Section 3. Interstellar absorption lines. 3.1 Equivalent width

Size: px
Start display at page:

Download "Section 3. Interstellar absorption lines. 3.1 Equivalent width"

Transcription

1 Setion 3 Interstellar absorption lines 3.1 Equivalent width We an study diuse interstellar louds through the absorption lines they produe in the spetra of bakground stars. Beause of the low temperatures of diuse interstellar louds, the thermal line broadening is small se ), and the lines are intrinsially narrow few km s 1 ) too narrow to be resolved by typial spetrographs. Sine we an't usually study the line proles, it's often neessary to haraterize the absorption by a simple line-strength measurement, the equivalent width. The equivalent width, W λ, is dened as ) I C W λ = λ I λ dλ = I C λ 1 I ) λ dλ 3.1) Iλ C where I C λ is the interpolated) ontinuum intensity and I λ the atual intensity through the line prole. An equivalent `frequeny' version an be onstruted: ) I C W ν = ν I ν dν = 1 I ) ν dν 3.2) Iν C Iν C where the formal integration limits apply only to the line of interest; in pratie, of ourse, the integration is trunated at the edges of the line, beause the term in brakets is zero elsewhere). Physially, W λ is the width in wavelength units) of the ontinuum that has the same area as the line. The total ux in the line is W λ Iλ C. By onvention and in agreement with the above denitions), absorption-line equivalent widths take positive values and emission lines negative ones). 1

2 Figure 3.1: Some interstellar lines in the UV spetrum of HD I/I Same area 1 W λ Wavelength Figure 3.2: The equivalent width. 11

3 Observationally, it is usual to use W λ, but in model alulations it is more onvenient to work in frequeny spae. To alulate theoretial values of W for omparison with observations requires a knowledge of line broadening mehanisms. 3.2 Line Broadening The primary proesses responsible for line broadening in diuse louds are: i) Natural broadening, intrinsi to the transition and resulting from the Heisenberg unertainty priniple. This gives rise to a Lorentzian prole. ii) Thermal Doppler) broadening, due to random thermal motions of th atoms. For a Maxwellian veloity distribution, the 1-D projeted veloity distribution is Gaussian. iii) `Turbulent' broadening a rude haraterization of marosopi motions e.g., a loud having internal motions) We should also reall that instrumental broadening is almost always signiant in interstellar-line observations. Beause equivalent width is onserved through instrumental broadening, that broadening is haraterized by a funtion φν) or φλ)) whih obeys the normalization φν) dν 1 that is, the equivalent width is just redistributed over frequeny, not hanged). 3.3) Natural Line Broadening From the Unertainty Priniple, some energy level i does not have a perfetly dened energy E i, but is rather a superposition of possible states spread around E i. The lassial result for an absorption ross-setion of a line, 1 aω), is: [ ] 8π e4 ω 4 aω) =, 3.4) 3m 2 e 4 ω 2 ω) γ 2 ω 2 1 Derived in PHAS2116? 12

4 where ω = 2πν ω = 2πν is the angular frequeny, and ν is the line-entre frequeny, and ) 2e 2 γ = ω 2 3m e 3 ) 8π 2 e 2 = ν 2 3m e 3 = ν 2 s 1 3.5) is the lassial damping onstant. This lassial form eqtn. 3.4) still applies in absorption lines, but with two modiations. First, we replae the lassial damping oeient, γ, by the quantum-mehanial damping onstant Γ, the sum of all transition probabilities for natural deay from eah of the lower and upper levels of the transition: Γ = Γ i + Γ j 3.6) where Γ i = l<i A il, 3.7) Γ j = l<j A jl and A il is the probability in units of s 1 ) of a transition from the upper level, i, to lower level l. The resulting absorption prole of a transition between two states reets the intrinsi energy widths of both states. Seondly, we inorporate the osillator strength, f, as a orretion/saling fator, whene eqtn. 3.4 beomes aν) = [ 8π e4 3m 2 e 4f ν 4 ν 2 ν 2 ) 2 + [Γ/2π)] 2 ν 2 ]. 3.8) Line absorption is only important near the resonane frequeny, ν, so we an simplify eqtn. 3.8 by substituting ν ν exept in the ν ν ) term; then using ν 2 ν 2 ) 2 = ν + ν ) 2 ν ν ) 2 2ν ) 2 ν ν ) 2 3.9) 13

5 we obtain aν) [ 2π e4 3m 2 e 4f We an express this as aν) = a φν); ν 2 ν ν ) 2 + [Γ/4π)] 2 ]. 3.1) i.e., a frequeny-independent set of physial onstants, and a frequeny-dependent absorption prole funtion, φν) = C ν ν ) 2 + [Γ/4π)] ) The onstant C an be determined from the normalization ondition, whene 1 = C φν) dν 1 whih is of the form where 1 = C 3.12) dν ν ν ) 2 + Γ/4π) ) dx x 2 + b ) x = ν ν ), b = Γ/4π. 3.15) We an solve this as standard integral, giving + C = Γ 4π 2 dx x 2 + b 2 = π b Thus from eqtn we obtain our nal result, 3.16) Γ/4π 2 ) φν) = 3.17) ν ν ) 2 + [Γ/4π)] 2 Γ = 4π 2 ν ν ) 2 + Γ/2) 2 whih is, as antiipated, a Natural, or Lorentz, prole. 14

6 Figure 3.3: The Lorentz prole showing full-width half maximum). Peak value and width We haraterize the funtion width by the full-width at half maximum, FWHM. The maximum value of φν) ours when ν = ν, giving φν ) = 4/Γ 3.18) from eqtn. 3.17). The value of the prole funtion at half-maximum is thus 2/Γ. We hoose the half-strength point to dene a width; substituting φν) = 2/Γ into eqtn we have Γ 2 2 = 4π2 ν 1/2 ν ) 2 + so ν 1/2 ν ) 2 = Γ 2 /16π 2, or ) 2 Γ 3.19) 2 ν 1/2 = ν ± Γ/4π; 3.2) that is, the full-width at half maximum FWHM), in frequeny units, is 2ν 1/2, or ν 1/2 = Γ 2π 3.21) 15

7 3.2.2 Thermal Line Broadening In a gas at kineti) temperature T k, individual atoms will have random motions away from or towards the observer, leading to red- or blue-wards frequeny shifts. If the natural) absorption oeient for a stationary atom is aν) at frequeny ν, then for an atom moving at veloity v the same absorption ours at frequeny ν D = ν 1 v ) = ν ν v where the `D' subsript an be taken as standing for Doppler, or Displaed). At a given frequeny, the total absorption oeient is the produt of the absorption oeient of atoms moving at some veloity, v, times the fration of atoms at that veloity, fv), integrated over veloity: aν) = + a ν ν v ) fv) dv. 3.22) We want to express aν) as a funtion of the observable quantity ν instead of v), whih requires some algebra. For a Maxwellian veloity distribution, the line-of-sight veloity distribution is gaussian, fv) dv = 1 ) 1/2 } m exp { mv2 dv 3.23) π 2kT k 2kT k for a partile of mass m at temperature T k ; and the line of sight) thermal doppler width is dened as v D [Note that 2kTk m 3.24) 1 2 m v 2 = 3 2 kt k so the higher the temperature, or the less the partile mass, the greater the spread in veloities.] Using eqtns and 3.24 in eqtn gives aν) = 1 1 a ν ν v ) } exp { mv2 dv 3.25) π v D 2kT k We now swith between veloity spae and frequeny spae by noting that ν = ν v 16

