DOCSLIB.ORG

Chapter 9 Angular Momentum Quantum Mechanical Angular

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

Chapter 9 Angular Momentum Quantum Mechanical Angular Momentum Operators Classical angular momentum is a vector quantity denoted L~ = ~r X ~p. A common mnemonic to calculate the components is ˆi ˆj kˆ ~ L = x y z = ypz zpy ˆi + zpx xpz ˆj + xpy ypx ˆj p p p x y z ˆ ˆ ˆ = Lxi + Lyj + Lzj. Let’s focus on one component of angular momentum, say Lx = ypz zpy. On the right side of the equation are two components of position and two components of linear momentum. Quantum mechanically, all four quantities are operators. Since the product of two operators is an operator, and the dierence of operators is another operator, we expect the components of angular momentum to be operators. In other words, quantum mechanically x = z y, y = x z, z = y x. L YP ZP L ZP X P L XP YP These are the components. Angular momentum is the vector sum of the components. The sum of operators is another operator, so angular momentum is an operator. We have not encountered an operator like this one, however, this operator is comparable to a vector sum of operators; it is essentially a ket with operator components. We might write x z y L YP ZP > = y = x z . (9 1) L L ZP X P z y x L XP YP A word of caution concerning common notation—this is usually written just , and the ket/vector nature of quantum mechanical angular momentum is not explicitly writtenL but implied. Equation (9-1) is in abstract Hilbert space and is completely devoid of a representation. We will want to pick a basis to perform a calculation. In position space, for instance x, y, and z, X → Y → Z → and ∂ ∂ ∂ x ih , y ih , and z ih . P → ∂x P → ∂y P → ∂z Equation (9–1) in position space would then be written ihy ∂ + ihz ∂ ∂z ∂y > = ihz ∂ + ihx ∂ . (9 2) L ∂x ∂z ihx ∂ + ihy ∂ ∂y ∂x The operator nature of the components promise diculty, because unlike their classical analogs which are scalars, the angular momentum operators do not commute. 300 Example 9–1: Show the components of angular momentum in position space do not commute. Let the commutator of any two components, say x, y , act on the function x. This means L L x, y x = x y y x x L L L L L L ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ihy + ihz ihz + ihx x ihz + ihx ihy + ihz x → ∂z ∂y ∂x ∂z ∂x ∂z ∂z ∂y ∂ ∂ ∂ ∂ = ihy + ihz ihz ihz + ihx 0 ∂z ∂y ∂x ∂z 2 = ih y = h2y = 0, 6 therefore x and y do not commute. Using functions which are simply appropriate posi- tion spaceL components,L other components of angular momentum can be shown not to commute similarly. Example 9–2: What is equation (9–1) in the momentum basis? In momentum space, the operators are ∂ ∂ ∂ ih , ih , and ih , X → ∂px Y → ∂py Z → ∂pz and x px, y py, and z pz. P → P → P → Equation (9–1) in momentum space would be written ∂ ∂ ih pz ih py ∂py ∂pz ∂ ∂ > = ih px ih pz . ∂pz ∂px L ∂ ∂ ih py ih px ∂px ∂py Canonical Commutation Relations in Three Dimensions We indicated in equation (9–3) the fundamental canonical commutator is , = ih. X P This is ne when working in one dimension, however, descriptions of angular momentum are generally three dimensional. The generalization to three dimensions2,3 is i, j = 0, (9 3) X X 2 Cohen-Tannoudji, Quantum Mechanics (John Wiley & Sons, New York, 1977), pp 149 – 151. 3 Sakurai, Modern Quantum Mechanics (Addison–Wesley Publishing Company, Reading, Mas- sachusetts; 1994), pp 44 – 51. 