Quantum Mechanics Li-Chung Wang February 8, 2014 Abstract The Dirac delta generalized function should not be treated as a function; we provide an easy way to bridge the gap between a function and a generalized function. Because the proofs given in Reif [93, §15 14] fail to get to the heart of the matter, we shall prove the equality between the ensemble average and· the time average using Chung [15, p.133, Theorem 5.4.2(8)] instead. Without the guide of the theory of Lie groups, the discussion about angular momentum given in Cohen-Tannoudji–Diu– Laloe¨ [17, vol. 1, chap. VI, §B, §D.1.a, Complement BVI] looks disorganized. In contrast, with the theory of Lie groups as a guide, Vvedensky [119, §7.4.1 & §7.4.2] clarifies the inner structure of angular momentum. Comparing Vvedensky [119, §7.3.1] with Marion–Thornton [77, p.35, l. 22–p.36, l. 1], the latter discussion is marginal. It is unnecessary to represent δθ by a vector [Marion–Thornton− − [77, p.262, l. 2–l. 1; p.263, Fig. 7-8]]; see Fomin–Gelfand [35, p.87, l. 18–l. 5]. In order to be natural, simple, and− easy− to understand, it is an improper approach to represent− a matrix− by a vector. Keywords. Calculus of variation, variation of a functional, Taylor’s polynomial formula, admissible functions, variational derivative, isoperimetric problems, holonomic problems, non-holonomic problems, inhomogeneous wave equations, extremal, natural frame field, Legendre transformation, duality, Hadamard global inverse function theorem, convex function, proper map, principle of least action, conservation of momentum and that of angular momentum, Lagrangian method, symmetry, canonical Euler equations, characteristic system associated with the Hamilton–Jacobi equation, Dirac delta function, solid angle, au- tonomous system, initial condition, Lagrangian, total derivative, infinitesimal displacement, infinitesimal transformations, finite transformations, Lie groups, infinitesimal rotation, complete integral, cyclic coor- dinates, Hamilton’s characteristic function, canonical transformations, generating functions, Hamiltonian, equipartition theorem, Lissajous curves, conic sections, Galilean transformations, Lorentz transformations, Michelson–Morley experiment, Maxwell’s equations, muon decay, special relativity, proper time, relativis- tic kinetic energy, relativistic Lagrangian, refraction index, phase velocity, group velocity, signal veloc- ity, inertial frame, timelike interval, spacelike interval, scattering angle, final velocity, total cross section, center-of-mass coordinate system, out of phase, coupled harmonic oscillators, normal modes, generalized coordinates, complete orthonormal set [orthonormal basis], node, closure relation, state space, Bessel func- tions, Leibniz rule for higher derivatives, indicial equation, generators of a one-sheet hyperboloid, ruled surface, developable surface, directrix, director sphere, orthogonal, argumented matrix, conjugate diame- ters, polar lines, polar planes, diametral planes of a paraboloid, umbilics of an ellipsoid, principal planes, conicoids of revolution, double tangent plane, osculating plane, osculating circle, osculating sphere, recip- rocal cone, consecutive points, contact of second or third order, envelope, edge of regression, inflectional tangent, ensemble, measurable, Laplace’s equation, confocal conicoids, equipotential surfaces, strain ten- sor, bounded variation, random variable, Markov or stopping time, expectation, variance, moment of inertia, 1 center of mass, uniformly integrable, Slutsky’s theorem, Laplace transform, characteristic functions, distribu- tion, density, indicator, inversion formula, distribution determining class, pushforward measure, convergence in probability, almost sure convergence, strong law of large numbers, central limit theorem, weak conver- gence, pointwise convergence, spin, Stern–Gerlach experiment, ensemble, ergodicity, Liouville’s theorem, chain rule, reversible processes, Stirling’s formula, dielectrics, postulate of equal a priori probabilities, mi- crostates, equation of state, Boltzmann’s constant, specific heats, entropy, dipole layer, Green’s theorem, point dipole, Jacobian, Wronskian, normal derivative, Lommel’s formula, Ritz method, Sturm–Liouville problems, direct methods, method of finite differences, Euler expression, Gaussian curvature, positive def- inite, extremal, equation of the vibrating membrane, vector triple product, method of Lagrange multipliers, uncernty principle, De Broglie wavelength, bound state, specific heat, zero-point energy, ground state, mag- netic moment, normal operators, observables, conjugacy, wave function, probability amplitude, probabil- ity distribution, testing functions, surface harmonics, principal normal, Mobius¨ strip, vector bundle, trivial bundle, n-plane bundle, Jordan-measurable, simply-connected regions, Stokes theorem, multiply-connected regions, potential flow, Kelvin’s inversion theorem, Whipple’s formula, contour integrals, dummy variable, branch points, Riemann’s differential