Rotational Invariance

7 Key excerpts on "Rotational Invariance"

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  Introduction to High Energy Physics

  • Donald H. Perkins(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  3 Invariance principles and conservation laws A very important concept in physics is the symmetry or invariance of the equations describing a physical system under an operation – which might be, for example, a translation or rotation in space. Intimately connected with such invariance properties are conservation laws – in the above cases, conservation of linear and angular momentum. Such conservation laws and the invariance principles and symmetries underlying them are the very backbone of particle physics. However, one must remember that their credibility rests entirely on experimental verification. A conservation law can be assumed to be absolute if there is no observational evidence to the contrary, but this assumption has to be accompanied by a limit set on possible violations by experiment. The transformations to be considered can be either continuous or discrete. A translation or rotation in space is an example of a continuous transformation, while spatial reflection through the origin of coordinates (the parity operation) is a discrete transformation. The associated conservation laws are additive and multiplicative, respectively. 3.1 Translation and rotation operators In an isolated physical system, free of any external forces, the total energy must be invariant under translations of the whole system in space. Since there are no external forces, the rate of change of momentum is zero and the momentum is constant. So invariance of the energy of a system under space translations corresponds to conservation of linear momentum. Similarly, invariance of the energy of a system under spatial rotations corresponds to conservation of angular momentum. 63 64 3 Invariance principles and conservation laws The effect of an infinitesimal translation δ r in space on a wavefunction ψ will be that it becomes ψ = ψ( r + δ r ) = ψ( r ) + δ r ∂ψ( r ) ∂ r = D ψ where D = 1 + δ r ∂ ∂ r (3.1) is an infinitesimal space translation operator.

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  Angular Momentum

  An Illustrated Guide to Rotational Symmetries for Physical Systems

  • William J. Thompson(Author)
  • 2008(Publication Date)
  • Wiley-VCH

    (Publisher)

  PREFACE If a physical system has only internal interactions and if space is isotropic, then in- trinsic properties of the system must be independent of its orientation and must be indistinguishable in all directions. From this fundamental rotational symmetry con- cept the theory of angular momentum has been developed into a sophisticated analy- tical and computational technique, especially when applied to quantum mechanics. I aim in this book to develop angular momentum theory in a pedagogically consistent way, starting from the geometrical concept of Rotational Invariance rather than from the dynamical idea of orbital angular momentum and its quantization. The latter ap- proach, though hallowed by tradition, needlessly confuses quantum mechanics with geometry. Topics are presented in an order so that new concepts are introduced and relevant formulas are derived in ways arising naturally in the treatment rather than by appeal- ing to unfamiliar concepts or ud hoc methods. Modern notation and terminology are used in a geometric and algebraic approach. Some concepts of group theory are in- troduced and are related to this approach, but knowledge of group theory is not re- quired. Those who plan to use continuous groups that are more abstract than the rotation group may thereby develop their insight and skills by practicing with rota- tions. I try to distinguish carefully results that depend only on rotational symmetry and are generally valid from those having their most fruitful interpretation from the viewpoint of quantum mechanics. Applications to quantum mechanics therefore usually appear toward the end of sections and chapters. Although Angular Momentum is intended to be pedagogically self-contained, the treatment is not encyclopedic, since broad-ranging surveys of angular momen- tum theory and extensive tabulations of formulas are now available. There is also a xi x i i PREFACE large research literature for further study, to which I direct you.

