Difference Between Angular Momentum and Pseudoangular
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Thermodynamics of Spacetime: the Einstein Equation of State
gr-qc/9504004 UMDGR-95-114 Thermodynamics of Spacetime: The Einstein Equation of State Ted Jacobson Department of Physics, University of Maryland College Park, MD 20742-4111, USA [email protected] Abstract The Einstein equation is derived from the proportionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat, entropy, and temperature. The key idea is to demand that this relation hold for all the local Rindler causal horizons through each spacetime point, with δQ and T interpreted as the energy flux and Unruh temperature seen by an accelerated observer just inside the horizon. This requires that gravitational lensing by matter energy distorts the causal structure of spacetime in just such a way that the Einstein equation holds. Viewed in this way, the Einstein equation is an equation of state. This perspective suggests that it may be no more appropriate to canonically quantize the Einstein equation than it would be to quantize the wave equation for sound in air. arXiv:gr-qc/9504004v2 6 Jun 1995 The four laws of black hole mechanics, which are analogous to those of thermodynamics, were originally derived from the classical Einstein equation[1]. With the discovery of the quantum Hawking radiation[2], it became clear that the analogy is in fact an identity. How did classical General Relativity know that horizon area would turn out to be a form of entropy, and that surface gravity is a temperature? In this letter I will answer that question by turning the logic around and deriving the Einstein equation from the propor- tionality of entropy and horizon area together with the fundamental relation δQ = T dS connecting heat Q, entropy S, and temperature T . -
Lecture Notes
Solid State Physics PHYS 40352 by Mike Godfrey Spring 2012 Last changed on May 22, 2017 ii Contents Preface v 1 Crystal structure 1 1.1 Lattice and basis . .1 1.1.1 Unit cells . .2 1.1.2 Crystal symmetry . .3 1.1.3 Two-dimensional lattices . .4 1.1.4 Three-dimensional lattices . .7 1.1.5 Some cubic crystal structures ................................ 10 1.2 X-ray crystallography . 11 1.2.1 Diffraction by a crystal . 11 1.2.2 The reciprocal lattice . 12 1.2.3 Reciprocal lattice vectors and lattice planes . 13 1.2.4 The Bragg construction . 14 1.2.5 Structure factor . 15 1.2.6 Further geometry of diffraction . 17 2 Electrons in crystals 19 2.1 Summary of free-electron theory, etc. 19 2.2 Electrons in a periodic potential . 19 2.2.1 Bloch’s theorem . 19 2.2.2 Brillouin zones . 21 2.2.3 Schrodinger’s¨ equation in k-space . 22 2.2.4 Weak periodic potential: Nearly-free electrons . 23 2.2.5 Metals and insulators . 25 2.2.6 Band overlap in a nearly-free-electron divalent metal . 26 2.2.7 Tight-binding method . 29 2.3 Semiclassical dynamics of Bloch electrons . 32 2.3.1 Electron velocities . 33 2.3.2 Motion in an applied field . 33 2.3.3 Effective mass of an electron . 34 2.4 Free-electron bands and crystal structure . 35 2.4.1 Construction of the reciprocal lattice for FCC . 35 2.4.2 Group IV elements: Jones theory . 36 2.4.3 Binding energy of metals . -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential -
The Equation of Radiative Transfer How Does the Intensity of Radiation Change in the Presence of Emission and / Or Absorption?