8 and by expressing the thermal doppler width in frequeny units, v D ν D = ν = ν 2kTk m giving, from eqtn. 3.25, aν) = 1 π 1 ν D 3.26) { ) } ν a ν ν 2 ) exp dν 3.27) ν D The natural line width is muh less than the doppler width and so an be approximated by a δ funtion i.e., an innitely narrow intrinsi line), aν) a δν ν ); this redues the integral to a single point and) gives our nal form, { aν) = a ) } 2 1 ν ν exp π ν D ν D 3.28) where all terms in eqtn are onstant exept ν ν ), for given m and T k ), or, equivalently, { aλ) = a ) } 2 λ) 1 λ λ exp π λ D λ D Eqtn again has the general form aν) = a φν) with { φν) = 1 ) } 2 1 ν ν exp π ν D ν D 3.29) Note that φν) is a normalized gaussian whih already satises φν) dν = 1 and that, in eet, the algebra we have arried out has simply onverted a gaussian veloity distribution of partiles into a gaussian absorption prole). 17

9 Peak value and width The peak value of φν), at ν = ν, is given by eqtn φν ) = 1 1 = 1 1 = 1 ) 1/2 m. 3.3) π ν D π ν v D π ν 2kT k At half maximum φν 1/2 ) =.5φν ): giving so that [ ] [ ] { ) } m 1 π = ν 2kT k m π exp m2 ν1/2 ν 2 ν 2kT k 2kT k ν ν1/2 ν ) 2 = ν 2 2kTk m 2 FWHM = 2ν ) ln 2 [ ] 1/2 2kTk m ln 2 = 2ln 2) 1/2 ν D = ν D 3.31) in frequeny units; or i.e., v = ν ν = ν D ν = v D ν ν 2kTk v = m ) 1/2 3.32) For example, for Hα n = 3 2), m = mh): at 6K, v = 1.7 km s 1 λ =.36nm) at 6K, λ =.36nm v = 17 km s 1 ). Heavier elements will have smaller thermal line widths. The natural width is nm, justifying our `δ funtion' approximation.) 18

10 3.2.3 `Turbulent' Broadening The nal broadening proess is not well dened physially; it is, if you like, `the other stu'. This might inlude internal motions within an interstellar loud, or even non-physial eets like unreognized overlapping or unresolved lines from dierent louds). Without a rm physial model for these additional soures of veloity dispersion, it is ustomary to adopt gaussian line-of-sight veloities, largely as a matter of onveniene and beause this assumption appears onsistent with observation in general whih doesn't make it true!). The root-mean-square rms) line-of-sight 1-D) doppler veloity dispersion arising from a Maxwellian veloity distribution is, from eqtn. 3.24, v D / ) 1/2 ktk 2 =. m We haraterize the turbulent assumed gaussian) veloity with an rms value v turb ; then the total 3.33) broadening is obtained by adding the thermal and turbulent widths in quadrature, expressed as [ ] 1/2 2kTk b = m + 2v2 turb = [ vd 2 + 2vturb] 2 1/2 3.34) in the interstellar ase; learly, as v D, b 2v turb v turb, b v D. As for thermal broadening, FWHM = 2ln 2) 1/2 b = 1.665b [ ] 1/2 2kTk = m + 2v2 turb 3.35) Can v turb and v D be separately determined? In priniple, yes, beause all elements should share the same turbulent broadening, but will undergo dierent mass-dependent) thermal broadening. In pratie, this is very diult, beause the turbulene is frequently the dominant term. 3.3 Interstellar Curve of Growth Theory The `urve of growth' is, essentially, a plot of line strength as a funtion of number of absorbers, and is a standard tool in interstellar absorption-line studies. 19

11 In astronomy, we measure line strength in terms of equivalent width. From eqtn. 3.2 ) I C W ν = ν I ν dν Iν C = 1 I ) ν dν Iν C or sine W λ = λ2 = λ2 = λ2 I ν = I C ν exp { τν)} ) I C ν I ν dν Iν C 1 I ) ν dν Iν C where τν) is the optial depth, τν) = d 3.36) 1 exp { τν)}) dν 3.37) κ i ν) ds = d aν)n i s) ds 3.38) with d the distane over whih absorption ours i.e., the `homogeneous slab' approximation), and where κ i ν) is the `length' opaity 2 for transition i, with a i ν) the absorption ross-setion m 2 ) and n i m 3 ) the number density so τ is dimensionless). Eqtn is the basi relationship between W λ and τ. For line transitions the absorption ross-setion for lower level i is a i ν) = 2 8πν 2 g j g i A ji φ s ν) 3.39) where g i, g j are statistial weights for the lower, upper levels; A ji is the emission transition probability Einstein oeient); and φ s ν) is the normalized prole funtion at position s. Substituting eqtn into eqtn and integrating, τν) dν = 2 g d ) j A 8πν 2 ji φ s ν)dν n i s)ds g i = 2 8πν 2 g j g i A ji N i if φ is not a funtion of s). 3.4) 2 In units of m 1 ; note that one an also dene opaity in terms of per unit mass. 2

12 It is usual, when onsidering absorption lines, to work not with the A oeient, but with the absorption osillator strength f ij ; the two are related by A ji = 8π2 e 2 ν 2 m e 3 g i g j f ij. 3.41) Thus eqtn. 3.4 beomes τν) dν = 2 8π 2 e 2 ν 2 f 8πν 2 m e 3 ij N i = πe2 m e N if ij. 3.42) Eqtn is the relationship between optial depth and olumn density; with eqtn it denes the relationship between W λ and N the urve of growth. We now want to onsider this relationship, i.e., the development of W λ with N Weak lines: optially thin limit For weak lines, τν) << 1; i.e., 1 exp τν)) τν), and so, from eqtn. 3.37, W λ λ2 or, from eqtn. 3.42, τν) dν W λ λ = πe2 m e 2N iλ f ij 3.43) A plot of logw λ /λ ) versus logn i f ij λ ) is a 1:1 straight line - this is the linear part of the CoG. Doubling N doubles W λ for an optially thin line, regardless of any details of the shape of the line prole. Unfortunately, optially thin lines are often too weak to be reliably measured General ase without damping at part of the CoG. As the line strength inreases, the line entre doppler ore) beomes `blak' all the available light has been removed from the line prole, and an inrease in N produes no signiant hange in W λ. This is the saturated part of the CoG. Eventually the damping wings beome important and the line strength again inreases.) The details do now depend on the line prole for an intrinsially broader line e.g., high T k ) the absorption is spread out over a greater veloity range, and a larger N is required before saturation beomes important. 21

