encodings:

BibTeX

…

16: 10.22 Integrals

… ►

10.22.37 $${\int }_0^{1}t{J}_{\nu }\left(j_{\nu ,\mathrm{\ell }}t\right)J_{\nu }\left(j_{\nu ,m}t\right)dt=\frac{1}{2}{\left(J_{\nu }^{\prime }\left(j_{\nu ,\mathrm{\ell }}\right)\right)}^2{\delta }_{\mathrm{\ell },m},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $m$: integer and $\nu$: complex parameter

A&S Ref:

11.4.5

Referenced by:

§10.22(ii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/10.22.E37

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

… ►

10.22.38 $${\int }_0^{1}t{J}_{\nu }\left({\alpha }_{\mathrm{\ell }}t\right)J_{\nu }\left({\alpha }_{m}t\right)dt=\left(\frac{a^2}{b^2}+{\alpha }_{\mathrm{\ell }}^2-{\nu }^2\right)\frac{{(J_{\nu }\left({\alpha }_{\mathrm{\ell }}\right))}^2}{2{\alpha }_{\mathrm{\ell }}^2}{\delta }_{\mathrm{\ell },m},$$

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, ${\delta }_{j,k}$: Kronecker delta, $dx$: differential of $x$ , $\int$: integral, $m$: integer and $\nu$: complex parameter

A&S Ref:

11.4.5

Referenced by:

§10.22(ii), Erratum (V1.0.17) for Equations (10.22.37), (10.22.38), (14.17.6)–(14.17.9)

Permalink:

http://dlmf.nist.gov/10.22.E38

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.17):

The Kronecker delta symbol has been moved furthest to the right.

See also:

Annotations for §10.22(ii), §10.22(ii), §10.22 and Ch.10

… ►Equation (10.22.70) also remains valid if the order $\nu +1$ of the $J$ functions on both sides is replaced by $\nu +2n-3$, $n=1,2,\mathrm{\dots }$, and the constraint $\mathrm{\Re }\nu >-\frac{3}{2}$ is replaced by $\mathrm{\Re }\nu >-n+\frac{1}{2}$. ►See also §1.17(ii) for an integral representation of the Dirac delta in terms of a product of Bessel functions. … ►

10.22.79 $$f(x)={\int }_0^{\mathrm{\infty }}{(xt)}^{\frac{1}{2}}\frac{c{J}_{\nu }\left(xt\right)+t^{2\nu }{J}_{-\nu }\left(xt\right)}{c^2+2c\cos \left(\nu \pi \right)t^{2\nu }+t^{4\nu }}{\int }_0^{\mathrm{\infty }}{(yt)}^{\frac{1}{2}}\left(c{J}_{\nu }\left(yt\right)+t^{2\nu }{J}_{-\nu }\left(yt\right)\right)f(y)dydt,$$ .

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $x$: real variable, $y$: real variable, $\nu$: complex parameter and $f(x)$: function

Referenced by:

§1.18(ix), §1.18(vi), §10.22(v), §10.22(vi), Erratum (V1.2.0) for Chapter 10 Additions

Permalink:

http://dlmf.nist.gov/10.22.E79

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §10.22(v), §10.22 and Ch.10

…

17: Bibliography F

… ►

The Lamé wave equation. Uspekhi Mat. Nauk 44 (1(265)), pp. 123–144, 248 (Russian).

ⓘ

Notes:

In Russian. English translation: Russian Math. Surveys, 44(1989), no. 1, pp. 153–180

External Links:

ISSN 0042-1316, Document, MathReview (R. G. Langebartel), ZentralBlatt

Referenced by:

§29.11.

Encodings:

BibTeX

… ►

On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

Encodings:

BibTeX

►

From continuous to discrete Painlevé equations. J. Math. Anal. Appl. 180 (2), pp. 342–360.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§32.7(ii).

Encodings:

BibTeX

… ►

Finite Differences and Difference Equations in the Real Domain. Clarendon Press, Oxford.

