Principles of Optics: 60th Anniversary Edition [7th Edition. 60th Anniversary Edition] 9781108477437

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Principles of Optics: 60th Anniversary Edition [7th Edition. 60th Anniversary Edition]
 9781108477437

Table of contents :
Contents
Foreword by Sir Peter Knight
Historical introduction
I Basic properties of the electromagnetic field
1.1 The electromagnetic field
1.1.1 Maxwell's equations
1.1.2 Material equations
1.1.3 Boundary conditions at a surface of discontinuity
1.1.4 The energy law of the electromagnetic field
1.2 The wave equation and the velocity of light
1.3 Scalar waves
1.3.1 Plane waves
1.3.2 Spherical waves
1.3.3 Harmonic waves. The phase velocity
1.3.4 Wave packets. The group velocity
1.4 Vector waves
1.4.1 The general electromagnetic plane wave
1.4.2 The harmonic electromagnetic plane wave
a) Elliptic polarization
b) Linear and circular polarization
c) Characterization of the state of polarization by Stokes parameters
1.4.3 Harmonic vector waves of arbitrary form
1.5 Reflection and refraction of a plane wave
1.5.1 The laws of reflection and refraction
1.5.2 Fresnel formulae
1.5.3 The reflectivity and transmissivity; polarization on reflection and refraction
1.5.4 Total reflection
1.6 Wave propagation in a stratified medium. Theory of dielectric films
1.6.1 The basic differential equations
1.6.2 The characteristic matrix of a stratified medium
a) A homogeneous dielectric film
b) A stratified medium as a pile of thin homogeneous films
1.6.3 The reflection and transmission coefficients
1.6.4 A homogeneous dielectric film
1.6.5 Periodically stratified media
II Electromagnetic potentials and polarization
2.1 The electrodynamic potentials in the vacuum
2.1.1 The vector and scalar potentials
2.1.2 Retarded potentials
2.2 Polarization and magnetization
2.2.1 The potentials in terms of polarization and magnetization
2.2.2 Hertz vectors
2.2.3 The field of a linear electric dipole
2.3 The Lorentz-Lorenz formula and elementary dispersion theory
2.3.1 The dielectric and magnetic susceptibilities
2.3.2 The effective field
2.3.3 The mean polarizability: the Lorentz-Lorenz formula
2.3.4 Elementary theory of dispersion
2.4 Propagation of electromagnetic waves treated by integral equations
2.4.1 The basic integral equation
2.4.2 The Ewald-Oseen extinction theorem and a rigorous derivation of the Lorentz-Lorenz formula
2.4.3 Refraction and reflection of a plane wave, treated with the help of the Ewald-Oseen extinction theorem
III Foundations of geometrical optics
3.1 Approximation for very short wavelengths
3.1.1 Derivation of the eikonal equation
3.1.2 The light rays and the intensity law of geometrical optics
3.1.3 Propagation of the amplitude vectors
3.1.4 Generalizations and the limits of validity of geometrical optics
3.2 General properties of rays
3.2.1 The differential equation of light rays
3.2.2 The laws of refraction and reflection
3.2.3 Ray congruences and their focal properties
3.3 Other basic theorems of geometrical optics
3.3.1 Lagrange's integral invariant
3.3.2 The principle of Fermat
3.3.3 The theorem of Malus and Dupin and some related theorems
IV Geometrical theory of optical imaging
4.1 The characteristic functions of Hamilton
4.1.1 The point characteristic
4.1.2 The mixed characteristic
4.1.3 The angle characteristic
4.1.4 Approximate form of the angle characteristic of a refracting surface of revolution
4.1.5 Approximate form of the angle characteristic of a reflecting surface of revolution
4.2 Perfect imaging
4.2.1 General theorems
4.2.2 Maxwell's 'fish-eye'
4.2.3 Stigmatic imaging of surfaces
4.3 Projective transformation (collineation) with axial symmetry
4.3.1 General formulae
4.3.2 The telescopic case
4.3.3 Classification of projective transformations
4.3.4 Combination of projective transformations
4.4 Gaussian optics
4.4.1 Refracting surface of revolution
4.4.2 Reflecting surface of revolution
4.4.3 The thick lens
4.4.4 The thin lens
4.4.5 The general centred system
4.5 Stigmatic imaging with wide-angle pencils
4.5.1 The sine condition
4.5.2 The Herschel condition
4.6 Astigmatic pencils of rays
4.6.1 Focal properties of a thin pencil
4.6.2 Refraction of a thin pencil
4.7 Chromatic aberration. Dispersion by a prism
4.7.1 Chromatic aberration
4.7.2 Dispersion by a prism
4.8 Radiometry and apertures
4.8.1 Basic concepts of radiometry
4.8.2 Stops and pupils
4.8.3 Brightness and illumination of images
4.9 Ray tracing
4.9.1 Oblique meridional rays
4.9.2 Paraxial rays
4.9.3 Skew rays
4.10 Design of aspheric surfaces
4.10.1 Attainment of axial stigmatism
4.10.2 Attainment of aplanatism
4.11 Image-reconstruction from projections (computerized tomography)
