This book is the fruit of for many years teaching the introduction to quantum mechanics to second-year students of physics at Oxford University. Wehave tried to convey to students that it is the use of probability amplitudesrather than probabilities that makes quantum mechanics the extraordinarything that it is, and to grasp that the theory’s mathematical structure followsalmost inevitably from the concept of a probability amplitude. We have alsotried to explain how classical mechanics emerges from quantum mechanics.Classical mechanics is about movement and change, while the strong emphasis on stationary states in traditional quantum courses makes the quantumworld seem static and irreconcilably different from the world of every-dayexperience and intuition. By stressing that stationary states are merely thetool we use to solve the time-dependent Schr¨odinger equation, and presentingplenty of examples of how interference between stationary states gives riseto familiar dynamics, we have tried to pull the quantum and classical worldsinto alignment, and to help students to extend their physical intuition intothe quantum domain.Traditional courses use only the position representation. If you stepback from the position representation, it becomes easier to explain that thefamiliar operators have a dual role: on the one hand they are repositories ofinformation about the physical characteristics of the associated observable,and on the other hand they are the generators of the fundamental symmetriesof space and time. These symmetries are crucial for, as we show already inChapter 4, they dictate the canonical commutation relations, from whichmuch follows.
55) Hence, the eigenfunctions of L2 and Lz for given l all have the same radial dependence, R(r). The function of , that multiplies R in lm is conventionally denoted Ym l and called a spherical harmonic. The normalisation of Ym l is chosen such that Z d2 | Ym l | 2 = 1 with d2 sin dd (7.56) the element of solid angle. We have shown that Yl l sinl eil and Yl 1 l sinl 1 cos ei(l 1). (7.57) The normalising constants can be determined by rst evaluating the integral
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Z d2 sin2l = 4 22l (l!)2 (2l + 1)! (7.58) involved in the normalisation of Yl equation (7.47b) each time L l, and then dividing by the factor is applied. The spherical harmonics Ym l 2 are listed in Table 7.1. Figures 7.4 and 7.5 show contour plots of several spherical harmonics. Since spherical harmonics are functions on the unit sphere, the gures show a series of balls with contours drawn on them. Since spherical harmonics are complex functions we had to decide whether to show the real part, the imaginary part, the modulus or the phase of the function. We decided it was most instructive to plot contours on which the real part is constant; when the real part is positive, the contour is full, and when it is negative, the contour is dotted. for l of 150 Chapter 7: Angular Momentum
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(Ym 15) on the unit sphere for m = 15 (left), m = 7 (centre) and Figure 7.4 Contours of (Ym 15) = 0 are the heavy curves, while contours m = 2 (right). The contours on which 15) < 0 are dotted. Contours of the imaginary part of Ym l would look the on which same except shifted in azimuth by half the distance between the heavy curves of constant azimuth. (Ym (Ym 1 ) for m = 1 (left) and 0 (right) with line styles Figure 7.5 Top row: contours of having the same meaning as in Figure 7.4. Contours of the imaginary part of Y1 l would look the same as the left panel but with the circles centred on the y axis. Bottom row: 2 ) for m = 2 (left), m = 1 (centre) and m = 0 (right). contours of (Ym For large l, Yl
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1, i.e., around l is signicantly non-zero only where sin the equator, = /2 the leftmost panel of Figure 7.4 illustrates this case. The rst l applications of L each introduce a term that contains one less power of sin and an extra power of cos . Consequently, as m diminishes from l to zero, the region of the sphere in which Ym is signicantly non-zero l gradually spreads from the equator toward the poles compare the leftmost and rightmost panels of Figure 7.4. These facts make good sense physically: Yl l is the wavefunction of a particle that has essentially all its orbital angular momentum parallel to the z axis, so the particle should not stray far from the xy plane. Hence Yl l, the amplitude to nd the particle at , should be small for signicantly dierent from /2. As m diminishes the orbital plane is becoming more inclined to the xy plane, so we are likely to nd the particle further and further from the plane. This is why Ym increases away from the l equator as m decreases.
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