# Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics

Pré-visualização50 páginas

and E3 ÿ 2ìB B (non-degenerate). Energy levels for s orbitals (l 0) are not affected by the application of the magnetic field. Energies for p orbitals (l 1) are split by the 6.5 Spectra 191 magnetic field into three levels. For d orbitals (l 2), the energies are split into five levels. This splitting of the energy levels by the magnetic field leads to the splitting of the lines in the atomic spectrum. The wave number ~í of the spectral line corresponding to a transition between the state jn1 l1 m1i and the state jn2 l2 m2i is ~í j\u2dcEj hc RZ2 1 n21 ÿ 1 n22 \ufffd \ufffd ìB B hc (m2 ÿ m1), n2 . n1 (6:91) Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which \u2dcl \ufffd1 and for which \u2dcm 0, \ufffd1. Thus, an observed spectral line ~í0 in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers ~í0 (ìB B=hc), ~í0, and ~í0 ÿ (ìB B=hc). Problems 6.1 Obtain equations (6.28) from equations (6.26). 6.2 Evaluate the commutator [A^º, B^º] where the operators A^º and B^º are those in equations (6.26). 6.3 Show explicitly by means of integration by parts that the operator H^ l in equation (6.18) is hermitian for a weighting function equal to r2. 6.4 Demonstrate by means of integration by parts that the operator H^9l in equation (6.36) is hermitian for a weighting function w(r) r. 6.5 Show that (A^º 1)Sº1, l aº1, lSº l and that (B^º 1)Sº l bº1, lSº1, l. 6.6 Derive equation (6.45) from equation (6.34). 6.7 Derive the relationship anl 1 0 SnlSnÿ1, lr2 drÿ bn1, l 1 0 SnlSn1, lr2 dr 1 6.8 Evaluate hrÿ1inl for the hydrogen-like atom using the properties of associated Laguerre polynomials. First substitute equations (6.22) and (6.55) into (6.69) for k ÿ1. Then apply equations (F.22) to obtain (6.71). 6.9 From equation (F.19) with í 2, show that 1 0 r2 l3eÿr[L2 l1n l (r)] 2 dr 2[3n 2 ÿ l(l 1)][(n l)!]3 (nÿ l ÿ 1)! Then show that hrinl is given by equation (6.72). 6.10 Show that hri2s 6a0=Z using the appropriate radial distribution function in equations (6.66). 6.11 Set º e9 in the Hellmann\u2013Feynman theorem (3.71) to obtain hrÿ1inl for the hydrogen-like atom. Note that a0 depends on e9. 192 The hydrogen atom 6.12 Show explicitly for a hydrogen atom in the 1s state that the total energy E1 is equal to one-half the expectation value of the potential energy of interaction between the electron and the nucleus. This result is an example of the quantum- mechanical virial theorem. 6.13 Calculate the frequency, wavelength, and wave number for the series limit of the Balmer series of the hydrogen-atom spectral lines. 6.14 The atomic spectrum of singly ionized helium He with n1 4, n2 5, 6, . . . is known as the Pickering series. Calculate the energy differences, wave numbers, and wavelengths for the first three lines in this spectrum and for the series limit. 6.15 Calculate the frequency, wavelength, and wave number of the radiation emitted from an electronic transition from the third to the first electronic level of Li2. Calculate the ionization potential of Li2 in electron volts. 6.16 Derive an expression in terms of R1 for the difference in wavelength, \u2dcº ºH ÿ ºD, between the first line of the Balmer series (n1 2) for a hydrogen atom and the corresponding line for a deuterium atom? Assume that the masses of the proton and the neutron are the same. Problems 193 7 Spin 7.1 Electron spin In our development of quantum mechanics to this point, the behavior of a particle, usually an electron, is governed by a wave function that is dependent only on the cartesian coordinates x, y, z or, equivalently, on the spherical coordinates r, \u141, j. There are, however, experimental observations that cannot be explained by a wave function which depends on cartesian coordinates alone. In a quantum-mechanical treatment of an alkali metal atom, the lone valence electron may be considered as moving in the combined field of the nucleus and the core electrons. In contrast to the hydrogen-like atom, the energy levels of this valence electron are found to depend on both the principal and the azimuthal quantum numbers. The experimental spectral line pattern corre- sponding to transitions between these energy levels, although more complex than the pattern for the hydrogen-like atom, is readily explained. However, in a highly resolved spectrum, an additional complexity is observed; most of the spectral lines are actually composed of two lines with nearly identical wave numbers. In an alkaline-earth metal atom, which has two valence electrons, many of the lines in a highly resolved spectrum are split into three closely spaced lines. The spectral lines for the hydrogen atom, as discussed in Section 6.5, are again observed to be composed of several very closely spaced lines, with equation (6.83) giving the average wave number of each grouping. The splitting of the spectral lines in the alkali and alkaline-earth metal atoms and in hydrogen cannot be explained in terms of the quantum-mechanical postulates that are presented in Section 3.7, i.e., they cannot be explained in terms of a wave function that is dependent only on cartesian coordinates. G. E. Uhlenbeck and S. Goudsmit (1925) explained the splitting of atomic spectral lines by postulating that the electron possesses an intrinsic angular momentum, which is called spin. The component of the spin angular momen- 194 tum in any direction has only the value "=2 or ÿ"=2. This spin angular momentum is in addition to the orbital angular momentum of the electronic motion about the nucleus. They further assumed that the spin imparts to the electron a magnetic moment of magnitude e"=2me, where ÿe and me are the electronic charge and mass. The interaction of an electron\u2019s magnetic moment with its orbital motion accounts for the splitting of the spectral lines in the alkali and alkaline-earth metal atoms. A combination of spin and relativistic effects is needed to explain the fine structure of the hydrogen-atom spectrum. The concept of spin as introduced by Uhlenbeck and Goudsmit may also be applied to the Stern\u2013Gerlach experiment, which is described in detail in Section 1.7. The explanation for the splitting of the beam of silver atoms into two separate beams by the external inhomogeneous magnetic field requires the introduction of an additional parameter to describe the behavior of the odd electron. Thus, the magnetic moment of the silver atom is attributed to the odd electron possessing an intrinsic angular momentum which can have one of only two distinct values. Following the hypothesis of electron spin by Uhlenbeck and Goudsmit, P. A. M. Dirac (1928) developed a quantum mechanics based on the theory of relativity rather than on Newtonian mechanics and applied it to the electron. He found that the spin angular momentum and the spin magnetic moment of the electron are obtained automatically from the solution of his relativistic wave equation without any further postulates. Thus, spin angular momentum is an intrinsic property of an electron (and of other elementary particles as well) just as are the charge and rest mass. In classical mechanics, a sphere moving under the influence of a central force has two types of angular momentum, orbital and spin. Orbital angular momentum is associated with the motion of the center of mass of the sphere about the origin of the central force. Spin angular momentum refers to the motion of the sphere about an axis through its center of mass. It is tempting to apply