ⓘ

External Links:

MathReview (W. J. Trjitzinsky), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

… ►

Generalized Fermi-Dirac integrals—FD, FDG, FDH. Comput. Phys. Comm. 39 (2), pp. 181–185.

ⓘ

External Links:

ISSN 0010-4655, Document, Code, MathReview

Referenced by:

4th item.

Encodings:

BibTeX

…

18: Bibliography M

… ►

Algorithm 779: Fermi-Dirac functions of order $-1/2$, $1/2$, $3/2$, $5/2$ . ACM Trans. Math. Software 24 (1), pp. 1–12.

ⓘ

Notes:

Double-precision Fortran, maximum accuracy 16D.

External Links:

ISSN 0098-3500, Document, GAMS (module 779; package TOMS), ZentralBlatt

Referenced by:

8th item.

Encodings:

BibTeX

… ►

On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

ⓘ

External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

Encodings:

BibTeX

►

The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

ⓘ

External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

Encodings:

BibTeX

… ►

Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

ⓘ

External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

Encodings:

BibTeX

… ►

The accurate evaluation of a particular Fermi-Dirac integral. Comput. Phys. Comm. 101 (1-2), pp. 47–53.

ⓘ

External Links:

Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

…

19: Bibliography N

… ►

Toda equation and its solutions in special functions. J. Phys. Soc. Japan 65 (6), pp. 1589–1597.

ⓘ

External Links:

ISSN 0031-9015, Document, MathReview, ZentralBlatt

Referenced by:

§18.38(ii).

Encodings:

BibTeX

… ►

On the numerical evaluation of the generalised Fermi-Dirac integrals. Comput. Phys. Comm. 76 (1), pp. 48–50.

ⓘ

External Links:

ISSN 0010-4655, Document, ZentralBlatt

Referenced by:

§25.18(i).

Encodings:

BibTeX

… ►

Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

Encodings:

BibTeX

… ►

Uniform Asymptotic Approximations of Solutions of Second-order Linear Differential Equations, with a Coalescing Simple Turning Point and Simple Pole. Ph.D. Thesis, University of Maryland, College Park, MD.

ⓘ

Referenced by:

§2.8(vi).

Encodings:

BibTeX

… ►

Solving equations exactly. J. Res. Nat. Bur. Standards Sect. B 71B, pp. 171–179.

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§27.15.

Encodings:

BibTeX

…

20: 14.30 Spherical and Spheroidal Harmonics

… ►For a series representation of the product of two Dirac deltas in terms of products of spherical harmonics see §1.17(iii). … ►As an example, Laplace’s equation ${\nabla }^{2}W=0$ in spherical coordinates (§1.5(ii)): … ►In the quantization of angular momentum the spherical harmonics $Y_{l,m}(\theta ,\varphi )$ are normalized solutions of the eigenvalue equations … ►

14.30.11_5 $${\mathrm{L}}_z{Y}_{l,m}=\mathrm{\hslash }m{Y}_{l,m},$$ $m=-l,-1+1,\mathrm{\dots },0,\mathrm{\dots },l-1,l$,

ⓘ

Symbols:

$Y_{l,m}(\theta ,\varphi )$: spherical harmonic, $m$: integer, $l$: nonnegative integer and ${\mathrm{L}}_z$: $z$ component of the angular momentum operator

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E11_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

… ►

14.30.13 $${\mathrm{L}}_z=-\mathrm{i}\mathrm{\hslash }\frac{\partial }{\partial \varphi };$$

ⓘ

Defines:

${\mathrm{L}}_z$: $z$ component of the angular momentum operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ and $z$: complex variable

Referenced by:

§14.30(iv), Erratum (V1.1.0) for Additions

Permalink:

http://dlmf.nist.gov/14.30.E13

Encodings:

pMML, png, TeX

Addition (effective with 1.1.0):

This equation was added.

See also:

Annotations for §14.30(iv), §14.30 and Ch.14

…