The equation of radiative transfer How does the intensity of radiation change in the presence of emission and / or absorption? Definition of solid angle and steradian Sphere radius r - area of a patch dS on the surface is: dS = rdq ¥ rsinqdf ≡ r2dW q dS dW is the solid angle subtended by the area dS at the center of the † sphere. Unit of solid angle is the steradian. 4p steradians cover whole sphere. ASTR 3730: Fall 2003 Definition of the specific intensity Construct an area dA normal to a light ray, and consider all the rays that pass through dA whose directions lie within a small solid angle dW. Solid angle dW dA The amount of energy passing through dA and into dW in time dt in frequency range dn is: dE = In dAdtdndW Specific intensity of the radiation. † ASTR 3730: Fall 2003 Compare with definition of the flux: specific intensity is very similar except it depends upon direction and frequency as well as location. Units of specific intensity are: erg s-1 cm-2 Hz-1 steradian-1 Same as Fn Another, more intuitive name for the specific intensity is brightness. ASTR 3730: Fall 2003 Simple relation between the flux and the specific intensity: Consider a small area dA, with light rays passing through it at all angles to the normal to the surface n: n o In If q = 90 , then light rays in that direction contribute zero net flux through area dA. q For rays at angle q, foreshortening reduces the effective area by a factor of cos(q). -
Energy Bands in Crystals
Energy Bands in Crystals This chapter will apply quantum mechanics to a one dimensional, periodic lattice of potential wells which serves as an analogy to electrons interacting with the atoms of a crystal. We will show that as the number of wells becomes large, the allowed energy levels for the electron form nearly continuous energy bands separated by band gaps where no electron can be found. We thus have an interesting quantum system which exhibits many dual features of the quantum continuum and discrete spectrum. several tenths nm The energy band structure plays a crucial role in the theory of electron con- ductivity in the solid state and explains why materials can be classified as in- sulators, conductors and semiconductors. The energy band structure present in a semiconductor is a crucial ingredient in understanding how semiconductor devices work. Energy levels of “Molecules” By a “molecule” a quantum system consisting of a few periodic potential wells. We have already considered a two well molecular analogy in our discussions of the ammonia clock. 1 V -c sin(k(x-b)) V -c sin(k(x-b)) cosh βx sinhβ x V=0 V = 0 0 a b d 0 a b d k = 2 m E β= 2m(V-E) h h Recall for reasonably large hump potentials V , the two lowest lying states (a even ground state and odd first excited state) are very close in energy. As V →∞, both the ground and first excited state wave functions become completely isolated within a well with little wave function penetration into the classically forbidden central hump. -
Area of Polygons and Complex Figures
Geometry AREA OF POLYGONS AND COMPLEX FIGURES Area is the number of non-overlapping square units needed to cover the interior region of a two- dimensional figure or the surface area of a three-dimensional figure. For example, area is the region that is covered by floor tile (two-dimensional) or paint on a box or a ball (three- dimensional). For additional information about specific shapes, see the boxes below. For additional general information, see the Math Notes box in Lesson 1.1.2 of the Core Connections, Course 2 text. For additional examples and practice, see the Core Connections, Course 2 Checkpoint 1 materials or the Core Connections, Course 3 Checkpoint 4 materials. AREA OF A RECTANGLE To find the area of a rectangle, follow the steps below. 1. Identify the base. 2. Identify the height. 3. Multiply the base times the height to find the area in square units: A = bh. A square is a rectangle in which the base and height are of equal length. Find the area of a square by multiplying the base times itself: A = b2. Example base = 8 units 4 32 square units height = 4 units 8 A = 8 · 4 = 32 square units Parent Guide with Extra Practice 135 Problems Find the areas of the rectangles (figures 1-8) and squares (figures 9-12) below. 1. 2. 3. 4. 2 mi 5 cm 8 m 4 mi 7 in. 6 cm 3 in. 2 m 5. 6. 7. 8. 3 units 6.8 cm 5.5 miles 2 miles 8.7 units 7.25 miles 3.5 cm 2.2 miles 9. -
Right Triangles and the Pythagorean Theorem Related?