301 which means any position component commutes with any other position component including itself, i, j = 0, (9 4) P P which means any linear momentum component commutes with any other linear momentum com- ponent including itself, i, j = ihi,j , (9 5) X P and the meaning of this equation requires some discussion. This means a position component will commute with an unlike component of linear momentum, , y = , z = , x = , z = , x = , y = 0, X P X P Y P Y P Z P Z P but a position component and a like component of linear momentum are canonical commutators, i.e., x, x = , y = , z = ih. X P Y P Z P Commutator Algebra In order to use the canonical commutators of equations (9–3) through (9–5), we need to develop some relations for commutators in excess of those discussed in chapter 3. For any operators , , and , the relations below, some of which we have used previously, may be a useful list. A B C , = 0 A A , = , A B B A , c = 0, for any scalar c, A , c = c , , for any scalar c, A B A B + , = , + , A B C A C B C , = , + , (9 6) A BC A B C B A C , , + , , + , , = 0. A B C B C A C A B You may have encounteredh relations i similarh to thesei inh classical mechanicsi where the brackets are Poisson brackets. In particular, the last relation is known as the Jacobi identity. We are interested in quantum mechanical commutators and there are two important dierences. Classical mechanics is concerned with quantities which are intrinsically real and are of nite dimension. Quantum mechanics is concerned with quantitites which are intrinsically complex and are generally of innite dimension. Equation (9–6) is a relation we want to develop further. Example 9–3: Prove equation (9–6). , = A BC A B C B C A = + A B C B A C B A C B C A = + A B B A C B A C C A = , + , , A B C B A C where we have added zero, in the form + , in the second line. B A C BAC 302 Example 9–4: Develop a relation for , in terms of commutators of individual operators. AB C , = AB C A B C C A B = + A B C A C B A C B C A B = + A B C C B A C C A B = , + , . A B C A C B Example 9–5: Develop a relation for , in terms of commutators of individual operators. AB CD Using the result of example 9–3, , = , + , , AB CD AB C D C AB D and using the result of example 9–4 on both of the commutators on the right, , = , + , + , + , AB CD A B C A C B D C A B D A D B = , + , + , + , , A B C D A C BD CA B D C A D B which is the desired result. Angular Momentum Commutation Relations Given the relations of equations (9–3) through (9–5), it follows that x, y = ih z, y, z = ih x, and z, x = ih y. (9 7) L L L L L L L L L Example 9–6: Show x, y = ih z. L L L x, y = z y, x z L L YP Z P ZP X P = z y x z x z z y YP Z P ZP X P ZP X P YP Z P = z x z z y x + y z x z + x y + z z z y YP ZP Y P XP Z P ZP ZP XP Z P YP ZP ZP XP YP X P ZP = z x x z + y z z y YP ZP Z P YP ZP XP X P ZP + x y y x + z z z z ZP ZP Z P ZP XP YP Y P XP = z, x + y , z + x, y + z, z . YP ZP ZP XP ZP ZP XP YP Using the result of example 9–5, the plan is to express these commutators in terms of individual operators, and then evaluate those using the commutation relations of equations (9–3) through (9– 5). In example 9–5, one commutator of the products of two operators turns into four commutators. Since we start with four commutators of the products of two operators, we are going to get 16 303 commutators in terms of individual operators. The good news is 14 of them are zero from equations (9–3), (9–4), and (9–5), so will be struck. x, y = z, x + , z x + z, x + , x z L L Y P Z P Y Z P P ZY P P Z Y P P + y, z + , y z + y, z + , z y Z P X P Z X P P XZ P P X Z P P + x, y + , x y + x, y + , y x Z P Z P Z Z P P ZZ P P Z Z P P + z, z + , z z + z, z + , z z X P Y P X Y P P YX P P Y X P P = z, x + , z y Y P Z P X Z P P = ih x + ih y Y P X P = ih y x XP Y P = ih z . L The other two relations, y, z = ih x and z, x = ih y can be calculated using similar procedures. L L L L L L A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators x, y, and L L z. In the form x, y, and z , these are abstract operators in an innite dimensional HilbertL space. RememberL L from chapterL 2 that a subspace is a specic subset of a general complex linear vector space. In this case, we are going to nd relations in a subspace C3 of an innite dimensional Hilbert space.
Recommended publications
  • 1 Notation for States