13 Further analytial analysis beomes umbersome, but we will illustrate the role of broadening on the line prole with an analyti demonstration that broader lines saturate at higher N values. From eqtns and 3.41 we have aν) = πe2 m e f ijφν) so, from eqtn. 3.38, τν) = d πe 2 m e f ij n i s)φν) ds = πe2 m e f ij φν) d = πe2 m e N i f ij φν) n i s) ds From eqtn. 3.3, for a thermally broadened line φν ) = 1 π ν 1 v D whih we an generalize to inlude gaussian turbulene eqtn. 3.34) to write φν ) = 1 π λ b Then the line-entre optial depth is τ τν ) = πe2 m e N if ij φν ) πe 2 N i f ij λ = m e b whih expliitly shows the dependene of the line prole on the broadening parameter b the larger the b value, the smaller the τ for given N, and the smaller the degree of saturation. Writing v/b/λ ) as x, then, from eqtn. 3.37, after some manipulation we obtain W λ = λ2 b λ ) 1 exp τ exp x 2 ) )) dx 3.45) For large τ the integral in eqtn whih has to be evaluated numerially) approahes W λ lnτ ) 1/2. The dependene of W λ on τ i.e., on N; eqtn. 3.44) is therefore very small, and this region is alled the at part of the CoG. 3 3 If we expand the rst of the exponents in eqtn in a Taylor series, we get W λ = π 1/2 τν ) bλ 1! 1 τν ) 2 2! 2 + τν ) 3 ) 3!

14 3.3.4 Damping dominates square root part of the CoG. For very strong lines the damping wings of the Lorentzian, φ = beome important. Γ 4π 2 ν ν ) 2 + Γ/2) 2, 3.17) In this ase it an be shown that W λ λ λ 2 N i ) 1/2 This delineates the square-root setion of the CoG. Note that this region of the CoG is again independent of b. In the diuse ISM, lines are not normally strong enough for damping wings to beome signiant beause stars behind large olumns are heavily extinguished and therefore hard to observe). However, the resonane line of hydrogen, at 121.6nm, is an important exeption, and is nearly always on the square root part of the CoG. 3.4 The Empirial Curve of Growth A theoretial urve of growth is, in pratie, a plot of logw λ /λ) vs. lognfλ). Observationally, we an measure W λ, and look up λ and f ij but we don't know the olumn density, N. In order to derive the abundanes of speies in the gas phase of the ISM, we onstrut an empirial CoG. From observations of equivalent widths of dierent lines of a given atomi speies, we plot logw λ /λ ) against logf ij λ ) for observed lines. In priniple, we should onstrut separate CoG for eah speies. This is beause eah element has a dierent mass; so for a given temperature, eah element will have a dierent b value. Also, we have no a priori reason to suppose that dierent ions of a given element are formed at the same temperatures, or neessarily, in the same plae. However, in pratie the prinipal broadening proess is often `turbulene' not thermal, and so we may ombine observations of dierent Lundenburg 193, Zs. Phys 65, 2) Then, using eqtn. 3.44, for small τν ) we get W λ λ = π 1/2 bτν ) = πe2 m e N if ij λ whih shows that this general form reovers the optially thin limit, eqtn

15 dominant ions in the diuse neutral ISM, those with IP<13.6eV). We do this by overlaying various empirial CoGs, and sliding them along the horizontal axis to obtain a smooth `global' urve. This gives a muh better sampling of the empirial CoG, sine normally we only have a few lines of any one speies, typially overing a small range in fλ. If the shape of the urve indiates that the weaker lines are on the linear part of the CoG N independent of b) then we an determine N for those lines. Thus the empirial CoG an, in priniple, yield olumn densities for all speies on the CoG, independently of any model assumptions. Unfortunately, lines weak enough to be ertainly unsaturated are, in pratie, often so weak as to be undetetable. We must then resort to overlaying the empirial CoG on a family of theoretial CoGs alulated for a range of b values). It is normally possible to nd a horizontal shift between the empirial and theoretial CoGs whih gives a `best t'. That shift yields log N, the olumn density; and, as a by-produt, b is also onstrained. As before, log N is most reliably determined if lines are observed on the steepest part of the CoG, i.e. on or near the linear part. If lines are present only on the at part of the CoG there will be large unertainties on log N; if there is only one line observed, the best we an do is use the linear formula to obtain a lower limit to N. 24

16 Figure 3.4: Calulated line proles for inreasing olumn densities and two dierent b values. Inreasing line strengths orrespond to olumn densities of 11, 13, 15, and 17 dex m 2. 25

17 Figure 3.5: Theoretial urves of growth, illustrating the dependene on b. Figure 3.6: An empirial urve of growth for dierent lines in the spetrum of HD

18 Figure 3.7: Depletions in the line of sight towards HD 5896 in two veloity systems. 27

Line Radiative Transfer

Line Radiative Transfer http://www.v.nrao.edu/ourse/astr534/ineradxfer.html ine Radiative Transfer Einstein Coeffiients We used armor's equation to estimate the spontaneous emission oeffiients A U for À reombination lines. A

More information

Simple Considerations on the Cosmological Redshift

Simple Considerations on the Cosmological Redshift Apeiron, Vol. 5, No. 3, July 8 35 Simple Considerations on the Cosmologial Redshift José Franiso Garía Juliá C/ Dr. Maro Mereniano, 65, 5. 465 Valenia (Spain) E-mail: jose.garia@dival.es Generally, the

More information

arxiv: v1 [physics.gen-ph] 5 Jan 2018

arxiv: v1 [physics.gen-ph] 5 Jan 2018 The Real Quaternion Relativity Viktor Ariel arxiv:1801.03393v1 [physis.gen-ph] 5 Jan 2018 In this work, we use real quaternions and the basi onept of the final speed of light in an attempt to enhane the

More information

Relativistic Dynamics

Relativistic Dynamics Chapter 7 Relativisti Dynamis 7.1 General Priniples of Dynamis 7.2 Relativisti Ation As stated in Setion A.2, all of dynamis is derived from the priniple of least ation. Thus it is our hore to find a suitable

More information

Chapter 9. The excitation process

Chapter 9. The excitation process Chapter 9 The exitation proess qualitative explanation of the formation of negative ion states Ne and He in He-Ne ollisions an be given by using a state orrelation diagram. state orrelation diagram is

More information

Wavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013

Wavetech, LLC. Ultrafast Pulses and GVD. John O Hara Created: Dec. 6, 2013 Ultrafast Pulses and GVD John O Hara Created: De. 6, 3 Introdution This doument overs the basi onepts of group veloity dispersion (GVD) and ultrafast pulse propagation in an optial fiber. Neessarily, it

More information

Diffuse Interstellar Medium

Diffuse Interstellar Medium Diffuse Interstellar Medium Basics, velocity widths H I 21-cm radiation (emission) Interstellar absorption lines Radiative transfer Resolved Lines, column densities Unresolved lines, curve of growth Abundances,

More information

ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES.

ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. ELECTROMAGNETIC NORMAL MODES AND DISPERSION FORCES. All systems with interation of some type have normal modes. One may desribe them as solutions in absene of soures; they are exitations of the system

More information

9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry

9 Geophysics and Radio-Astronomy: VLBI VeryLongBaseInterferometry 9 Geophysis and Radio-Astronomy: VLBI VeryLongBaseInterferometry VLBI is an interferometry tehnique used in radio astronomy, in whih two or more signals, oming from the same astronomial objet, are reeived

More information

On the Quantum Theory of Radiation.

On the Quantum Theory of Radiation. Physikalishe Zeitshrift, Band 18, Seite 121-128 1917) On the Quantum Theory of Radiation. Albert Einstein The formal similarity between the hromati distribution urve for thermal radiation and the Maxwell

More information

Lecture 2 Line Radiative Transfer for the ISM

Lecture 2 Line Radiative Transfer for the ISM Lecture 2 Line Radiative Transfer for the ISM Absorption lines in the optical & UV Equation of transfer Absorption & emission coefficients Line broadening Equivalent width and curve of growth Observations

More information

Lecture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY

Lecture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.286: The Early Universe Otober 1, 218 Prof. Alan Guth Leture Notes 4 MORE DYNAMICS OF NEWTONIAN COSMOLOGY THE AGE OF A FLAT UNIVERSE: We

More information

3: Interstellar Absorption Lines: Radiative Transfer in the Interstellar Medium. James R. Graham University of California, Berkeley

3: Interstellar Absorption Lines: Radiative Transfer in the Interstellar Medium. James R. Graham University of California, Berkeley 3: Interstellar Absorption Lines: Radiative Transfer in the Interstellar Medium James R. Graham University of California, Berkeley Interstellar Absorption Lines Example of atomic absorption lines Structure

More information

The Laws of Acceleration

The Laws of Acceleration The Laws of Aeleration The Relationships between Time, Veloity, and Rate of Aeleration Copyright 2001 Joseph A. Rybzyk Abstrat Presented is a theory in fundamental theoretial physis that establishes the

More information

RESEARCH ON RANDOM FOURIER WAVE-NUMBER SPECTRUM OF FLUCTUATING WIND SPEED

RESEARCH ON RANDOM FOURIER WAVE-NUMBER SPECTRUM OF FLUCTUATING WIND SPEED The Seventh Asia-Paifi Conferene on Wind Engineering, November 8-1, 9, Taipei, Taiwan RESEARCH ON RANDOM FORIER WAVE-NMBER SPECTRM OF FLCTATING WIND SPEED Qi Yan 1, Jie Li 1 Ph D. andidate, Department

More information

Millennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion

Millennium Relativity Acceleration Composition. The Relativistic Relationship between Acceleration and Uniform Motion Millennium Relativity Aeleration Composition he Relativisti Relationship between Aeleration and niform Motion Copyright 003 Joseph A. Rybzyk Abstrat he relativisti priniples developed throughout the six

More information

F = c where ^ı is a unit vector along the ray. The normal component is. Iν cos 2 θ. d dadt. dp normal (θ,φ) = dpcos θ = df ν

F = c where ^ı is a unit vector along the ray. The normal component is. Iν cos 2 θ. d dadt. dp normal (θ,φ) = dpcos θ = df ν INTRODUCTION So far, the only information we have been able to get about the universe beyond the solar system is from the eletromagneti radiation that reahes us (and a few osmi rays). So doing Astrophysis

More information

Introduction to Quantum Chemistry

Introduction to Quantum Chemistry Chem. 140B Dr. J.A. Mak Introdution to Quantum Chemistry Without Quantum Mehanis, how would you explain: Periodi trends in properties of the elements Struture of ompounds e.g. Tetrahedral arbon in ethane,

More information

Name Solutions to Test 1 September 23, 2016

Name Solutions to Test 1 September 23, 2016 Name Solutions to Test 1 September 3, 016 This test onsists of three parts. Please note that in parts II and III, you an skip one question of those offered. Possibly useful formulas: F qequb x xvt E Evpx

More information

Advanced Computational Fluid Dynamics AA215A Lecture 4

Advanced Computational Fluid Dynamics AA215A Lecture 4 Advaned Computational Fluid Dynamis AA5A Leture 4 Antony Jameson Winter Quarter,, Stanford, CA Abstrat Leture 4 overs analysis of the equations of gas dynamis Contents Analysis of the equations of gas

More information

The Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge

The Concept of Mass as Interfering Photons, and the Originating Mechanism of Gravitation D.T. Froedge The Conept of Mass as Interfering Photons, and the Originating Mehanism of Gravitation D.T. Froedge V04 Formerly Auburn University Phys-dtfroedge@glasgow-ky.om Abstrat For most purposes in physis the onept

More information

Copyright 2018 Society of Photo-Optical Instrumentation Engineers (SPIE). One print or electronic copy may be made for personal use only.

Copyright 2018 Society of Photo-Optical Instrumentation Engineers (SPIE). One print or electronic copy may be made for personal use only. Copyright 018 Soiety of Photo-Optial Instrumentation Engineers (SPIE) One print or eletroni opy may be made for personal use only Systemati reprodution and distribution, dupliation of any material in this

More information

UTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker.

UTC. Engineering 329. Proportional Controller Design. Speed System. John Beverly. Green Team. John Beverly Keith Skiles John Barker. UTC Engineering 329 Proportional Controller Design for Speed System By John Beverly Green Team John Beverly Keith Skiles John Barker 24 Mar 2006 Introdution This experiment is intended test the variable

More information

Relativistic Addition of Velocities *

Relativistic Addition of Velocities * OpenStax-CNX module: m42540 1 Relativisti Addition of Veloities * OpenStax This work is produed by OpenStax-CNX and liensed under the Creative Commons Attribution Liense 3.0 Abstrat Calulate relativisti

More information

Where as discussed previously we interpret solutions to this partial differential equation in the weak sense: b

Where as discussed previously we interpret solutions to this partial differential equation in the weak sense: b Consider the pure initial value problem for a homogeneous system of onservation laws with no soure terms in one spae dimension: Where as disussed previously we interpret solutions to this partial differential

More information

Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms

Non-Markovian study of the relativistic magnetic-dipole spontaneous emission process of hydrogen-like atoms NSTTUTE OF PHYSCS PUBLSHNG JOURNAL OF PHYSCS B: ATOMC, MOLECULAR AND OPTCAL PHYSCS J. Phys. B: At. Mol. Opt. Phys. 39 ) 7 85 doi:.88/953-75/39/8/ Non-Markovian study of the relativisti magneti-dipole spontaneous

More information

(E B) Rate of Absorption and Stimulated Emission. π 2 E 0 ( ) 2. δ(ω k. p. 59. The rate of absorption induced by the field is. w k

(E B) Rate of Absorption and Stimulated Emission. π 2 E 0 ( ) 2. δ(ω k. p. 59. The rate of absorption induced by the field is. w k p. 59 Rate of Absorption and Stimulated Emission The rate of absorption indued by the field is π w k ( ω) ω E 0 ( ) k ˆ µ δω ( k ω) The rate is learly dependent on the strength of the field. The variable

More information

THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE?

THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? THEORETICAL PROBLEM No. 3 WHY ARE STARS SO LARGE? The stars are spheres of hot gas. Most of them shine beause they are fusing hydrogen into helium in their entral parts. In this problem we use onepts of

More information

23.1 Tuning controllers, in the large view Quoting from Section 16.7:

23.1 Tuning controllers, in the large view Quoting from Section 16.7: Lesson 23. Tuning a real ontroller - modeling, proess identifiation, fine tuning 23.0 Context We have learned to view proesses as dynami systems, taking are to identify their input, intermediate, and output

More information

QUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1

QUANTUM MECHANICS II PHYS 517. Solutions to Problem Set # 1 QUANTUM MECHANICS II PHYS 57 Solutions to Problem Set #. The hamiltonian for a lassial harmoni osillator an be written in many different forms, suh as use ω = k/m H = p m + kx H = P + Q hω a. Find a anonial

More information

An Adaptive Optimization Approach to Active Cancellation of Repeated Transient Vibration Disturbances

An Adaptive Optimization Approach to Active Cancellation of Repeated Transient Vibration Disturbances An aptive Optimization Approah to Ative Canellation of Repeated Transient Vibration Disturbanes David L. Bowen RH Lyon Corp / Aenteh, 33 Moulton St., Cambridge, MA 138, U.S.A., owen@lyonorp.om J. Gregory

More information

Green s function for the wave equation

Green s function for the wave equation Green s funtion for the wave equation Non-relativisti ase January 2019 1 The wave equations In the Lorentz Gauge, the wave equations for the potentials are (Notes 1 eqns 43 and 44): 1 2 A 2 2 2 A = µ 0

More information

Theory. Coupled Rooms

Theory. Coupled Rooms Theory of Coupled Rooms For: nternal only Report No.: R/50/TCR Prepared by:. N. taey B.., MO Otober 00 .00 Objet.. The objet of this doument is present the theory alulations to estimate the reverberant

More information

). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become

). In accordance with the Lorentz transformations for the space-time coordinates of the same event, the space coordinates become Relativity and quantum mehanis: Jorgensen 1 revisited 1. Introdution Bernhard Rothenstein, Politehnia University of Timisoara, Physis Department, Timisoara, Romania. brothenstein@gmail.om Abstrat. We first

More information

19 Lecture 19: Cosmic Microwave Background Radiation

19 Lecture 19: Cosmic Microwave Background Radiation PHYS 652: Astrophysis 97 19 Leture 19: Cosmi Mirowave Bakground Radiation Observe the void its emptiness emits a pure light. Chuang-tzu The Big Piture: Today we are disussing the osmi mirowave bakground

More information

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E')

22.54 Neutron Interactions and Applications (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') 22.54 Neutron Interations and Appliations (Spring 2004) Chapter 6 (2/24/04) Energy Transfer Kernel F(E E') Referenes -- J. R. Lamarsh, Introdution to Nulear Reator Theory (Addison-Wesley, Reading, 1966),

More information

Supplementary information for: All-optical signal processing using dynamic Brillouin gratings

Supplementary information for: All-optical signal processing using dynamic Brillouin gratings Supplementary information for: All-optial signal proessing using dynami Brillouin gratings Maro Santagiustina, Sanghoon Chin 2, Niolay Primerov 2, Leonora Ursini, Lu Thévena 2 Department of Information

More information

ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW. P. М. Меdnis

ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW. P. М. Меdnis ELECTROMAGNETIC WAVES WITH NONLINEAR DISPERSION LAW P. М. Меdnis Novosibirs State Pedagogial University, Chair of the General and Theoretial Physis, Russia, 636, Novosibirs,Viljujsy, 8 e-mail: pmednis@inbox.ru

More information

Processi di Radiazione e MHD

Processi di Radiazione e MHD Proessi di Radiazione e MHD 0. Overview of elestial bodies and sky at various frequenies 1. Definition of main astrophysial observables. Radiative transfer 3. Blak body radiation 4. basi theory of radiation

More information

Supplementary Information. Infrared Transparent Visible Opaque Fabrics (ITVOF) for Personal Cooling

Supplementary Information. Infrared Transparent Visible Opaque Fabrics (ITVOF) for Personal Cooling Supplementary Information Infrared Transparent Visible Opaque Fabris (ITVOF) for Personal Cooling Jonathan K. Tong 1,Ɨ, Xiaopeng Huang 1,Ɨ, Svetlana V. Boriskina 1, James Loomis 1, Yanfei Xu 1, and Gang

More information

Phase Diffuser at the Transmitter for Lasercom Link: Effect of Partially Coherent Beam on the Bit-Error Rate.

Phase Diffuser at the Transmitter for Lasercom Link: Effect of Partially Coherent Beam on the Bit-Error Rate. Phase Diffuser at the Transmitter for Laserom Link: Effet of Partially Coherent Beam on the Bit-Error Rate. O. Korotkova* a, L. C. Andrews** a, R. L. Phillips*** b a Dept. of Mathematis, Univ. of Central

More information

UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS. V. N. Matveev and O. V. Matvejev

UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS. V. N. Matveev and O. V. Matvejev UNCERTAINTY RELATIONS AS A CONSEQUENCE OF THE LORENTZ TRANSFORMATIONS V. N. Matveev and O. V. Matvejev Joint-Stok Company Sinerta Savanoriu pr., 159, Vilnius, LT-315, Lithuania E-mail: matwad@mail.ru Abstrat

More information

Four-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field

Four-dimensional equation of motion for viscous compressible substance with regard to the acceleration field, pressure field and dissipation field Four-dimensional equation of motion for visous ompressible substane with regard to the aeleration field, pressure field and dissipation field Sergey G. Fedosin PO box 6488, Sviazeva str. -79, Perm, Russia

More information

Acoustic Waves in a Duct

Acoustic Waves in a Duct Aousti Waves in a Dut 1 One-Dimensional Waves The one-dimensional wave approximation is valid when the wavelength λ is muh larger than the diameter of the dut D, λ D. The aousti pressure disturbane p is

More information

Determination of the reaction order

Determination of the reaction order 5/7/07 A quote of the wee (or amel of the wee): Apply yourself. Get all the eduation you an, but then... do something. Don't just stand there, mae it happen. Lee Iaoa Physial Chemistry GTM/5 reation order

More information

A simple expression for radial distribution functions of pure fluids and mixtures

A simple expression for radial distribution functions of pure fluids and mixtures A simple expression for radial distribution funtions of pure fluids and mixtures Enrio Matteoli a) Istituto di Chimia Quantistia ed Energetia Moleolare, CNR, Via Risorgimento, 35, 56126 Pisa, Italy G.