Activity Assess 9-6 EXPLORE & REASON Right Triangles and Consider △ ABC with altitude CD‾ as shown. the Pythagorean B Theorem D PearsonRealize.com A 45 C 5√2 I CAN… prove the Pythagorean Theorem using A. What is the area of △ ABC? Of △ACD? Explain your answers. similarity and establish the relationships in special right B. Find the lengths of AD‾ and AB‾ . triangles. C. Look for Relationships Divide the length of the hypotenuse of △ ABC VOCABULARY by the length of one of its sides. Divide the length of the hypotenuse of △ACD by the length of one of its sides. Make a conjecture that explains • Pythagorean triple the results. ESSENTIAL QUESTION How are similarity in right triangles and the Pythagorean Theorem related? Remember that the Pythagorean Theorem and its converse describe how the side lengths of right triangles are related. THEOREM 9-8 Pythagorean Theorem If a triangle is a right triangle, If... △ABC is a right triangle. then the sum of the squares of the B lengths of the legs is equal to the square of the length of the hypotenuse. c a A C b 2 2 2 PROOF: SEE EXAMPLE 1. Then... a + b = c THEOREM 9-9 Converse of the Pythagorean Theorem 2 2 2 If the sum of the squares of the If... a + b = c lengths of two sides of a triangle is B equal to the square of the length of the third side, then the triangle is a right triangle. c a A C b PROOF: SEE EXERCISE 17. Then... △ABC is a right triangle. -
Calculus Formulas and Theorems
Formulas and Theorems for Reference I. Tbigonometric Formulas l. sin2d+c,cis2d:1 sec2d l*cot20:<:sc:20 +.I sin(-d) : -sitt0 t,rs(-//) = t r1sl/ : -tallH 7. sin(A* B) :sitrAcosB*silBcosA 8. : siri A cos B - siu B <:os,;l 9. cos(A+ B) - cos,4cos B - siuA siriB 10. cos(A- B) : cosA cosB + silrA sirrB 11. 2 sirrd t:osd 12. <'os20- coS2(i - siu20 : 2<'os2o - I - 1 - 2sin20 I 13. tan d : <.rft0 (:ost/ I 14. <:ol0 : sirrd tattH 1 15. (:OS I/ 1 16. cscd - ri" 6i /F tl r(. cos[I ^ -el : sitt d \l 18. -01 : COSA 215 216 Formulas and Theorems II. Differentiation Formulas !(r") - trr:"-1 Q,:I' ]tra-fg'+gf' gJ'-,f g' - * (i) ,l' ,I - (tt(.r))9'(.,') ,i;.[tyt.rt) l'' d, \ (sttt rrJ .* ('oqI' .7, tJ, \ . ./ stll lr dr. l('os J { 1a,,,t,:r) - .,' o.t "11'2 1(<,ot.r') - (,.(,2.r' Q:T rl , (sc'c:.r'J: sPl'.r tall 11 ,7, d, - (<:s<t.r,; - (ls(].]'(rot;.r fr("'),t -.'' ,1 - fr(u") o,'ltrc ,l ,, 1 ' tlll ri - (l.t' .f d,^ --: I -iAl'CSllLl'l t!.r' J1 - rz 1(Arcsi' r) : oT Il12 Formulas and Theorems 2I7 III. Integration Formulas 1. ,f "or:artC 2. [\0,-trrlrl *(' .t "r 3. [,' ,t.,: r^x| (' ,I 4. In' a,,: lL , ,' .l 111Q 5. In., a.r: .rhr.r' .r r (' ,l f 6. sirr.r d.r' - ( os.r'-t C ./ 7. /.,,.r' dr : sitr.i'| (' .t 8. tl:r:hr sec,rl+ C or ln Jccrsrl+ C ,f'r^rr f 9. -
Distortion Correction and Momentum Representation of Angle-Resolved
Distortion Correction and Momentum Representation of Angle-Resolved Photoemission Data Jonathan Adam Rosen 13.Sc., University of California at Santa Cruz, 2006 A THESIS SUBMIYI’ED 1N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Physics) THE UNIVERSITY OF BRETISH COLUMBIA (Vancouver) October 2008 © Jonathan Adam Rosen, 2008 Abstract Angle Resolve Photoemission Spectroscopy (ARPES) experiments provides a map of intensity as function of angles and electron kinetic energy to measure the many-body spectral function, but the raw data returned by standard apparatus is not ready for analysis. An image warping based distortion correction from slit array calibration is shown to provide the relevant information for construction of ARPES intensity as a function of electron momentum. A theory is developed to understand the calculation and uncertainty of the distortion corrected angle space data and the final momentum data. An experimental procedure for determination of the electron analyzer focal point is described and shown to be in good agreement with predictions. The electron analyzer at the Quantum Materials Laboratory at UBC is found to have a focal point at cryostat position 1.09mm within 1.00 mm, and the systematic error in the angle is found to be 0.2 degrees. The angular error is shown to be proportional to a functional form of systematic error in the final ARPES data that is highly momentum dependent. 11 Table .of Contents Abstract ii Table of Contents iii List of Tables v -
The American Society of Echocardiography