    1 Notation for States

    The S-Matrix1 D. E. Soper2 University of Oregon Physics 634, Advanced Quantum Mechanics November 2000 1 Notation for states In these notes we discuss scattering nonrelativistic quantum mechanics. We will use states with the nonrelativistic normalization h~p |~ki = (2π)3δ(~p − ~k). (1) Recall that in a relativistic theory there is an extra factor of 2E on the right hand side of this relation, where E = [~k2 + m2]1/2. We will use states in the “Heisenberg picture,” in which states |ψ(t)i do not depend on time. Often in quantum mechanics one uses the Schr¨odinger picture, with time dependent states |ψ(t)iS. The relation between these is −iHt |ψ(t)iS = e |ψi. (2) Thus these are the same at time zero, and the Schrdinger states obey d i |ψ(t)i = H |ψ(t)i (3) dt S S In the Heisenberg picture, the states do not depend on time but the op- erators do depend on time. A Heisenberg operator O(t) is related to the corresponding Schr¨odinger operator OS by iHt −iHt O(t) = e OS e (4) Thus hψ|O(t)|ψi = Shψ(t)|OS|ψ(t)iS. (5) The Heisenberg picture is favored over the Schr¨odinger picture in the case of relativistic quantum mechanics: we don’t have to say which reference 1Copyright, 2000, D. E. Soper [email protected] 1 frame we use to define t in |ψ(t)iS. For operators, we can deal with local operators like, for instance, the electric field F µν(~x, t).
  • Motion of the Reduced Density Operator

    Motion of the Reduced Density Operator

    Motion of the Reduced Density Operator Nicholas Wheeler, Reed College Physics Department Spring 2009 Introduction. Quantum mechanical decoherence, dissipation and measurements all involve the interaction of the system of interest with an environmental system (reservoir, measurement device) that is typically assumed to possess a great many degrees of freedom (while the system of interest is typically assumed to possess relatively few degrees of freedom). The state of the composite system is described by a density operator ρ which in the absence of system-bath interaction we would denote ρs ρe, though in the cases of primary interest that notation becomes unavailable,⊗ since in those cases the states of the system and its environment are entangled. The observable properties of the system are latent then in the reduced density operator ρs = tre ρ (1) which is produced by “tracing out” the environmental component of ρ. Concerning the specific meaning of (1). Let n) be an orthonormal basis | in the state space H of the (open) system, and N) be an orthonormal basis s !| " in the state space H of the (also open) environment. Then n) N) comprise e ! " an orthonormal basis in the state space H = H H of the |(closed)⊗| composite s e ! " system. We are in position now to write ⊗ tr ρ I (N ρ I N) e ≡ s ⊗ | s ⊗ | # ! " ! " ↓ = ρ tr ρ in separable cases s · e The dynamics of the composite system is generated by Hamiltonian of the form H = H s + H e + H i 2 Motion of the reduced density operator where H = h I s s ⊗ e = m h n m N n N $ | s| % | % ⊗ | % · $ | ⊗ $ | m,n N # # $% & % &' H = I h e s ⊗ e = n M n N M h N | % ⊗ | % · $ | ⊗ $ | $ | e| % n M,N # # $% & % &' H = m M m M H n N n N i | % ⊗ | % $ | ⊗ $ | i | % ⊗ | % $ | ⊗ $ | m,n M,N # # % &$% & % &'% & —all components of which we will assume to be time-independent.
  • An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]

    An S-Matrix for Massless Particles Arxiv:1911.06821V2 [Hep-Th]