More information

Nuclear Shell Structure Evolution Theory

Nuclear Shell Structure Evolution Theory Nulear Shell Struture Evolution Theory Zhengda Wang (1) Xiaobin Wang () Xiaodong Zhang () Xiaohun Wang () (1) Institute of Modern physis Chinese Aademy of SienesLan Zhou P. R. China 70000 () Seagate Tehnology

More information

Time and Energy, Inertia and Gravity

Time and Energy, Inertia and Gravity Time and Energy, Inertia and Gravity The Relationship between Time, Aeleration, and Veloity and its Affet on Energy, and the Relationship between Inertia and Gravity Copyright 00 Joseph A. Rybzyk Abstrat

More information

General Equilibrium. What happens to cause a reaction to come to equilibrium?

General Equilibrium. What happens to cause a reaction to come to equilibrium? General Equilibrium Chemial Equilibrium Most hemial reations that are enountered are reversible. In other words, they go fairly easily in either the forward or reverse diretions. The thing to remember

More information

The Electromagnetic Radiation and Gravity

The Electromagnetic Radiation and Gravity International Journal of Theoretial and Mathematial Physis 016, 6(3): 93-98 DOI: 10.593/j.ijtmp.0160603.01 The Eletromagneti Radiation and Gravity Bratianu Daniel Str. Teiului Nr. 16, Ploiesti, Romania

More information

Brazilian Journal of Physics, vol. 29, no. 3, September, Classical and Quantum Mechanics of a Charged Particle

Brazilian Journal of Physics, vol. 29, no. 3, September, Classical and Quantum Mechanics of a Charged Particle Brazilian Journal of Physis, vol. 9, no. 3, September, 1999 51 Classial and Quantum Mehanis of a Charged Partile in Osillating Eletri and Magneti Fields V.L.B. de Jesus, A.P. Guimar~aes, and I.S. Oliveira

More information

Aharonov-Bohm effect. Dan Solomon.

Aharonov-Bohm effect. Dan Solomon. Aharonov-Bohm effet. Dan Solomon. In the figure the magneti field is onfined to a solenoid of radius r 0 and is direted in the z- diretion, out of the paper. The solenoid is surrounded by a barrier that

More information

The Unified Geometrical Theory of Fields and Particles

The Unified Geometrical Theory of Fields and Particles Applied Mathematis, 014, 5, 347-351 Published Online February 014 (http://www.sirp.org/journal/am) http://dx.doi.org/10.436/am.014.53036 The Unified Geometrial Theory of Fields and Partiles Amagh Nduka

More information

Tutorial 8: Solutions

Tutorial 8: Solutions Tutorial 8: Solutions 1. * (a) Light from the Sun arrives at the Earth, an average of 1.5 10 11 m away, at the rate 1.4 10 3 Watts/m of area perpendiular to the diretion of the light. Assume that sunlight

More information

Wave Propagation through Random Media

Wave Propagation through Random Media Chapter 3. Wave Propagation through Random Media 3. Charateristis of Wave Behavior Sound propagation through random media is the entral part of this investigation. This hapter presents a frame of referene

More information

Casimir self-energy of a free electron

Casimir self-energy of a free electron Casimir self-energy of a free eletron Allan Rosenwaig* Arist Instruments, In. Fremont, CA 94538 Abstrat We derive the eletromagneti self-energy and the radiative orretion to the gyromagneti ratio of a

More information

Answers to Coursebook questions Chapter J2

Answers to Coursebook questions Chapter J2 Answers to Courseook questions Chapter J 1 a Partiles are produed in ollisions one example out of many is: a ollision of an eletron with a positron in a synhrotron. If we produe a pair of a partile and

More information

The Corpuscular Structure of Matter, the Interaction of Material Particles, and Quantum Phenomena as a Consequence of Selfvariations.

The Corpuscular Structure of Matter, the Interaction of Material Particles, and Quantum Phenomena as a Consequence of Selfvariations. The Corpusular Struture of Matter, the Interation of Material Partiles, and Quantum Phenomena as a Consequene of Selfvariations. Emmanuil Manousos APM Institute for the Advanement of Physis and Mathematis,

More information

Analysis of discretization in the direct simulation Monte Carlo

Analysis of discretization in the direct simulation Monte Carlo PHYSICS OF FLUIDS VOLUME 1, UMBER 1 OCTOBER Analysis of disretization in the diret simulation Monte Carlo iolas G. Hadjionstantinou a) Department of Mehanial Engineering, Massahusetts Institute of Tehnology,

More information

Evaluation of effect of blade internal modes on sensitivity of Advanced LIGO

Evaluation of effect of blade internal modes on sensitivity of Advanced LIGO Evaluation of effet of blade internal modes on sensitivity of Advaned LIGO T0074-00-R Norna A Robertson 5 th Otober 00. Introdution The urrent model used to estimate the isolation ahieved by the quadruple

More information

Electromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution.

Electromagnetic radiation of the travelling spin wave propagating in an antiferromagnetic plate. Exact solution. arxiv:physis/99536v1 [physis.lass-ph] 15 May 1999 Eletromagneti radiation of the travelling spin wave propagating in an antiferromagneti plate. Exat solution. A.A.Zhmudsky November 19, 16 Abstrat The exat

More information

arxiv:hep-ph/ v2 30 May 1998

arxiv:hep-ph/ v2 30 May 1998 Ref. SISSA 31/98/EP hep ph/9805262 8 May, 1998 Diffrative-Like (or Parametri-Resonane-Like?) Enhanement of the Earth (Day-Night) Effet arxiv:hep-ph/9805262v2 30 May 1998 for Solar Neutrinos Crossing the

More information

Lecture 3 - Lorentz Transformations

Lecture 3 - Lorentz Transformations Leture - Lorentz Transformations A Puzzle... Example A ruler is positioned perpendiular to a wall. A stik of length L flies by at speed v. It travels in front of the ruler, so that it obsures part of the

More information

STATISTICAL MECHANICS & THERMODYNAMICS

STATISTICAL MECHANICS & THERMODYNAMICS UVA PHYSICS DEPARTMENT PHD QUALIFYING EXAM PROBLEM FILE STATISTICAL MECHANICS & THERMODYNAMICS UPDATED: NOVEMBER 14, 212 1. a. Explain what is meant by the density of states, and give an expression for