equation, Euler transforms, Fuchsian type, Borel measures, hyper- geometric, confluent, Chu–Vandermonde identity, Kummer’s first formula, regular singularity, Riemann’s P-equation, Lebesgue dominated convergence theorem, methodical solutions, Weber–Schafheitlin integral, Hankel inversion theorem, Riemann–Lebesgue theorem, Riemann surfaces, Lommel’s theorem, Sturm’s the- orem, meromorphic functions, essential singularities, Cauchy–Kowalewski theorem, microstates, flux den- sity, rate of flow, Plancherel’s theorem, transient, steady state, inductance, dimensions, polarization, solid angle, inertia tensor, harmonic function, polarization, magnetization, potential, vector potential, divergence, curl, solenoid, toroid, polarizability, Clausius–Mossotti relation, Larmor frequency, diamagnetic, Debye equation, Curie law for paramagnetic material, ferroelectric, ferromagnetic, Curie temperature, Maxwell velocity distribution function, Lorenz–Lorentz law, cophasal surface, continuity equation 1 Prerequisites Rome wasn’t built in a day. To lay a solid foundation, there are no shortcuts except by studying classics. Because of its fragmentariness, a cyclopidea is useful only in the following cases: After reading a classical textbook, it may help review and organize the material; after a long while, it may help recover a clear image. It is unnecessary to rewrite an entire classsical textbook. We need only pointout its shortcomings, and propose the methods of improvement. For example, one may update its reference or quote more direct or detailed proofs. Sometimes one rewrites an entire classical book, only to lose its originality or damage its structural delicacy. 1. Geometry (2-dimensional analytic geometry: Fine–Thompson [33]; 3-dimensional analytic geometry: Bell [2]; differential geometry: Carmo [12]; Courant–John [22, vol. 2, chap. 5]; Deo [25]; Hicks [55]; Kreyszig [66]; Lee [73] [For its solutions to exercises and problems, see https://wj32.org/wp/ wp-content/uploads/2012/12/Introduction-to-Smooth-Manifolds.pdf]; Munkres [84]; Munkres [85]; O’neill [86]; Spivak [108]; Weatherburn [124]; Willmore [126]) Remark 1. (Point dividing a chord) Fine–Thompson [33, p.46, Example 2] Proof. We may assume the center of the circle is (0,0). Rotate the coordinate axes so that P =(x ,y ), Q =(x ,y1), and R =(x , y1). ′ ′ ′ ′ − 2 Remark 2. Parallel lines project into parallel lines [Fine–Thompson [33, p.90, l. 14]]. − Proof. Let l1,l2 be parallel lines that cut plane γ at points P1,P2. Draw the normal Ni of γ at Pi. Plane Nili cuts γ in line li′. By the definition of cross product, plane N1l1 and plane N2l2 are parallel. Consequently, l1′ ,l2′ are parallel. Remark 3. By Fine–Thompson [33, p.34, (3)], (ae1,0),(a/e1,b/e2),(0,be2) are on a straight line [Fine– Thompson [33, p.105, l. 2–l. 1]]. − − Remark 4. (The equaton of hyperbola referred to its asymptotes) If the lines ax + by + c = 0 and a′x + b′y + c′ = 0 are neither identical nor parallel, then ,by Fine–Thompson [33, p.131, l. 15–l. 12], (ax + by + c)(a′x + b′y + c′)= d (where d is a constant) is a hyperbola [Fine–Thompson− − [33, p.114, l.13–l.15]]. Remark 5. Fine–Thompson [33, p.121, (3)] Proof. For (3)(i), consider the perpendiculars to the y-axis of x,x1,y1. For (3)(ii), consider the perpendiculars to the x-axis of y,x1,y1. x 2y 14 Remark 6. − − + 8√5 = 0 [Fine–Thompson [33, p.133, l. 13]]. √5 − Proof. Let (x,y) be an arbitrary point on the directrix. (x,y) and O are on the same side of Cx1. The distance of O from Cx1 is negative, so is the distance of (x,y) from Cx1 [Fine–Thompson x 2y 14 [33, p.32, (3)]]. Consequently, − − = CD = 8√5. √5 −| | − Remark 7. Let U = 0 be a conic and l = 0 a line. If U = 0 and l = 0 intersect at two points P and Q. Then U + λl2 = 0 (where λ is a constant) touches U = 0 at P and Q [Fine–Thompson [33, p.142, l. 10–l. 1]]. − − Proof. Suppose U = 0 and (l = 0) (l +ε = 0) intersect at P,P ,Q,Q , where P P and Q Q ∪ ′ ′ ′ → ′ → as ε 0. Then PP approaches a tangent line to U = 0 at P as ε 0. The same discussion applies → ′ → to U + λl(l + ε)= 0 [Fine–Thompson [33, p.142, l.1–l.4]]. Remark 8. The equation of tangent to y2 = 4ax in terms of its slope is y = mx + a/m [Fine–Thompson [33, p.181, l. 10–l. 9]]. − − 2 Proof. Let P(x′,y′) be a point on y = 4ax. Then the tangent at P is yy ′ = 2a(x + x ′) [Fine– Thompson [33, p.61, (6)]]. Namely, y = 2a x + 2ax ′ . y ′ y ′ 2 2a 2ax ′ y ′ /2 m = = = y ′/2 = a/m. y ′ ⇒ y ′ y ′ Remark 9. Every plane section of a conicoid is a conic [Fine–Thompson [33, p.236, l. 9]]. − Proof. Bell [2, p.74, Ex. 2] gives one proof. Another proof is as follows: Take a point (x0,y0,z0) on the plane as the new origin and then take a basis A,B of the plane. { } By substituting (x,y,z)=(x0,y0,z0)+ uA + vB into the equation F(x,y,z)= 0 of the conicoid, we obtain an equation of the second degree in u,v. Remark 10. The proof of the statement given in Fine–Thompson [33, p.250, l.3–l.6] is similar to the proof of the statement given in Fine–Thompson [33, p.150, l.2–l.4]. 3 Remark 11.