    An S-Matrix for Massless Particles Holmfridur Hannesdottir and Matthew D. Schwartz Department of Physics, Harvard University, Cambridge, MA 02138, USA Abstract The traditional S-matrix does not exist for theories with massless particles, such as quantum electrodynamics. The difficulty in isolating asymptotic states manifests itself as infrared divergences at each order in perturbation theory. Building on insights from the literature on coherent states and factorization, we construct an S-matrix that is free of singularities order-by-order in perturbation theory. Factorization guarantees that the asymptotic evolution in gauge theories is universal, i.e. independent of the hard process. Although the hard S-matrix element is computed between well-defined few particle Fock states, dressed/coherent states can be seen to form as intermediate states in the calculation of hard S-matrix elements. We present a framework for the perturbative calculation of hard S-matrix elements combining Lorentz-covariant Feyn- man rules for the dressed-state scattering with time-ordered perturbation theory for the asymptotic evolution. With hard cutoffs on the asymptotic Hamiltonian, the cancella- tion of divergences can be seen explicitly. In dimensional regularization, where the hard cutoffs are replaced by a renormalization scale, the contribution from the asymptotic evolution produces scaleless integrals that vanish. A number of illustrative examples are given in QED, QCD, and N = 4 super Yang-Mills theory. arXiv:1911.06821v2 [hep-th] 24 Aug 2020 Contents 1 Introduction1 2 The hard S-matrix6 2.1 SH and dressed states . .9 2.2 Computing observables using SH ........................... 11 2.3 Soft Wilson lines . 14 3 Computing the hard S-matrix 17 3.1 Asymptotic region Feynman rules .
  • Vectors, Matrices and Coordinate Transformations

    Vectors, Matrices and Coordinate Transformations

    S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between them, physical laws can often be written in a simple form. Since we will making extensive use of vectors in Dynamics, we will summarize some of their important properties. Vectors For our purposes we will think of a vector as a mathematical representation of a physical entity which has both magnitude and direction in a 3D space. Examples of physical vectors are forces, moments, and velocities. Geometrically, a vector can be represented as arrows. The length of the arrow represents its magnitude. Unless indicated otherwise, we shall assume that parallel translation does not change a vector, and we shall call the vectors satisfying this property, free vectors. Thus, two vectors are equal if and only if they are parallel, point in the same direction, and have equal length. Vectors are usually typed in boldface and scalar quantities appear in lightface italic type, e.g. the vector quantity A has magnitude, or modulus, A = |A|. In handwritten text, vectors are often expressed using the −→ arrow, or underbar notation, e.g. A , A. Vector Algebra Here, we introduce a few useful operations which are defined for free vectors. Multiplication by a scalar If we multiply a vector A by a scalar α, the result is a vector B = αA, which has magnitude B = |α|A. The vector B, is parallel to A and points in the same direction if α > 0.
  • Position Management System Online Subject Area

    Position Management System Online Subject Area

    USDA, NATIONAL FINANCE CENTER INSIGHT ENTERPRISE REPORTING Business Intelligence Delivered Insight Quick Reference | Position Management System Online Subject Area What is Position Management System Online (PMSO)? • This Subject Area provides snapshots in time of organization position listings including active (filled and vacant), inactive, and deleted positions. • Position data includes a Master Record, containing basic position data such as grade, pay plan, or occupational series code. • The Master Record is linked to one or more Individual Positions containing organizational structure code, duty station code, and accounting station code data. History • The most recent daily snapshot is available during a given pay period until BEAR runs. • Bi-Weekly snapshots date back to Pay Period 1 of 2014. Data Refresh* Position Management System Online Common Reports Daily • Provides daily results of individual position information, HR Area Report Name Load which changes on a daily basis. Bi-Weekly Organization • Position Daily for current pay and Position Organization with period/ Bi-Weekly • Provides the latest record regardless of previous changes Management PII (PMSO) for historical pay that occur to the data during a given pay period. periods *View the Insight Data Refresh Report to determine the most recent date of refresh Reminder: In all PMSO reports, users should make sure to include: • An Organization filter • PMSO Key elements from the Master Record folder • SSNO element from the Incumbent Employee folder • A time filter from the Snapshot Time folder 1 USDA, NATIONAL FINANCE CENTER INSIGHT ENTERPRISE REPORTING Business Intelligence Delivered Daily Calendar Filters Bi-Weekly Calendar Filters There are three time options when running a bi-weekly There are two ways to pull the most recent daily data in a PMSO report: PMSO report: 1.
  • Chapters, in This Chapter We Present Methods Thatare Not Yet Employed by Industrial Robots, Except in an Extremely Simplifiedway