4.11.1 Introduction
4.11.2 Beam propagation in an absorbing medium
4.11.3 Ray integrals and projections
4.11.4 The N-dimensional Radon transform
4.11.5 Reconstruction of cross-sections and the projection-slice theorem of computerized tomography
V Geometrical theory of aberrations
5.1 Wave and ray aberrations; the aberration function
5.2 The perturbation eikonal of Schwarzschild
5.3 The primary (Seidel) aberrations
a) Spherical aberration (B ≠ 0)
b) Coma (F ≠ 0)
c) Astigmatism (C ≠ 0) and curvature of field (D ≠ 0)
d) Distortion (E ≠ 0)
5.4 Addition theorem for the primary aberrations
5.5 The primary aberration coefficients of a general centred lens system
5.5.1 The Seidel formulae in terms of two paraxial rays
5.5.2 The Seidel formulae in terms of one paraxial ray
5.5.3 Petzval's theorem
5.6 Example: The primary aberrations of a thin lens
5.7 The chromatic aberration of a general centred lens system
VI Image-forming instruments
6.1 The eye
6.2 The camera
6.3 The refracting telescope
6.4 The reflecting telescope
6.5 Instruments of illumination
6.6 The microscope
VII Elements of the theory of interference and interferometers
7.1 Introduction
7.2 Interference of two monochromatic waves
7.3 Two-beam interference: division of wave-front
7.3.1 Young's experiment
7.3.2 Fresnel's mirrors and similar arrangements
7.3.3 Fringes with quasi-monochromatic and white light
7.3.4 Use of slit sources; visibility of fringes
7.3.5 Application to the measurement of optical path difference: the Rayleigh interferometer
7.3.6 Application to the measurement of angular dimensions of sources: the Michelson stellar interferometer
7.4 Standing waves
7.5 Two-beam interference: division of amplitude
7.5.1 Fringes with a plane-parallel plate
7.5.2 Fringes with thin films; the Fizeau interferometer
7.5.3 Localization of fringes
7.5.4 The Michelson interferometer
7.5.5 The Twyman-Green and related interferometers
7.5.6 Fringes with two identical plates: the Jamin interferometer and interference microscopes
7.5.7 The Mach-Zehnder interferometer; the Bates wave-front shearing interferometer
7.5.8 The coherence length; the application of two-beam interference to the study of the fine structure of spectral lines
7.6 Multiple-beam interference
7.6.1 Multiple-beam fringes with a plane-parallel plate
7.6.2 The Fabry-Perot interferometer
7.6.3 The application of the Fabry-Perot interferometer to the study of the fine structure of spectral lines
7.6.4 The application of the Fabry-Perot interferometer to the comparison of wavelengths
7.6.5 The Lummer-Gehrcke interferometer
7.6.6 Interference filters
7.6.7 Multiple-beam fringes with thin films
7.6.8 Multiple-beam fringes with two plane-parallel plates
a) Fringes with monochromatic and quasi-monochromatic light
b) Fringes of superposition
7.7 The comparison of wavelengths with the standard metre
VIII Elements of the theory of diffraction
8.1 Introduction
8.2 The Huygens-Fresnel principle
8.3 Kirchhoff's diffraction theory
8.3.1 The integral theorem of Kirchhoff
8.3.2 Kirchhoff's diffraction theory
8.3.3 Fraunhofer and Fresnel diffraction
8.4 Transition to a scalar theory
8.4.1 The image field due to a monochromatic oscillator
8.4.2 The total image field
8.5 Fraunhofer diffraction at apertures of various forms
8.5.1 The rectangular aperture and the slit
8.5.2 The circular aperture
8.5.3 Other forms of aperture
8.6 Fraunhofer diffraction in optical instruments
8.6.1 Diffraction gratings
a) The principle of the diffraction grating
b) Types of grating
c) Grating spectrographs
8.6.2 Resolving power of image-forming systems
8.6.3 Image formation in the microscope
a) Incoherent illumination
b) Coherent illumination - Abbe's theory
c) Coherent illumination - Zernike's phase contrast method of observation
8.7 Fresnel diffraction at a straight edge
8.7.1 The diffraction integral
8.7.2 Fresnel's integrals
8.7.3 Fresnel diffraction at a straight edge
8.8 The three-dimensional light distribution near focus
8.8.1 Evaluation of the diffraction integral in terms of Lommel functions
8.8.2 The distribution of intensity
a) Intensity in the geometrical focal plane
b) Intensity along the axis
c) Intensity along the boundary of the geometrical shadow
8.8.3 The integrated intensity
8.8.4 The phase behaviour
8.9 The boundary diffraction wave
8.10 Gabor's method of imaging by reconstructed wave-fronts (holography)
8.10.1 Producing the positive hologram
8.10.2 The reconstruction
8.11 The Rayleigh-Sommerfeld diffraction integrals
8.11.1 The Rayleigh diffraction integrals
8.11.2 The Rayleigh-Sommerfeld diffraction integrals
IX The diffraction theory of aberrations
9.1 The diffraction integral in the presence of aberrations
