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Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.1 Chapter 8 Biographies, Figures, and Tables FIGURES FIGURE 8.1 [= old 9.1 XX] The two-dimensional, rigid square box. The particle is confined by perfectly rigid walls to the unshaded, square region, within which it moves freely. FIGURE 8.2 [= old 9.2 XX] The energy levels of a particle in a two-dimensional, square, rigid box. The lowest allowed energy is 2E0; the line at E = 0 is merely to show the zero of the energy scale. The degeneracies, listed on the right, refer to the number of independent wave functions with the same energy. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.2 FIGURE 8.3 [= old 9.3 XX] Contour map of the probability density ||2 for the ground state of the square box, The percentages shown give the value of ||2 as a percentage of its maximum value. FIGURE 8.4 [= old 9.4 XX] Contour maps of ||2 for three excited states of the square box. The two numbers under each picture are nx and ny . The dashed lines are nodal lines, where ||2 vanishes; these occur where passes through zero as it oscillates from positive to negative values. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.3 FIGURE 8.5 [= old 9.5 XX] The first five levels and their degeneracies for a particle in a three-dimensional cubical box. FIGURE 8.6 [= old 9.6 XX] A central force points exactly toward, or away from O. If the particle undergoes a displacement perpendicular to the radius vector OP, the force does no work and the potential energy U is therefore constant. FIGURE 8.7 [= old 9.7 XX] Definition of the polar coordinates r and in two dimensions. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.4 FIGURE 8.8 [= old 9.8 XX] If m is an integer, the function cos m repeats itself each time increases by 2 ; for intermediate values it does not. FIGURE 8.9 [= old 9.9 XX] If we move through an angle on a circle of radius r, the distance traveled is s = r. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.5 FIGURE 8.10 [= old 9.10 XX] General appearance of the energy levels for a two-dimensional central force. Energy is plotted upward and the angular-momentum quantum number is plotted to the right. For each value of m, there may be several possible energies, which we label with a quantum number n = 1, 2, 3, . . . For each pair of values n and m, we denote the corresponding energy by En, m . Notice that the variable plotted horizontally is the absolute value of m, since En, m depends only on the magnitude of the angular momentum, not its direction. FIGURE 8.11 [= old 9.11 XX] The spherical polar coordinates of a point P are (r, , ), where r is the distance OP, is the angle between OP and the z axis, and is the angle between the xz plane and the vertical plane containing OP. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.6 FIGURE 8.12 [= old 9.12 XX] If we fix r and and let vary, we move around a circle of radius = r sin . FIGURE 8.13 (a) [= old 9.13 XX] If the constant k has the form l(l + 1) with l an integer greater than or equal to |m|, then the equation (8.53) has one acceptable solution, finite for all from 0 to . The picture shows this acceptable solution for the case m = 0, l = 2. (b) Otherwise, every solution of the equation is infinite at = 0 or or both. The picture shows a solution that is finite at = 0 but infinite at = for the case m = 0 and l = 1.75. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.7 FIGURE 8.14 [= old 9.14 XX] Classical representation of the quantized values of angular momentum L for the case l = 2. The z component has (2l + 1) = 5 possible values, Lz = m h with m = 2, 1, 0, –1, –2. The magnitude of L is L = l(l + 1) h = 2 ´ 3 h » 2.4 h in all five cases. FIGURE 8.15 [= old 9.15 XX] The quantum properties of angular momentum can be visualized by imagining the vector L randomly distributed on the cone shown. This represents the quantum situation, where L and Lz have definite values but Lx and Ly do not. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.8 FIGURE 8.16 [= old 9.16 XX] Energy-level diagram for the hydrogen atom, with energy plotted upward and angular momentum to the right. The letters s, p, d, f, are code letters traditionally used to indicate l = 0, 1, 2, 3, . (Energy spacing not to scale.) FIGURE 8.17 [= old 9.17 XX] The wave function (8.81) for the ground state of hydrogen, as a function of r. FIGURE 8.18 [= old 9.18 XX] The probability of finding the electron a distance r from the nucleus is given by the radial probability density P(r). For the 1s or ground state of hydrogen P(r) is maximum at r = aB. The density P(r) has the dimensions of inverse length and is shown here in units of 1/aB. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.9 FIGURE 8.19 [= old 9.19 XX] The radial distribution P(r) for the 2s state (solid curve). The most probable radius is r 5.2 aB, with a small secondary maximum at r 0.76 aB. For comparison, the dashed curve shows the 1s distribution on the same scale. (Vertical axis in units of 1/aB.) FIGURE 8.20 (a) [= old 9.20 XX] Contour map of ||2 in the xz plane for the 2p (m = 0) state. The density is maximum at the points z = ±2aB , on the z axis and zero in the xy plane. The contours shown are for ||2 equal to 75%, 50%, and 25% of its maximum value. (b) A three-dimensional view of the 75% contour, obtained by rotating the 75% contour of (a) about the z axis. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.10 FIGURE 8.21 [= old 9.21 XX] Perspective views of the 75% contours of ||2 for the 2pz, 2px, and 2py, wave functions. FIGURE 8.22 [= old 9.22 XX] The radial probability density for the 2p states (solid curve). The most probable radius is r = 4 aB. For comparison the dashed curves show the 1s and 2s distributions to the same scale. 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.11 FIGURE 8.23 [= old 9.23 XX] The radial distributions for the n = 1, 2, and 3 states in hydrogen. The numbers shown are the most probable radii in units of aB . FIGURE 8.24 [= old 9.24 XX] The most probable radius for any n = 3 state in hydrogen is between 9aB and 13.1aB. The corresponding range for the n = 2 states is from 4aB to 5.2aB. The most probable radius for the n = 1 state is aB. This numbers define the “spatial shells” within which an electron with quantum number n is most likely to be found. FIGURE 8.25 [= old 9.25 XX] (Problem 8.6) 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc Ch. 8, Bios, Figs, & Tables, final [= old Ch 9] BFT 8.12 TABLES TABLE 8.1 The first few angular functions Ql ,m ( q) . The functions with m negative are given by Ql ,- m = (- 1)m Ql ,m . l=0 l=1 l=2 m=0 1/ 4p 3/ 4p cos q 5/ 16p (3 cos2q - 1) m=1 - 3/ 8p sin q - 15/ 8p sin q cos q m=2 15/ 32p sin 2q TABLE 8.2 The first few radial functions Rlm (r ) for the hydrogen atom. The variable r is an abbreviation for r = r / a B and a stands for aB. n=1 n=2 n=3 l= 0 1 1 ö æ 1 ÷ - r/2 2 æ 2 2 2ö - r / 3 e- r ç1 - r ÷e ç ç1 - r + ç r ÷e ÷ 2a 3 è 2 ø 27a 3 è 27 ø 3 2 a 3 l= 1 1 8 ö æ 1 ÷ - r/3 r e- r / 2 ç1 - r ÷r e ç 27 6a 3 è 6 ø 3 24a l= 2 4 3 r 2e - r / 3 81 30a 03/25/11 eec5fb28-0658-464d-aea2-798cccb65b08.doc

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posted: | 3/25/2011 |

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