    Chapters, in This Chapter We Present Methods Thatare Not Yet Employed by Industrial Robots, Except in an Extremely Simplifiedway

    C H A P T E R 11 Force control of manipulators 11.1 INTRODUCTION 11.2 APPLICATION OF INDUSTRIAL ROBOTS TO ASSEMBLY TASKS 11.3 A FRAMEWORK FOR CONTROL IN PARTIALLY CONSTRAINED TASKS 11.4 THE HYBRID POSITION/FORCE CONTROL PROBLEM 11.5 FORCE CONTROL OFA MASS—SPRING SYSTEM 11.6 THE HYBRID POSITION/FORCE CONTROL SCHEME 11.7 CURRENT INDUSTRIAL-ROBOT CONTROL SCHEMES 11.1 INTRODUCTION Positioncontrol is appropriate when a manipulator is followinga trajectory through space, but when any contact is made between the end-effector and the manipulator's environment, mere position control might not suffice. Considera manipulator washing a window with a sponge. The compliance of thesponge might make it possible to regulate the force applied to the window by controlling the position of the end-effector relative to the glass. If the sponge isvery compliant or the position of the glass is known very accurately, this technique could work quite well. If, however, the stiffness of the end-effector, tool, or environment is high, it becomes increasingly difficult to perform operations in which the manipulator presses against a surface. Instead of washing with a sponge, imagine that the manipulator is scraping paint off a glass surface, usinga rigid scraping tool. If there is any uncertainty in the position of the glass surfaceor any error in the position of the manipulator, this task would become impossible. Either the glass would be broken, or the manipulator would wave the scraping toolover the glass with no contact taking place. In both the washing and scraping tasks, it would bemore reasonable not to specify the position of the plane of the glass, but rather to specifya force that is to be maintained normal to the surface.
  • 5 the Dirac Equation and Spinors

    5 the Dirac Equation and Spinors

    5 The Dirac Equation and Spinors In this section we develop the appropriate wavefunctions for fundamental fermions and bosons. 5.1 Notation Review The three dimension differential operator is : ∂ ∂ ∂ = , , (5.1) ∂x ∂y ∂z We can generalise this to four dimensions ∂µ: 1 ∂ ∂ ∂ ∂ ∂ = , , , (5.2) µ c ∂t ∂x ∂y ∂z 5.2 The Schr¨odinger Equation First consider a classical non-relativistic particle of mass m in a potential U. The energy-momentum relationship is: p2 E = + U (5.3) 2m we can substitute the differential operators: ∂ Eˆ i pˆ i (5.4) → ∂t →− to obtain the non-relativistic Schr¨odinger Equation (with = 1): ∂ψ 1 i = 2 + U ψ (5.5) ∂t −2m For U = 0, the free particle solutions are: iEt ψ(x, t) e− ψ(x) (5.6) ∝ and the probability density ρ and current j are given by: 2 i ρ = ψ(x) j = ψ∗ ψ ψ ψ∗ (5.7) | | −2m − with conservation of probability giving the continuity equation: ∂ρ + j =0, (5.8) ∂t · Or in Covariant notation: µ µ ∂µj = 0 with j =(ρ,j) (5.9) The Schr¨odinger equation is 1st order in ∂/∂t but second order in ∂/∂x. However, as we are going to be dealing with relativistic particles, space and time should be treated equally. 25 5.3 The Klein-Gordon Equation For a relativistic particle the energy-momentum relationship is: p p = p pµ = E2 p 2 = m2 (5.10) · µ − | | Substituting the equation (5.4), leads to the relativistic Klein-Gordon equation: ∂2 + 2 ψ = m2ψ (5.11) −∂t2 The free particle solutions are plane waves: ip x i(Et p x) ψ e− · = e− − · (5.12) ∝ The Klein-Gordon equation successfully describes spin 0 particles in relativistic quan- tum field theory.
  • Chapter 5 ANGULAR MOMENTUM and ROTATIONS