9.1.1 The diffraction integral
9.1.2. The displacement theorem. Change of reference sphere
9.1.3. A relation between the intensity and the average deformation of wave-fronts
9.2 Expansion of the aberration function
9.2.1 The circle polynomials of Zernike
9.2.2 Expansion of the aberration function
9.3 Tolerance conditions for primary aberrations
9.4 The diffraction pattern associated with a single aberration
9.4.1 Primary spherical aberration
9.4.2 Primary coma
9.4.3 Primary astigmatism
9.5 Imaging of extended objects
9.5.1 Coherent illumination
9.5.2 Incoherent illumination
X Interference and diffraction with partially coherent light
10.1 Introduction
10.2 A complex representation of real polychromatic fields
10.3 The correlation functions of light beams
10.3.1 Interference of two partially coherent beams. The mutual coherence function and the complex degree of coherence
10.3.2 Spectral representation of mutual coherence
10.4 Interference and diffraction with quasi-monochromatic light
10.4.1 Interference with quasi-monochromatic light. The mutual intensity
10.4.2 Calculation of mutual intensity and degree of coherence for light from an extended incoherent quasi-monochromatic source
a) The van Cittert-Zernike theorem
b) Hopkins' formula
10.4.3 An example
10.4.4 Propagation of mutual intensity
10.5 Interference with broad-band light and the spectral degree of coherence. Correlation-induced spectral changes
10.6 Some applications
10.6.1 The degree of coherence in the image of an extended incoherent quasi-monochromatic source
10.6.2 The influence of the condenser on resolution in a microscope
a) Critical illumination
b) Kohler's illumination
10.6.3 Imaging with partially coherent quasi-monochromatic illumination
a) Transmission of mutual intensity through an optical system
b) Images of transilluminated objects
10.7 Some theorems relating to mutual coherence
10.7.1 Calculation of mutual coherence for light from an incoherent source
10.7.2 Propagation of mutual coherence
10.8 Rigorous theory of partial coherence
10.8.1 Wave equations for mutual coherence
10.8.2 Rigorous formulation of the propagation law for mutual coherence
10.8.3 The coherence time and the effective spectral width
10.9 Polarization properties of quasi-monochromatic light
10.9.1 The coherency matrix of a quasi-monochromatic plane wave
a) Completely unpolarized light (natural light)
b) Complete polarized light
10.9.2 Some equivalent representations. The degree of polarization of a light wave
10.9.3 The Stokes parameters of a quasi-monochromatic plane wave
XI Rigorous diffraction theory
11.1 Introduction
11.2 Boundary conditions and surface currents
11.3 Diffraction by a plane screen: electromagnetic form of Babinet's principle
11.4 Two-dimensional diffraction by a plane screen
11.4.1 The scalar nature of two-dimensional electromagnetic fields
11.4.2 An angular spectrum of plane waves
11.4.3 Formulation in terms of dual integral equations
11.5 Two-dimensional diffraction of a plane wave by a half-plane
11.5.1 Solution of the dual integral equations for E-polarization
11.5.2 Expression of the solution in terms of Fresnel integrals
11.5.3 The nature of the solution
11.5.4 The solution for H-polarization
11.5.5 Some numerical calculations
11.5.6 Comparison with approximate theory and with experimental results
11.6 Three-dimensional diffraction of a plane wave by a half-plane
11.7 Diffraction of a field due to a localized source by a half-plane
11.7.1 A line-current parallel to the diffracting edge
11.7.2 A dipole
11.8 Other problems
11.8.1 Two parallel half-planes
11.8.2 An infinite stack of parallel, staggered half-planes
11.8.3 A strip
11.8.4 Further problems
11.9 Uniqueness of solution
XII Diffraction of light by ultrasonic waves
12.1 Qualitative description of the phenomenon and summary of theories based on Maxwell's differential equations
12.1.1 Qualitative description of the phenomenon
12.1.2 Summary of theories based on Maxwell's equations
12.2 Diffraction of light by ultrasonic waves as treated by the integral equation method
12.2.1 Integral equation for E-polarization
12.2.2 The trial solution of the integral equation
12.2.3 Expressions for the amplitudes of the light waves in the diffracted and reflected spectra
12.2.4 Solution of the equations by a method of successive approximations
12.2.5 Expressions for the intensities of the first and second order lines for some special cases
12.2.6 Some qualitative results
12.2.7 The Raman-Nath approximation
XIII Scattering from inhomogeneous media
13.1 Elements of the scalar theory of scattering
13.1.1 Derivation of the basic integral equation
13.1.2 The first-order Born approximation
13.1.3 Scattering from periodic potentials