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  Advanced Modern Physics

  Theoretical Foundations

  • John Dirk Walecka(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  126 Advanced Modern Physics Lorentz, or Poincar´ e, group . 1 We postpone a discussion of the full in-homogeneous Lorentz group, and here return to our previous analysis of rotations , which form a subgroup of the full Lorentz group. 6.2 Rotational Invariance To the best of our current knowledge, space is isotropic. This implies that the hamiltonian must be invariant under rotations. The chapter on angular momentum investigated in detail the implications of this observation. The generators of rotations are the components of the angular momentum oper-ator ˆ J = ( ˆ J 1 , ˆ J 2 , ˆ J 3 ), and Rotational Invariance implies that the hamiltonian commutes with these generators [ ˆ J , ˆ H ] = 0 ; Rotational Invariance (6.1) The eigenstates of the hamiltonian can be labeled with the eigenvalues of ( ˆ J 2 , ˆ J 3 ), in our notation | jm angbracketright , and the first important observation is that if the hamiltonian is invariant under rotations, then the energy eigenvalues will be independent of m , that is, independent of the orientation of the system in space. Thus If the hamiltonian is invariant under rotations, the energy eigen-values for a given j will exhibit a (2 j + 1) -fold degeneracy in m . We saw this m -degeneracy in all our central-field applications in Vol. I. 2 The commutation relations between components of the angular momen-tum provide the basis for the angular momentum theory in chapter 3 [ ˆ J i , ˆ J j ] = iepsilon1 ijk ˆ J k ; angular momentum ( i,j,k ) = 1 , 2 , 3 (6.2) This algebra of the generators is closed under commutation. The funda-mental basis for angular momentum is | 1 2 m angbracketright , since any other angular mo-mentum can be obtained from direct products of these states.

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  An Introduction to Mechanics

  • Daniel Kleppner, Robert Kolenkow(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  Although fields lie beyond the scope of particle mechanics, it turns out that fields can possess energy, momentum, and angular momentum. When the angular momentum of the field is taken into account, the angular momentum of the entire particle–field system is conserved. The situation, in brief, is that Newtonian physics is incapable of pre-dicting conservation of angular momentum but no isolated system has yet been encountered experimentally for which the total angular mo-mentum is not conserved. We conclude that conservation of angular mo-mentum is an independent physical law, and until a contradiction is dis-covered, our physical understanding must be guided by it. 8.6 Rigid Body Rotation and the Tensor of Inertia The governing equation τ = d L / dt for rigid body motion bears a formal resemblance to the translational equation of motion F = d P / dt . There is, however, an essential di ff erence between them. Linear momentum and center of mass motion are simply related by the vector equation P = M V where M is a scalar, a simple number. P and V are therefore always parallel. The connection between L and ω is not so simple. For fixed axis rotation, L = I ω , and it is tempting to suppose that for any general rotation L = I ω , where I is a scalar. However, this cannot be correct, since we know from our study of the rotating skew rod, Example 8.4 , that L and ω are not necessarily parallel. In this section, we shall develop the general relation between angular momentum and angular velocity, and in Section 8.7 we shall attack the problem of solving the equations of motion. 8.6.1 Angular Momentum and the Tensor of Inertia We begin by showing that to analyze the rotational motion of a rigid body, we really need consider only the angular momentum about the center of mass as origin.

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  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  rotational energetics ,an analysis of the ability of bodies to do work because they are rotating. Energetics offers an occasion to introduce the important concept of rotational inertia, which will be especially useful for topics on rotational motion that will be covered later. In discussing bodies undergoing pure rotation, we will define and develop the notion of angular momentum, a fundamental quantity of considerable physical significance.

  Then we will consider a more general form of motion of an extended body—the combination of rotation and translation, which involves rotation of a body about an axis that is itself moving through space. We will concentrate on two specific cases of simultaneous rotation and translation: bodies with circular symmetry that are rolling without slipping along a plane surface and bodies with circular symmetry with motions that are determined by a string wound about the body.

  Finally, we will develop and illustrate a rotational conservation principle, the conservation of angular momentum. This rotational principle is of comparable stature in physics to that of the conservation of linear momentum, a principle introduced in Chapter 6 .

  7.1 Rotation About a Fixed Axis

  An extended body is composed of many particles. If the distance between each particle of the body remains constant, we call that body a rigid body . Rigid bodies are an idealization—they don’t exist in any state—but actual bodies can flex. We use the notion of a rigid body to facilitate the analysis of the motion of actual physical bodies. In the analysis of many physical bodies, the rigid-body assumption is an excellent approximation. In this chapter we will assume that all extended bodies are rigid bodies.

  Let us now consider the description of the purely rotational motion of a rigid body turning about an axis fixed in space.