1 THE AMERICAN SOCIETY OF ECHOCARDIOGRAPHY RECOMMENDATIONS FOR CARDIAC CHAMBER QUANTIFICATION IN ADULTS: A QUICK REFERENCE GUIDE FROM THE ASE WORKFLOW AND LAB MANAGEMENT TASK FORCE Accurate and reproducible assessment of cardiac chamber size and function is essential for clinical care. A standardized methodology creates a common approach to the assessment of cardiac structure and function both within and between echocardiography labs. This facilitates better communication and improves the ability to compare results between studies as well as differentiate normal from abnormal findings in an individual patient. This document summarizes key points from the 2015 ASE Chamber Quantification Guideline and is meant to serve as quick reference for sonographers and interpreting physicians. It is designed to provide guidance on chamber quantification for adult patients; a separate ASE Guidelines document that details recommended quantification methods in the pediatric age group has also been published and should be used for patients <18 years of age (3). (1) For details of the methodology and the rationale for current recommendations, the interested reader is referred to the complete Guideline statement. Figures and tables are reproduced from ASE Guidelines. (1,2) Table of Contents: 1. Left Ventricle (LV) Size and Function p. 2 a. LV Size p. 2 i. Linear Measurements p. 2 ii. Volume Measurements p. 2 iii. LV Mass Calculations p. 3 b. Left Ventricular Function Assessment p. 4 i. Global Systolic Function Parameters p. 4 ii. Regional Function p. 5 2. Right Ventricle (RV) Size and Function p. 6 a. RV Size p. 6 b. RV Function p. 8 3. Atria p. -
Residential Square Footage Guidelines
R e s i d e n t i a l S q u a r e F o o t a g e G u i d e l i n e s North Carolina Real Estate Commission North Carolina Real Estate Commission P.O. Box 17100 • Raleigh, North Carolina 27619-7100 Phone 919/875-3700 • Web Site: www.ncrec.gov Illustrations by David Hall Associates, Inc. Copyright © 1999 by North Carolina Real Estate Commission. All rights reserved. 7,500 copies of this public document were printed at a cost of $.000 per copy. • REC 3.40 11/1/2013 Introduction It is often said that the three most important factors in making a home buying decision are “location,” “location,” and “location.” Other than “location,” the single most-important factor is probably the size or “square footage” of the home. Not only is it an indicator of whether a particular home will meet a homebuyer’s space needs, but it also affords a convenient (though not always accurate) method for the buyer to estimate the value of the home and compare it to other properties. Although real estate agents are not required by the Real Estate License Law or Real Estate Commission rules to report the square footage of properties offered for sale (or rent), when they do report square footage, it is essential that the information they give prospective purchasers (or tenants) be accurate. At a minimum, information concerning square footage should include the amount of living area in the dwelling. The following guidelines and accompanying illustrations are designed to assist real estate brokers in measuring, calculating and reporting (both orally and in writing) the living area contained in detached and attached single-family residential buildings. -
Schrödinger Correspondence Applied to Crystals Arxiv:1812.06577V1
Schr¨odingerCorrespondence Applied to Crystals Eric J. Heller∗,y,z and Donghwan Kimy yDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 zDepartment of Physics, Harvard University, Cambridge, MA 02138 E-mail: [email protected] arXiv:1812.06577v1 [cond-mat.other] 17 Dec 2018 1 Abstract In 1926, E. Schr¨odingerpublished a paper solving his new time dependent wave equation for a displaced ground state in a harmonic oscillator (now called a coherent state). He showed that the parameters describing the mean position and mean mo- mentum of the wave packet obey the equations of motion of the classical oscillator while retaining its width. This was a qualitatively new kind of correspondence princi- ple, differing from those leading up to quantum mechanics. Schr¨odingersurely knew that this correspondence would extend to an N-dimensional harmonic oscillator. This Schr¨odingerCorrespondence Principle is an extremely intuitive and powerful way to approach many aspects of harmonic solids including anharmonic corrections. 1 Introduction Figure 1: Photocopy from Schr¨odinger's1926 paper In 1926 Schr¨odingermade the connection between the dynamics of a displaced quantum ground state Gaussian wave packet in a harmonic oscillator and classical motion in the same harmonic oscillator1 (see figure 1). The mean position of the Gaussian (its guiding position center) and the mean momentum (its guiding momentum center) follows classical harmonic oscillator equations of motion, while the width of the Gaussian remains stationary if it initially was a displaced (in position or momentum or both) ground state. This classic \coherent state" dynamics is now very well known.2 Specifically, for a harmonic oscillator 2 2 1 2 2 with Hamiltonian H = p =2m + 2 m! q , a Gaussian wave packet that beginning as A0 2 i i (q; 0) = exp i (q − q0) + p0(q − q0) + s0 (1) ~ ~ ~ becomes, under time evolution, At 2 i i (q; t) = exp i (q − qt) + pt(q − qt) + st : (2) ~ ~ ~ where pt = p0 cos(!t) − m!q0 sin(!t) qt = q0 cos(!t) + (p0=m!) sin(!t) (3) (i.e.