13.1.4 Multiple scattering
13.2 Principles of diffraction tomography for reconstruction of the scattering potential
13.2.1 Angular spectrum representation of the scattered field
13.2.2 The basic theorem of diffraction tomography
13.3 The optical cross-section theorem
13.4 A reciprocity relation
13.5 The Rytov series
13.6 Scattering of electromagnetic waves
13.6.1 The integro-differential equations of electromagnetic scattering theory
13.6.2 The far field
13.6.3 The optical cross-section theorem for scattering of electromagnetic waves
XIV Optics of metals
14.1 Wave propagation in a conductor
14.2 Refraction and reflection at a metal surface
14.3 Elementary electron theory of the optical constants of metals
14.4 Wave propagation in a stratified conducting medium. Theory of metallic films
14.4.1 An absorbing film on a transparent substrate
14.4.2 A transparent film on an absorbing substrate
14.5 Diffraction by a conducting sphere; theory of Mie
14.5.1 Mathematical solution of the problem
a) Representation of the field in terms of Debye's potentials
b) Series expansions for the field components
c) Summary of formulae relating to the associated Legendre functions and to the cylindrical functions
14.5.2 Some consequences of Mie's formulae
a) The partial waves
b) Limiting cases
c) Intensity and polarization of the scattered light
14.5.3 Total scattering and extinction
a) Some general considerations
b) Computational results
XV Optics of crystals
15.1 The dielectric tensor of an anisotropic medium
15.2 The structure of a monochromatic plane wave in an anisotropic medium
15.2.1 The phase velocity and the ray velocity
15.2.2 Fresnel's formulae for the propagation of light in crystals
15.2.3 Geometrical constructions for determining the velocities of propagation and the directions of vibration
a) The ellipsoid of wave normals
b) The ray ellipsoid
c) The normal surface and the ray surface
15.3 Optical properties of uniaxial and biaxial crystals
15.3.1 The optical classification of crystals
15.3.2 Light propagation in uniaxial crystals
15.3.3 Light propagation in biaxial crystals
15.3.4 Refraction in crystals
a) Double refraction
b) Conical refraction
15.4 Measurements in crystal optics
15.4.1 The Nicol prism
15.4.2 Compensators
a) The quarter-wave plate
b) Babinet's compensator
c) Soleil's compensator
d) Berek's compensator
15.4.3 Interference with crystal plates
15.4.4 Interference figures from uniaxial crystal plates
15.4.5 Interference figures from biaxial crystal plates
15.4.6 Location of optic axes and determination of the principal refractive indices of a crystalline medium
15.5 Stress birefringence and form birefringence
15.5.1 Stress birefringence
15.5.2 Form birefringence
15.6 Absorbing crystals
15.6.1 Light propagation in an absorbing anisotropic medium
15.6.2 Interference figures from absorbing crystal plates
a) Uniaxial crystals
b) Biaxial crystals
15.6.3 Dichroic polarizers
Appendices
I The Calculus of variations
1 Euler's equations as necessary conditions for an extremum
2 Hilbert's independence integral and the Hamilton-Jacobi equation
3 The field of extremals
4 Determination of all extremals from the solution of the Hamilton-Jacobi equation
5 Hamilton's canonical equations
6 The special case when the independent variable does not appear explicitly in the integrand
7 Discontinuities
8 Weierstrass' and Legendre's conditions (sufficiency conditions for an extremum)
9 Minimum of the variational integral when one end point is constrained to a surface
10 Jacobi's criterion for a minimum
11 Example I: Optics
12 Example II: Mechanics of material points
II Light optics, electron optics and wave mechanics
1 The Hamiltonian analogy in elementary form
2 The Hamiltonian analogy in variational form
3 Wave mechanics of free electrons
4 The application of optical principles to electron optics
III Asymptotic approximations to integrals
1 The method of steepest descent
2 The method of stationary phase
3 Double integrals
IV The Dirac delta function
V A mathematical lemma used in the rigorous derivation of the Lorentz-Lorenz formula (§2.4.2)
VI Propagation of discontinuities in an electromagnetic field (§3.1.1)
1 Relations connecting discontinuous changes in field vectors
2 The field on a moving discontinuity surface
VII The circle polynomials of Zernike (§9.2.1)
1 Some general considerations
2 Explicit expressions for the radial polynomials
VIII Proof of the inequality for the spectral degree of coherence (§10.5)
IX Proof of a reciprocity inequality (§10.8.3)
X Evaluation of two integrals (§12.2.2)
XI Energy conservation in scalar wavefields (§13.3)
XII Proof of Jones' lemma (§13.3)
Author index
Subject index