    Chapter 5 ANGULAR MOMENTUM and ROTATIONS

    Chapter 5 ANGULAR MOMENTUM AND ROTATIONS In classical mechanics the total angular momentum L~ of an isolated system about any …xed point is conserved. The existence of a conserved vector L~ associated with such a system is itself a consequence of the fact that the associated Hamiltonian (or Lagrangian) is invariant under rotations, i.e., if the coordinates and momenta of the entire system are rotated “rigidly” about some point, the energy of the system is unchanged and, more importantly, is the same function of the dynamical variables as it was before the rotation. Such a circumstance would not apply, e.g., to a system lying in an externally imposed gravitational …eld pointing in some speci…c direction. Thus, the invariance of an isolated system under rotations ultimately arises from the fact that, in the absence of external …elds of this sort, space is isotropic; it behaves the same way in all directions. Not surprisingly, therefore, in quantum mechanics the individual Cartesian com- ponents Li of the total angular momentum operator L~ of an isolated system are also constants of the motion. The di¤erent components of L~ are not, however, compatible quantum observables. Indeed, as we will see the operators representing the components of angular momentum along di¤erent directions do not generally commute with one an- other. Thus, the vector operator L~ is not, strictly speaking, an observable, since it does not have a complete basis of eigenstates (which would have to be simultaneous eigenstates of all of its non-commuting components). This lack of commutivity often seems, at …rst encounter, as somewhat of a nuisance but, in fact, it intimately re‡ects the underlying structure of the three dimensional space in which we are immersed, and has its source in the fact that rotations in three dimensions about di¤erent axes do not commute with one another.
  • Solving the Geodesic Equation

    Solving the Geodesic Equation

    Solving the Geodesic Equation Jeremy Atkins December 12, 2018 Abstract We find the general form of the geodesic equation and discuss the closed form relation to find Christoffel symbols. We then show how to use metric independence to find Killing vector fields, which allow us to solve the geodesic equation when there are helpful symmetries. We also discuss a more general way to find Killing vector fields, and some of their properties as a Lie algebra. 1 The Variational Method We will exploit the following variational principle to characterize motion in general relativity: The world line of a free test particle between two timelike separated points extremizes the proper time between them. where a test particle is one that is not a significant source of spacetime cur- vature, and a free particles is one that is only under the influence of curved spacetime. Similarly to classical Lagrangian mechanics, we can use this to de- duce the equations of motion for a metric. The proper time along a timeline worldline between point A and point B for the metric gµν is given by Z B Z B µ ν 1=2 τAB = dτ = (−gµν (x)dx dx ) (1) A A using the Einstein summation notation, and µ, ν = 0; 1; 2; 3. We can parame- terize the four coordinates with the parameter σ where σ = 0 at A and σ = 1 at B. This gives us the following equation for the proper time: Z 1 dxµ dxν 1=2 τAB = dσ −gµν (x) (2) 0 dσ dσ We can treat the integrand as a Lagrangian, dxµ dxν 1=2 L = −gµν (x) (3) dσ dσ and it's clear that the world lines extremizing proper time are those that satisfy the Euler-Lagrange equation: @L d @L − = 0 (4) @xµ dσ @(dxµ/dσ) 1 These four equations together give the equation for the worldline extremizing the proper time.
  • Position, Velocity, and Acceleration

