Pictures in Mechanics

  1. [Mer2, p.219, l.-4-p.220, l.8] gives a brief summary of [Coh, Complement CIII, 2, pp.287-288].

  2. Drawn figures should be based on the basic principles rather than data plugged in from ready-made formulas.
        Cohen-Tanoudji draws [Coh, p.502, Fig. 4] by plugging in data from formulas [Coh, p.501, (C-27)].  Levine draws [Lev2, p.72, Fig. 4.3] based on [Lev2, pp.69-70, (4.48), (4.49)]. By following Levine's method of construction, we learn more about Hermite polynomials and need not memorize the complicated formula [Coh, p.501, (C-27)].

  3. If one considers a problem from only one perspective, his solution will be superficial, incomplete and incorrect.
    Example (The explanation of Hund's rule). Compare [Lan3, p.253, l.-7-l.-1] with [Lev2, p.328, l.-7-p.329, l.16].

  4. The quantum-mechanical proof in [Ashc, pp.134-135, First proof of Bloch's theorem] emphasizes detailed calculations and concise arguments, but fails to identify its essential point. Any simultaneous eigenfunction of all the translation operators has the form specified by [Mer2, p.71, Exercise 4.14]. Since the Hamiltonian commutes with translation operators [Ashc, p.134, (8.9)], we may choose eigenfunctions of the Hamiltonian that have the above specified form ([Coh, p.140, Theorem III] or [Jaco, vol. 2, p.134, Theorem 7]).

  5. The picture in [Mer2, p.235, Fig. 11.1(a)] says a lot. The proof of [Mer2, p.234, (11.8)] immediately follows. [Mer2, p.234, (11.9)] evaluates the position change Dr = r'-r, while [Go2, p.165, (4-92)] evaluates the final position r'. In the light of [Mer2, p.236, (11.12)], only Dr is a useful quantity because it is simpler and more natural. However, if we consider the relationship between angular momentum and rotations, then r' is more natural than Dr (compare [Coh, p.694, (19)] with [Mer2, p.234, l.13]).

  6. Graphical design using ray tracing: [Jen, p.34, l.10-l.-5].

  7. [Ashc, chap.5] fails to connect the concept of a reciprocal lattice to Fourier analysis, so Ashcroft's version of a reciprocal lattice has only a narrow meaning. In contrast, [Kit2, p.32, (9)] traces the origin of the concept of a reciprocal lattice back to the Fourier expansion of a periodic function, so we can see the big picture and the role that a reciprocal lattice plays in it.
        It is easy to derive [Kit2, p.29, (1)] without using Fourier analysis. However, Fourier analysis is required to explain why the angles that produce significant constructive interferences [Kit2, p.36, l.8] must satisfy [Kit2, p.29, (1)] and why any angle that does not satisfy [Kit2, p.29, (1)] will result in a destructive interference [Kit2, p.35, l.6]. Thus, Fourier analysis helps us gain deep insight into diffraction. Any interpretation of diffraction that does not use Fourier analysis is superficial.

  8. A figure must reflect reality.
        Whether the charge distribution is dense or sparse in [Wangs, p.149, Fig. 10-9] is determined by [Wangs, p.148, (10-27)]. When Choudhury discusses polarizability, the charge distribution in [Chou, p.75, Fig. 2.35] fails to reflect the truth of [Wangs, p.148, (10-27)]. In contrast, [Wangs, p.549, Fig. B-2] is also used to discuss polarizability. but the figure still preserves the truth of [Wangs, p.148, (10-27)].

  9. The existences of the maximum and the minimum in [Cor, p.107, 6.2.1] are assumed by an axiom. The proof given in [Chou, p.83, l.-2-p.86, l.5] is guided by a picture. Thus, the logical foundation of the proof in [Cor, p.107, 6.2.1] is not as solid as that of the proof given in [Chou, p.83, l.-2-p.86, l.5].  Indeed, the latter proof reveals more insight into the theorem.

  10. When the pictures serve to illustrate a subtle point, we must clearly explain the theory behind them rather than simply add additional pictures.
        [Hec, pp.190-192, Fig. 5.61-Fig. 5.68] give many figures, but Hecht fails to give an explanation about the orientations of the images. In contrast, [Bro, p.170, Fig. 4.24; p.171, Fig. 4.26] are good enough to illustrate the subtle point required for the above explanations.

  11. [Morg, p.61, Fig. 5.3] appears in almost every textbook in optics. For example, [Born, p.163, Fig. 4.15] and [Hec, p.244, Fig. 6.3]. However, only [Fur, p.161, l.-1-p.162, l.6] and [Morg, p.61, l.3-l.-9] give a good theoretical explanation.

  12. Sometimes, a figure is better than a thousand words. The prose explanation in [Born, p.12, l.-8-l.-3] is not as clear as the explanation provided by [Hec, p.100, Fig. 4.19]. The prose explanation in [Born, p.15, l.13-l.16] is not a good as the explanation provided by [Natv, p.25, Fig. 2; p.26, Fig. 3].