Citation preview

 O  Principles of Optics is one of the most highly cited and most influential physics books ever published, and one of the classic science books of the twentieth century. To celebrate the 60th anniversary of this remarkable book’s first publication, the seventh expanded edition has been reprinted with a special foreword by Sir Peter Knight. The seventh edition was the first thorough revision and expansion of this definitive text. Amongst the material introduced in the seventh edition is a section on CAT scans, a chapter on scattering from inhomogeneous media, including an account of the principles of diffraction tomography, an account of scattering from periodic potentials, and a section on the so-called Rayleigh-Sommerfield diffraction theory. This expansive and timeless book continues to be invaluable to advanced undergraduates, graduate students and researchers working in all areas of optics.

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34,

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11.4 Two-dimensional diffraction by a plane screen

11.4.1 The scalar nature of two-dimensional electromagnetic ®elds 5         (    z        5  $                              '       ì > + '  U /ì0       

                  jìj ! 1  /:0     P/ì0  ˆ ) ì:

) U/ì0 ) :ð U/ ì+ 0 /ì ‡ ì+ 0

/B0

            ì ˆ ì+       ! ))6 !             /B0      ì ˆ ì+    )=/:ð0    (            /:0     / kxì+ 0      /B0    ) U/ì0  )‡ì :ð U / ì+ 0

P/ì0 p /ì ‡ ì+ 0 ˆ ) ì

/C0

       /C0     !       

