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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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)

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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

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