More information

Gravitation is a Gradient in the Velocity of Light ABSTRACT

Gravitation is a Gradient in the Velocity of Light ABSTRACT 1 Gravitation is a Gradient in the Veloity of Light D.T. Froedge V5115 @ http://www.arxdtf.org Formerly Auburn University Phys-dtfroedge@glasgow-ky.om ABSTRACT It has long been known that a photon entering

More information

the cold 5-eV electrons. The energy E* of the

the cold 5-eV electrons. The energy E* of the JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 88, NO. A10, PAGES 8097-8102, OCTOBER 1, 1983 THE NON-MAXWELLIAN ENERGY DISTRIBUTION OF IONS IN THE WARM IO TORUS John D. Rihardson and George L. Sisoe Department

More information

Quantum Mechanics: Wheeler: Physics 6210

Quantum Mechanics: Wheeler: Physics 6210 Quantum Mehanis: Wheeler: Physis 60 Problems some modified from Sakurai, hapter. W. S..: The Pauli matries, σ i, are a triple of matries, σ, σ i = σ, σ, σ 3 given by σ = σ = σ 3 = i i Let stand for the

More information

The Hanging Chain. John McCuan. January 19, 2006

The Hanging Chain. John McCuan. January 19, 2006 The Hanging Chain John MCuan January 19, 2006 1 Introdution We onsider a hain of length L attahed to two points (a, u a and (b, u b in the plane. It is assumed that the hain hangs in the plane under a

More information

Time Domain Method of Moments

Time Domain Method of Moments Time Domain Method of Moments Massahusetts Institute of Tehnology 6.635 leture notes 1 Introdution The Method of Moments (MoM) introdued in the previous leture is widely used for solving integral equations

More information

Lecture 2 Interstellar Absorption Lines: Line Radiative Transfer

Lecture 2 Interstellar Absorption Lines: Line Radiative Transfer Lecture 2 Interstellar Absorption Lines: Line Radiative Transfer 1. Atomic absorption lines 2. Application of radiative transfer to absorption & emission 3. Line broadening & curve of growth 4. Optical/UV

More information

Blackbody radiation and Plank s law

Blackbody radiation and Plank s law lakbody radiation and Plank s law blakbody problem: alulating the intensity o radiation at a given wavelength emitted by a body at a speii temperature Max Plank, 900 quantization o energy o radiation-emitting

More information

Chapter 14. The Concept of Equilibrium and the Equilibrium Constant. We have for the most part depicted reactions as going one way.

Chapter 14. The Concept of Equilibrium and the Equilibrium Constant. We have for the most part depicted reactions as going one way. Chapter 14 The Conept of Equilibrium and the Equilibrium Constant In hapter 1 we dealt with Physial Equilibrium Physial Changes HO 2 (l) HO 2 (g) In hapter 14 we will learn about Chemial Equilibrium. We

More information

11.4 Molecular Orbital Description of the Hydrogen Molecule Electron Configurations of Homonuclear Diatomic Molecules

11.4 Molecular Orbital Description of the Hydrogen Molecule Electron Configurations of Homonuclear Diatomic Molecules Chap Moleular Eletroni Struture Table of Contents. The orn-oppenheimer pproximation -. The Hydrogen Moleule Ion.3 Calulation of the Energy of the Hydrogen Moleule Ion.4 Moleular Orbital Desription of the

More information

IMPACT MODELLING OF THE COEFFICIENT OF RESTITUTION OF POTATOES BASED ON THE KELVIN- VOIGHT PAIR

IMPACT MODELLING OF THE COEFFICIENT OF RESTITUTION OF POTATOES BASED ON THE KELVIN- VOIGHT PAIR Bulletin of the Transilvania University of Braşov Series II: Forestry Wood Industry Agriultural Food Engineering Vol. 9 (58) No. - 06 IMPACT MODELLING OF THE COEFFICIENT OF RESTITUTION OF POTATOES BASED

More information

Chapter 26 Lecture Notes

Chapter 26 Lecture Notes Chapter 26 Leture Notes Physis 2424 - Strauss Formulas: t = t0 1 v L = L0 1 v m = m0 1 v E = m 0 2 + KE = m 2 KE = m 2 -m 0 2 mv 0 p= mv = 1 v E 2 = p 2 2 + m 2 0 4 v + u u = 2 1 + vu There were two revolutions

More information

Module 5: Red Recedes, Blue Approaches. UNC-TFA H.S. Astronomy Collaboration, Copyright 2012

Module 5: Red Recedes, Blue Approaches. UNC-TFA H.S. Astronomy Collaboration, Copyright 2012 Objetives/Key Points Module 5: Red Reedes, Blue Approahes UNC-TFA H.S. Astronomy Collaboration, Copyright 2012 Students will be able to: 1. math the diretion of motion of a soure (approahing or reeding)

More information

PHY 108: Optical Physics. Solution to Midterm Test

PHY 108: Optical Physics. Solution to Midterm Test PHY 108: Optial Physis Solution to Midterm Test TA: Xun Jia 1 May 14, 2008 1 Email: jiaxun@physis.ula.edu Spring 2008 Physis 108 Xun Jia (May 14, 2008) Problem #1 For a two mirror resonant avity, the resonane

More information

General Closed-form Analytical Expressions of Air-gap Inductances for Surfacemounted Permanent Magnet and Induction Machines

General Closed-form Analytical Expressions of Air-gap Inductances for Surfacemounted Permanent Magnet and Induction Machines General Closed-form Analytial Expressions of Air-gap Indutanes for Surfaemounted Permanent Magnet and Indution Mahines Ronghai Qu, Member, IEEE Eletroni & Photoni Systems Tehnologies General Eletri Company

More information

Physics 486. Classical Newton s laws Motion of bodies described in terms of initial conditions by specifying x(t), v(t).

Physics 486. Classical Newton s laws Motion of bodies described in terms of initial conditions by specifying x(t), v(t). Physis 486 Tony M. Liss Leture 1 Why quantum mehanis? Quantum vs. lassial mehanis: Classial Newton s laws Motion of bodies desribed in terms of initial onditions by speifying x(t), v(t). Hugely suessful

More information

Gravity from the Uncertainty Principle.

Gravity from the Uncertainty Principle. Gravity from the Unertainty Priniple. M.E. MCulloh Otober 29, 2013 Abstrat It is shown here that Newton's gravity law an be derived from the unertainty priniple. The idea is that as the distane between

More information

Atomic and Nuclear Physics

Atomic and Nuclear Physics Atomi and Nulear Physis X-ray physis Compton effet and X-ray physis LD Physis Leaflets P6.3.7. Compton effet: Measuring the energy of the sattered photons as a funtion of the sattering angle Objets of

More information

1 sin 2 r = 1 n 2 sin 2 i

1 sin 2 r = 1 n 2 sin 2 i Physis 505 Fall 005 Homework Assignment #11 Solutions Textbook problems: Ch. 7: 7.3, 7.5, 7.8, 7.16 7.3 Two plane semi-infinite slabs of the same uniform, isotropi, nonpermeable, lossless dieletri with

More information

Metric of Universe The Causes of Red Shift.