    Position, Velocity, and Acceleration

    Position,Position, Velocity,Velocity, andand AccelerationAcceleration Mr.Mr. MiehlMiehl www.tesd.net/miehlwww.tesd.net/miehl [email protected]@tesd.net Position,Position, VelocityVelocity && AccelerationAcceleration Velocity is the rate of change of position with respect to time. ΔD Velocity = ΔT Acceleration is the rate of change of velocity with respect to time. ΔV Acceleration = ΔT Position,Position, VelocityVelocity && AccelerationAcceleration Warning: Professional driver, do not attempt! When you’re driving your car… Position,Position, VelocityVelocity && AccelerationAcceleration squeeeeek! …and you jam on the brakes… Position,Position, VelocityVelocity && AccelerationAcceleration …and you feel the car slowing down… Position,Position, VelocityVelocity && AccelerationAcceleration …what you are really feeling… Position,Position, VelocityVelocity && AccelerationAcceleration …is actually acceleration. Position,Position, VelocityVelocity && AccelerationAcceleration I felt that acceleration. Position,Position, VelocityVelocity && AccelerationAcceleration How do you find a function that describes a physical event? Steps for Modeling Physical Data 1) Perform an experiment. 2) Collect and graph data. 3) Decide what type of curve fits the data. 4) Use statistics to determine the equation of the curve. Position,Position, VelocityVelocity && AccelerationAcceleration A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. a) Where is the crab after 2 seconds? b) How fast is it moving at that instant (2 seconds)? Position,Position, VelocityVelocity && AccelerationAcceleration A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given by P (t ) = t 2 + t. a) Where is the crab after 2 seconds? 2 P()22=+ () ( 2) P()26= feet Position,Position, VelocityVelocity && AccelerationAcceleration A crab is crawling along the edge of your desk.
  • Chapter 3 Motion in Two and Three Dimensions

    Chapter 3 Motion in Two and Three Dimensions

    Chapter 3 Motion in Two and Three Dimensions 3.1 The Important Stuff 3.1.1 Position In three dimensions, the location of a particle is specified by its location vector, r: r = xi + yj + zk (3.1) If during a time interval ∆t the position vector of the particle changes from r1 to r2, the displacement ∆r for that time interval is ∆r = r1 − r2 (3.2) = (x2 − x1)i +(y2 − y1)j +(z2 − z1)k (3.3) 3.1.2 Velocity If a particle moves through a displacement ∆r in a time interval ∆t then its average velocity for that interval is ∆r ∆x ∆y ∆z v = = i + j + k (3.4) ∆t ∆t ∆t ∆t As before, a more interesting quantity is the instantaneous velocity v, which is the limit of the average velocity when we shrink the time interval ∆t to zero. It is the time derivative of the position vector r: dr v = (3.5) dt d = (xi + yj + zk) (3.6) dt dx dy dz = i + j + k (3.7) dt dt dt can be written: v = vxi + vyj + vzk (3.8) 51 52 CHAPTER 3. MOTION IN TWO AND THREE DIMENSIONS where dx dy dz v = v = v = (3.9) x dt y dt z dt The instantaneous velocity v of a particle is always tangent to the path of the particle. 3.1.3 Acceleration If a particle’s velocity changes by ∆v in a time period ∆t, the average acceleration a for that period is ∆v ∆v ∆v ∆v a = = x i + y j + z k (3.10) ∆t ∆t ∆t ∆t but a much more interesting quantity is the result of shrinking the period ∆t to zero, which gives us the instantaneous acceleration, a.
  • Geodetic Position Computations

    Geodetic Position Computations

    GEODETIC POSITION COMPUTATIONS E. J. KRAKIWSKY D. B. THOMSON February 1974 TECHNICALLECTURE NOTES REPORT NO.NO. 21739 PREFACE In order to make our extensive series of lecture notes more readily available, we have scanned the old master copies and produced electronic versions in Portable Document Format. The quality of the images varies depending on the quality of the originals. The images have not been converted to searchable text. GEODETIC POSITION COMPUTATIONS E.J. Krakiwsky D.B. Thomson Department of Geodesy and Geomatics Engineering University of New Brunswick P.O. Box 4400 Fredericton. N .B. Canada E3B5A3 February 197 4 Latest Reprinting December 1995 PREFACE The purpose of these notes is to give the theory and use of some methods of computing the geodetic positions of points on a reference ellipsoid and on the terrain. Justification for the first three sections o{ these lecture notes, which are concerned with the classical problem of "cCDputation of geodetic positions on the surface of an ellipsoid" is not easy to come by. It can onl.y be stated that the attempt has been to produce a self contained package , cont8.i.ning the complete development of same representative methods that exist in the literature. The last section is an introduction to three dimensional computation methods , and is offered as an alternative to the classical approach. Several problems, and their respective solutions, are presented. The approach t~en herein is to perform complete derivations, thus stqing awrq f'rcm the practice of giving a list of for11111lae to use in the solution of' a problem.