  Rotational Kinematics

  Consider the circular disk of Figure 7.1(a) . It is a rigid body attached to a rod lying along an axis passing through the center of the disk normal to the face of the disk. An axis of rotation of a body is a line in space about which the particles within the body maintain a constant distance and, therefore, move in a circular path about the axis. Because the disk is a rigid body, the rotational motion of the disk may be described by the motion of an arbitrary particle within the disk. Look, then, at the particle P on the face of the disk shown in Figure 7.1(b)

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  Introduction to Continuous Symmetries

  From Space-Time to Quantum Mechanics

  • Franck Laloë, Nicole Ostrowsky, Daniel Ostrowsky(Authors)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  • Translational invariance Let b be an arbitrary constant vector. Consider the transformation of the positions at every time t:    r 1 T = ⇒ r 1 + b r 2 T = ⇒ r 2 + b (I-2) (the two velocities are then unchanged). It transforms a possible motion into another possible motion (with the same potential U ). Since the transformation does not change the accelerations, we have:  f 1 (r 1 + b, r 2 + b ; t) = f 1 (r 1 , r 2 ; t) f 2 (r 1 + b, r 2 + b ; t) = f 2 (r 1 , r 2 ; t) (I-3) This means that, when the two variables increase by a quantity b, the gradients of the potential function U with respect to these two variables remains constant. It follows that, in this change of the two variables, U only varies by a constant. This constant could be time-dependent, but has no consequence on the particle’s motion since it does not depend on their positions. If we furthermore require U to go to zero at infinity, this constant is necessarily equal to zero. This means that the potential U is invariant in the translation of the two variables, and is therefore only a function of r 1 − r 2 : U (r 1 + b, r 2 + b ; t) ≡ U (r 1 , r 2 ; t) (I-4) This restricts the possible potentials, thus imposing a constraint on the form of the physical laws: U ≡ U (r 1 − r 2 ; t) (I-5) In addition, it is easy to show that f 1 = −f 2 , which leads to: d dt (m 1 ˙ r 1 + m 2 ˙ r 2 ) = 0 (I-6) 6 B. SYMMETRIES IN CLASSICAL MECHANICS Accordingly, when translations are symmetry transformations, the total momen- tum is a constant of the motion. • Rotational Invariance If, in addition, rotations are symmetry transformations, other properties appear. Following the same line of reasoning as above, we see that the field of forces f 1 (or f 2 ), considered as a function of r = r 1 − r 2 , is invariant under any rotation of the vector r. It is thus a field such that f 1 and f 2 are parallel to r 1 − r 2 and have a modulus that only depends on |r|.

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  Symmetry And Complexity: The Spirit And Beauty Of Nonlinear Science

  The Spirit and Beauty of Nonlinear Science

  • Klaus Mainzer(Author)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  Chapter 3 Symmetry and Complexity in Physical Sciences The symmetries of the laws and theories of physics and of the nat-ural sciences in general became clear only after the application of group theory methods in the 19th and 20th centuries. In particular, F. Klein’s “Erlanger Program”, according to which the objective va-lidity of geometric laws is defined by their invariance under certain groups of transformations, turned out to be the key concept for the mathematical explanation of the objectivity and invariance of the laws and theories of physics. Lie’s continuous groups became a valu-able resource for classical physics. In classical mechanics, invariance under groups of transformations leads to important consequences. If the Lagrange equations of a physical problem are invariant with respect to an n -parameter symmetry group in the Lie sense, the n conservation quantities can be indicated explicitly. Conservation theorems of physical quantities, which have a long tradition in the philosophy of nature, as in the case of the conversation of mass, can now be traced to space-time symmetries. These general rela-tionships between symmetry groups and conservation quantities are later found in an analogous fashion in the theory of relativity and quantum mechanics. Important characteristics of the modern concept of symmetry were already clear in Maxwell’s electrodynamics: a unified theory explained different physical phenomena which had been considered completely unrelated as recently as in the late 18th century, e.g. elec-trostatic charge, the effect of a magnet on a compass needle and 107 108 Symmetry and Complexity the light from a candle. But in terms of group theory, electrody-namics does not have the space-time symmetry of classical mechan-ics (“Galilean invariance”), but what is termed Lorentz invariance. Since Einstein’s theory of special relativity, Lorentz invariance also determines the space-time symmetry of relativistically corrected me-chanics.

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