Metric of Universe The Causes of Red Shift. Metri of Universe The Causes of Red Shift. ELKIN IGOR. ielkin@yande.ru Annotation Poinare and Einstein supposed that it is pratially impossible to determine one-way speed of light, that s why speed of

More information

Spontaneous Emission, Stimulated Emission, and Absorption

Spontaneous Emission, Stimulated Emission, and Absorption Chapter Six Spontaneous Emission, Stimulated Emission, and Absorption In this chapter, we review the general principles governing absorption and emission of radiation by absorbers with quantized energy

More information

Quantum Theory of Two-Photon Wavepacket Interference in a Beam Splitter

Quantum Theory of Two-Photon Wavepacket Interference in a Beam Splitter Quantum Theory of Two-Photon Wavepaket Interferene in a Beam Splitter Kaige Wang CCAST (World Laboratory), P. O. Box 8730, Beijing 100080, and Department of Physis, Applied Optis Beijing Area Major Laboratory,

More information

THE REFRACTION OF LIGHT IN STATIONARY AND MOVING REFRACTIVE MEDIA

THE REFRACTION OF LIGHT IN STATIONARY AND MOVING REFRACTIVE MEDIA HDRONIC JOURNL 24, 113-129 (2001) THE REFRCTION OF LIGHT IN STTIONRY ND MOVING REFRCTIVE MEDI C. K. Thornhill 39 Crofton Road Orpington, Kent, BR6 8E United Kingdom Reeived Deember 10, 2000 Revised: Marh

More information

Towards an Absolute Cosmic Distance Gauge by using Redshift Spectra from Light Fatigue.

Towards an Absolute Cosmic Distance Gauge by using Redshift Spectra from Light Fatigue. Towards an Absolute Cosmi Distane Gauge by using Redshift Spetra from Light Fatigue. Desribed by using the Maxwell Analogy for Gravitation. T. De Mees - thierrydemees @ pandora.be Abstrat Light is an eletromagneti

More information

Homework Set 4. gas B open end

Homework Set 4. gas B open end Homework Set 4 (1). A steady-state Arnold ell is used to determine the diffusivity of toluene (speies A) in air (speies B) at 298 K and 1 atm. If the diffusivity is DAB = 0.0844 m 2 /s = 8.44 x 10-6 m

More information

Study of Discrete-Particle Effects in a One-Dimensional Plasma Simulation with the Krook Type Collision Model

Study of Discrete-Particle Effects in a One-Dimensional Plasma Simulation with the Krook Type Collision Model Study of Disrete-Partile Effets in a One-Dimensional Plasma Simulation with the Kroo Type Collision Model Po-Yen Lai 1, Liu Chen 2, 3, Y. R. Lin-Liu 1, 4 and Shih-Hung Chen 1, 4, * 1 Department of Physis,

More information

Blackbody radiation (Text 2.2)

Blackbody radiation (Text 2.2) Blabody radiation (Text.) How Raleigh and Jeans model the problem:. Next step is to alulate how many possible independent standing waves are there per unit frequeny (ν) per unit volume (of avity). It is

More information

Monte Carlo Simulation of Electron and Radiative Emission from Silicon Diodes

Monte Carlo Simulation of Electron and Radiative Emission from Silicon Diodes SIMULATION OF SEMICONDUCTOR DEVICES AND PROCESSES Vol. 4 Edited by W. Fihtner, D. Aemmer - Zurih (Switzerland) September 12-14,1991 - Hartung-Gorre 521 Monte Carlo Simulation of Eletron and Radiative Emission

More information

A model for measurement of the states in a coupled-dot qubit

A model for measurement of the states in a coupled-dot qubit A model for measurement of the states in a oupled-dot qubit H B Sun and H M Wiseman Centre for Quantum Computer Tehnology Centre for Quantum Dynamis Griffith University Brisbane 4 QLD Australia E-mail:

More information

Doppler redistribution of anisotropic radiation and resonance polarization in moving scattering media

Doppler redistribution of anisotropic radiation and resonance polarization in moving scattering media Astron. Astrophys. 34 579 592 (1998) Doppler redistribution of anisotropi radiation and resonane polarization in moving sattering media I. Theory revisited in the density matrix formalism S. Sahal-Bréhot

More information

The gravitational phenomena without the curved spacetime

The gravitational phenomena without the curved spacetime The gravitational phenomena without the urved spaetime Mirosław J. Kubiak Abstrat: In this paper was presented a desription of the gravitational phenomena in the new medium, different than the urved spaetime,

More information

arxiv:physics/ v1 [physics.class-ph] 8 Aug 2003

arxiv:physics/ v1 [physics.class-ph] 8 Aug 2003 arxiv:physis/0308036v1 [physis.lass-ph] 8 Aug 003 On the meaning of Lorentz ovariane Lszl E. Szab Theoretial Physis Researh Group of the Hungarian Aademy of Sienes Department of History and Philosophy

More information

+Ze. n = N/V = 6.02 x x (Z Z c ) m /A, (1.1) Avogadro s number

+Ze. n = N/V = 6.02 x x (Z Z c ) m /A, (1.1) Avogadro s number In 1897, J. J. Thomson disovered eletrons. In 1905, Einstein interpreted the photoeletri effet In 1911 - Rutherford proved that atoms are omposed of a point-like positively harged, massive nuleus surrounded

More information

estimated pulse sequene we produe estimates of the delay time struture of ripple-red events. Formulation of the Problem We model the k th seismi trae

estimated pulse sequene we produe estimates of the delay time struture of ripple-red events. Formulation of the Problem We model the k th seismi trae Bayesian Deonvolution of Seismi Array Data Using the Gibbs Sampler Eri A. Suess Robert Shumway, University of California, Davis Rong Chen, Texas A&M University Contat: Eri A. Suess, Division of Statistis,

More information

Temperature-Gradient-Driven Tearing Modes

Temperature-Gradient-Driven Tearing Modes 1 TH/S Temperature-Gradient-Driven Tearing Modes A. Botrugno 1), P. Buratti 1), B. Coppi ) 1) EURATOM-ENEA Fusion Assoiation, Frasati (RM), Italy ) Massahussets Institute of Tehnology, Cambridge (MA),

More information

Experiment 03: Work and Energy

Experiment 03: Work and Energy MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physis Department Physis 8.01 Purpose of the Experiment: Experiment 03: Work and Energy In this experiment you allow a art to roll down an inlined ramp and run into

More information

Application of the Dyson-type boson mapping for low-lying electron excited states in molecules

Application of the Dyson-type boson mapping for low-lying electron excited states in molecules Prog. Theor. Exp. Phys. 05, 063I0 ( pages DOI: 0.093/ptep/ptv068 Appliation of the Dyson-type boson mapping for low-lying eletron exited states in moleules adao Ohkido, and Makoto Takahashi Teaher-training

More information