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Symmetries in Mechanics

  1. A physical ket is antisymmetric
    Pauli's exclusion principle.
    Proof.
    : [Coh, p.1389, (C-7)].
    :
    By [Coh, p.1386, the symmetrization postulate], a physical ket can be represented by G|1:φ;2:ψ>, where G =S or A [Coh, p.1380, (B-19 a,b)].
    By Pauli's exclusion principle, G|1:φ;2:φ>=0,
    G|1:ψ;2:ψ>=0, and G|1:φ+ψ;2:φ+ψ>=0.
    Therefore, G|1:φ;2:ψ>=−G|1:ψ;2:φ>.
    Namely, G=A.
    Remark. For the equivalence in terms of wave functions, see [Pat, p.128, l.−12].

  2. Since Maxwells equations are symmetric with respect to (E, e ) and (H, - m ), the theorems for TM waves can be derived from the corresponding theorems for TE waves by substituting (E, e ) with (H, - m ) [Born, p.52, l.19].

  3. Symmetry (i.e. constant motion) vs. degeneracy.
    1. [J, H]=0 (algebraic symmetry) means that H is invariant under rotation (geometric symmetry: [Coh, p.704, (94) &(95)]). In other words, H is symmetric with respect to J. Similarly, [A, H]=0 (i.e. A is a constant motion) H is symmetric with respect to A.
    2. Remove degeneracy Add a perturbation to the Hamiltonian to destroy its symmetry.
      Example. [Mer2, p.268, l.10-l.19; Lan3, p.127, Problem 3].

  4. Descriptions of momentum in terms of translational symmetry.
    1. (Classical mechanics) The Lagrangian of a closed system is invariant under any infinitesimal translation [Lan1, p.15, l.6-l.9]. We define momentum as a vector in [Lan1, p.15, (7.2)].
    2. (Quantum mechanics) The Hamiltonion of a closed system is invariant under any infinitesimal translation. In view of [Lan3, p.42, (15.1)], we define momentum as an operator in [Lan3, p.42, (15.2)].
    3. (Solid state) We view momentum as a quantum number characteristic of the fuller translational symmetry of free space [Ashc, p.139, l.-8-l.-5].

  5. How symmetry simplifies the calculation of an integral over an area.
    Examples.
    1. Uniform infinite line charge [Wangs, p.60, l.-6-p.61, l.20].
    2. Uniform infinite plane sheet [Wangs, p.62, l.10-p.63, l.17].
    3. Spherically symmetric spherical charge distribution [Wangs, p.63, l.-19-p.65, l.3].

  6. Group representation theory is is the most suitable language to describe the impact of the symmetry on various concepts in quantum mechanics.
    1. Selection rule [Hei, p.43, l.19-l.21].
    2. Classify the wave functions using angular momentum quantum numbers [Hei, p.43, l.22-24].
      Remark 1. [Hei, p.58, l.9] shows that it is impossible to reduce the state space further with the full rotation group. In this sense, group representation theory helps strengthen the structure of quantum mechanics.  It is possible to use other algebraic languages to deal with symmetry [Hei, p.viii, l.16-l.17], but the use of group representation theory gives the clearest image in the most organized manner by using the most economical  words. This is because the definitions in group representation theory are designed mainly for such a purpose.
      Remark 2. (Addition of angular momenta) [Hei, pp.67-71, 9 Reduction of the Product Representations] and [Coh, chap. X, Addition of Angular Momenta] basically discuss the same material. By comparison, Cohen-Tannoudji's exposition has the following drawbacks:
      1. Heine points out that the key idea of adding two angular momenta is the reduction of product representation. It would be difficult to describe this key idea in one sentence by using an algebraic language other than group representation theory.
      2. Heine provides a blue print of how to reduce the product representation [Hei, p.67, l.-7-p.68, l.7], while Cohen-Tannoudji does not.
      3. Unless we trace back to rotations (compare [Hei, p.49, (7.3); p.68, (9.1a)] with [Coh, p.1014, (C-24)]), we would not recognize the role the axial rotation group plays in the early stage of the reduction of the product representation.
      4. Unless we trace back to rotations [Hei, p.69, l.-14; p.66, Problem 8.10], we would not understand the precise meaning of the formal expression J-=J1-+J2- given in [Coh, p.1020, l.-1].
    3. Degeneracy [Hei, p.44, Theorem 2].
    4. A perturbation splits an energy level [Hei, p.43, Theorem 3].


  7. Descriptions of axial symmetry.
    1. In terms of geometry: It is symmetric with respect to an axis.
    2. In terms of mathematical analysis: Associated Legendre functions reduce to Legendre polynomials (i.e., [Chou, p.145, l.2 & (3.122) reduces to [Chou, p.146, (3.123)]).
    3. In terms of quantum mechanics: m=0 [Chou, p.145, l.2 & l.-1].

  8. (Symmetry vs. solutions) We start with Coulomb's law to develop the theory of electrostatics and start with Ampre's law [Wangs p.218, l.8] to develop the the theory of magnetostatics. However, if we proceed in separate ways like this, we would never obtain the complete picture of electromagnetism because the above two theories are somehow related to each other. Therefore, it is more proper to treat the macroscopic Maxwell equations as axioms. For a given distribution of charges and currents, it is difficult to find their solutions for the general case [Fan, p.20, l.1]. However, we may find interesting solutions when the sources have spherical or cylindrical symmetry. For example, Coulomb's law [Fan, p.20, (1.19)] can be derived from [Fan, p.19, (1.17)] using spherical symmetry; [Fan, p.22, (1.21); Wangs, p.243, (15-19) & (15-20)] can be derived as the solution of [Fan, p.16, (1.13)] using cylindrical symmetry. Similarly using this correct set of tools (axioms), we may derive both Biot-Savart's law and Ampre's law from the Maxwell equations [Sad, 7.8; Fan, p.124, (4.70)]. Thus, the above formulas are derived from solid bases (axioms) rather than regarded as empirical laws (trial-and-error bases). To have the correct big picture is also the reason why [Fan, p.11, l.-7] says that axioms should be discussed only after the presentation of a theory.

  9. Links {1, 2}.