                                    '                          ì J                 ì P/ì0 ! +  jìj ! 1             '           ì ˆ ì+      /C0 

µ0 1

1 0

! ))6 '        ì   7 '  K/r0         z ˆ  /ö0                               )=: Brr+ :)   /è ‡ è0  /)30 K/r0 ˆ :ðj ):/è+ ‡ è0j rB :r: : + R) :

           /    r0         = /)+0     : : )ð  kR) 1 r kR) r B  r /)60   J /è+ 0 ˆ  ð Kr: /R) R90=R) LKr: /R) ‡ R90=R) L +      ):/è+ ‡ è0 _ + !   

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) Bð

1 : ì  kR9 p  ì ì: ‡ :kR9 k/ R) R90

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/),0

))6 8

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2    /)0         : : )ð  kR 1 ì  ì  B  ð ì: ‡ :kR 1

334

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 mˆ m9 ˆ

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      ):/è+ ‡ è0 _ +:  /:)0

'           /:+0  8                 ( {         2  p               kr+  / kr+ 0   r+ ! 1 '     H  1  

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        !                            

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Bð  kr+  kr J /è+ 0 ˆ p  ):/è+ ‡ è0 p p % kr+ kr : :ð    k/R)

/:40

R0  ) )

Bð J / è+ 0 ˆ p  ):/è+ : :ð

 kr+  kr è0 p p : kr+ kr

/:B0

' 

                 =  8  Proc. Lond. Math. Soc. 14 /)*)C0 B)+ { = 2 (  Proc. Lond. Math. Soc. 30 /),**0 ):)

33B

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          hyperbolae k/R) R90 ˆ )  k/R) R0 ˆ )      è ‡ è+ ˆ ð  è è+ ˆ ð  '                 }))C4               !                /:+0  reciprocal            r+  è+  rè '

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          { ,ð % á ˆ ô% : ) }*' 1$+5 ô% ˆ

'1 * $5 : * ‡ *

1,5

12.2.2 The trial solution of the integral equation 6         )        "            1'5      {  : K * * 1ù . 51ù*  * .* 5$=*

ù* = 1ù . 51  5> ? ** 1ù . 51ù*  * .* 5$=*

1$%5

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/E,

C 8          

15

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15

ù* = 1ù . 5 > f= 1ù . 5> ** 1ù . 51ù*  * .* 5$=*

$g?

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‡

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:                                        .      

12.2.4 Solution of the equations by a method of successive approximations  }$**,  }$**&           Ä        1$E5D1*%5                !                                     }$**/  !  }$**+ 1$E5D1*%5                   @  ˆ %

1 ˆ % $ * . . .5

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12.2.5 Expressions for the intensities of the ®rst and second order lines for some special cases 67 ä8î  9   î    &.                       $   *   !            

           ‡ %%   %%    -   * =â1î  '*5> 3 ‡ 1î  $*5 1î  '*5

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            13.6 Scattering of electromagnetic waves

                                                     ;                                                                   { 13.6.1 The integro-differential equations of electromagnetic scattering theory 7 E ! " !r " ˆ RfE ! " !r ù"

ù 

H ! " !r " ˆ RfH ! " !r ù"

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   x @n n ˆ % s s @x

   y @n n ˆ : s s @y

         z%      z @n n ˆ % s s @z     %  

    3   3  3 x y z ‡ ‡ ˆ : s s s

!0"

!0"

!"

