Principles of Quantum Mechanics   as Applied to Chemistry and Chemical Physics

Principles of Quantum Mechanics as Applied to Chemistry and Chemical Physics


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and E3 ÿ 2ìB B
(non-degenerate). Energy levels for s orbitals (l ˆ 0) are not affected by the
application of the magnetic field. Energies for p orbitals (l ˆ 1) are split by the
6.5 Spectra 191
magnetic field into three levels. For d orbitals (l ˆ 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 ˆ j\u2dcEj
hc
ˆ 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 ˆ \ufffd1 and for which \u2dcm ˆ 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) ˆ r.
6.5 Show that (A^º ‡ 1)Sº‡1, l ˆ aº‡1, lSº l and that (B^º ‡ 1)Sº l ˆ bº‡1, lSº‡1, l.
6.6 Derive equation (6.45) from equation (6.34).
6.7 Derive the relationship
anl
…1
0
SnlSnÿ1, lr2 drÿ bn‡1, l
…1
0
SnlSn‡1, lr2 dr ˆ 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 ˆ ÿ1. Then apply equations (F.22) to obtain (6.71).
6.9 From equation (F.19) with í ˆ 2, show that…1
0
r2 l‡3eÿr[L2 l‡1n‡ l (r)]
2 dr ˆ 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 ˆ 6a0=Z using the appropriate radial distribution function in
equations (6.66).
6.11 Set º ˆ 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 ˆ 4, n2 ˆ 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 ˆ ºH ÿ ºD, between the first line of the Balmer series (n1 ˆ 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