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5   

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        n           %   !0"     

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5 mc 3       45"                 2 pv ˆ r

e9í .  V ::

475

*  v   V     p     4(5   4>5        í .  p ˆ ˆ í . 4 p5: r p

45

*- 45 is identical with } ( 4;5 for the curvature of rays in a medium of refractive index n"        p"                            =      p                         ?                 "   4>5 1             1         "                               "                      8  p   p         "            V 1  -                     "      "    0  -  @  "        =          "         &                       p"     p m "   # %   p  ˆ p m ‡ eA"

4'5

 A         1      "              p m       "                  p       1       "                              &            A + = 4KoÈnigl. Gess. Wiss. GoÈttingen, Math. Phys. Kl." 3 4'

$  

2 The Hamiltonian analogy in variational form

1           8 %   4}  (5"                P   P( &         " … P( n s ˆ  : 4              D    > 1          2 K !    5 4 # 5 6   6   > L 9 -    ,'  0E 1ˆ

      '   ‡   ' ‡   ' '

    ˆ @  =@   ˆ @ '  =@ '    4       "     ˆ   ˆ &   ˆ á         1 ˆ áâ

ã' :

Appendix IV

The Dirac delta function he                                                                               9 äx ˆ    x 6ˆ    = … ‡1   äxx ˆ :  ; 1

!   äx              {                                            "        äx          #         äx ì        ì               x$            

     ì … ‡1 äx ìx ˆ : % 1

!         &  ' ì äx ì ˆ p  ð

ì% x %

:

(

 * +    impulse function "         , -   The Principles of Quantum Mechanics . #   , )(    +                  /      . 0    1 2   ,  0 2 Operational Calculus Based on the Two-Sided Laplace Integral #   #   3   , )4  5%655 {                                     7 1     TheÂorie des distributions 8,   0

 #  9 "  )4 9 ""  )4 : *       1   ;       ? 7   An Introduction to Fourier Analysis and Generalised Functions #   #   3   , )4'

')%

  -      

')(

δ(x, µ) 1.8

1.4 µ4 1.0 µ2 0.6

µ1

0.2 1.6

1.2

0.8

0.4

x 0

0.4

0.8

1.2

1.6

&  ' "          -              ì % % äx ì ˆ p  ì x  ì ˆ  % @: ð             

"         -               ì ! 1          äx ì       x 0 … ‡1 7  äx ìx @ ì!1

1

          We interpret any operation involving äx as implying that this operation is to be performed with a function äx ì of a suitable chosen set such as (, and that the limit ì ! 1 is taken at the end of the calculation A                       äx ì                           *      -         $  sifting property      … ‡1 f xäx ax ˆ f a: 4 1

0 f x         x        4      äx a     äx a ì                    ì !     ì    … ‡1 f xäx a ìx 5 1

         f x           x ˆ a          f x  f a          + 

')@

*   "9

ì      3    4                    äx a         x            a          *                 +  1  ‡1 "               x ˆ a               f xäx

a ˆ f aäx

a

=

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'

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a%  ˆ

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äa

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)

äx

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b:

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1

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 n f  n :

 @

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ä9x

 4

xä9x ˆ

äx:

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ˆ f x%  ˆ f x% 

… x% 

x

f 9xx

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ˆ f : "   x ˆ y a f x ˆ f  y a ˆ F y       x ! 1 x% ! 1   … ‡1 F yU 9 y a y ˆ Fa 1

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*   "9

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a ì ˆ

…ì %ð

ì



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k ˆ

 ìx a ðx a

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%

% 

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1

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1                                           "                       äx y z ˆ äxä yäz

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       är   r         x y z               9 äx y z ˆ    x 6ˆ  y 6ˆ  z 6ˆ    > > > = ‡1 ……… %5 äx y zx  y z ˆ :  > > > ; 1

              ‡1 ………

f x y zäx

a y

b z

cx  y z ˆ f a b c

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1

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  -      

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äx y z ˆ ‡1 ………

1

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9  R n m (r)      r   n        r   m      

  

    n    

   R        m

m  n m R n (r) ˆ t Q  (t)

(3)

 t ˆ r  Q( n m)= (t)      t   (n (6)  Q           m m      t Qk (t)Q k9 (t) t ˆ a n ä kk9 0        k ˆ (n m) k9 ˆ (n9 m):

m) ;

   

(4)

      Q0 (t) Q (t) . . . Qk (t) . . .       ,   %      t t  . . . t k . . .

(1)

      w(t) ˆ t m     0 < t <  #  E!  F  (  )  { Gk ( p q t) ˆ ˆ

(q )!  q t ( (q ‡ k )!

t) q

p

k q Gt

t k

‡ k

(

t) p

k k!(q )!  ( p ‡ k ‡ s )! t s ( ) s ( p ‡ k )! sˆ0 (k s)!s!(q ‡ s )!

q‡ k

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(5) (A)

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06

;  D

w(t) ˆ t q  (

t) p

q

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 bk ( p q) ˆ

k!G(q )!H G p q ‡ kH! : Gq  ‡ kH!G p  ‡ kH!G p ‡ kH

()

(&     bk Gk ( p q 0) ˆ    k) -   (6)  (4)   {  m a  n Qk (t) ˆ (0) Gk (m ‡  m ‡  t): bk (m ‡  m ‡ ) 2 (3)  (0)                 F   ?  m a m n r m Gk (m ‡  m ‡  r ) R n (r) ˆ Gk ˆ (n m)H: () bk (m ‡  m ‡ ) 2   !    ,     n  m m R n () ˆ :

()

  ()  ()  bk (m ‡  m ‡ ) ˆ Gk (m ‡  m ‡  ): m a n

(3)

   Gk (m ‡  m ‡  )              F   { & 

 m  1 z  ‡  z( r ) ‡ z   m‡s

 ˆ Gs (m ‡  m ‡  r )z s : (4) s (zr ) m  z( r ) ‡ z  sˆ0 2 r ˆ   E       ( ‡ z)                      E  ? Gs (m ‡  m ‡  ) ˆ

( ) s : m‡s s

(1)

2 (1)  (3)     > + The Fundamental Principles of Quantum Mechanics (# C! @ =E* 3A)  14 {     %    E    (0)     (5)  { 9 : 7   B * loc. cit. D   

       ! (})

0

   m a n m (n ‡ m) n ˆ ( )   bk (m ‡  m ‡ ) (n m)

(5)



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n m m  R n (r) ˆ (A) (r )  (r )   n m

(r ) !r m  ˆ

 (n m) sˆ0

( ) s n‡ m s! 

(n

s)!  n s !

m 

 rn s !

s

:

(6)

>                   14 m  ,    a      (5)  ()? n m a ˆ n

 : n ‡ 

()

                s     k ˆ (n m)=  m ‡ s     n  ()  (5)     (4) &  

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 ˆ z s R m‡s (r): (30) (zr) m  ‡ z( r ) ‡ z  sˆ0 2         Rm n (r)J m (vr)r r 0

      7 I         !J#$

      &   R m n (r)    (A)    "   J m                   Rm n (r)J m (vr)r r 0

ˆ n  

 

m

1 

 !

sˆ0



 m‡s ( ) s v n m n‡ m n m f s    s!(s ‡ m)! 

  p

f (s p q r) ˆ u s Gu q (u

u 0

) r H u

 : 7   B * loc. cit. D   464

(3)

(3)

0

;  D

p q r s          (3)        

p   p 

s q r s 

f (s p q r) ˆ u Gu (u ) H s u Gu q (u ) r H u:

u

u 0 0

(33)

#  r > p  s ‡ q

p > 0          

f (s p q r) ˆ

sf (s

 p

 q r):

(34)

&       s > p  s , p & s > p      (34) p  f (s p q r) ˆ ( ) p s(s

)(s )    (s p ‡ ) f (s  ( ) p‡ r s!  s‡q p u ( u) r u: ˆ (s p)! 0

p 0 q r) (31)

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(s

s !(s ‡ q p)!r ! : p) !(s ‡ q ‡ r p ‡ ) !

(35)

#       s , p ;  (34) s     f (s p q r) ˆ ( ) s s(s )    f (0 p s q r)    p s 

ˆ ( ) s s! Gu q (u ) r H

u 0 ˆ 0:

(3A)

&    (35)  (3A)   (3)         l    s ˆ (n m) ‡ l &  

 n‡ l‡ 3( n m) 1  ( )   ( ) l v Rm (r)J (vr)r

r ˆ : (36) m n v l!(n ‡ l ‡ )!  0 lˆ0          ,       J n‡ (v) 9  n m      ( )3( n m)=      ( )( n m)=       n m J n‡ (v)  Rm : (3) n (r)J m (vr)r r ˆ ( ) v 0  : 7   B * loc. cit. D   463

 

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