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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the magnetic field gradient is along
the y-axis, then again only two spots are observed on the detection plate, but
are now located on the horizontal axis.
The Stern\u2013Gerlach experiment shows that the magnetic moment of each
Figure 1.11 A cross-section of the magnet in Figure 1.10.
1.7 Stern\u2013Gerlach experiment 27
silver atom is found only in one of two orientations, either parallel or
antiparallel to the magnetic field gradient, even though the magnetic moments
of the atoms are randomly oriented when they emerge from the oven. Thus, the
possible orientations of the atomic magnetic moment are quantized, i.e., only
certain discrete values are observed. Since the direction of the quantization is
determined by the direction of the magnetic field gradient, the experimental
process itself influences the result of the measurement. This feature occurs in
other experiments as well and is characteristic of quantum behavior.
If the beam of silver atoms is allowed to pass sequentially between the poles
of two or three magnets, additional interesting phenomena are observed. We
describe here three such related experimental arrangements. In the first
arrangement the collimated beam passes through a magnetic field gradient
pointing in the positive x-direction. One of the two exiting beams is blocked
(say the one with antiparallel orientation), while the other (with parallel
orientation) passes through a second magnetic field gradient which is parallel
to the first. The atoms exiting the second magnet are deposited on a detection
plate. In this case only one spot is observed, because the magnetic moments of
the atoms entering the second magnetic field are all oriented parallel to the
gradient and remain parallel until they strike the detection plate.
The second arrangement is the same as the first except that the gradient of
the second magnetic field is along the positive y-axis, i.e., it is perpendicular to
the gradient of the first magnetic field. For this arrangement, two spots of silver
atoms appear on the detection plate, one to the left and one to the right of the
vertical x-axis. The beam leaving the first magnet with all the atomic magnetic
moments oriented in the positive x-direction is now split into two equal beams
with the magnetic moments oriented parallel and antiparallel to the second
magnetic field gradient.
The third arrangement adds yet another vertical inhomogeneous magnetic
field to the setup of the second arrangement. In this new arrangement the
collimated beam of silver atoms coming from the oven first encounters a
magnetic field gradient in the positive x-direction, which splits the beam
vertically into two parts. The lower beam is blocked and the upper beam passes
through a magnetic field gradient in the positive y-direction. This beam is split
horizontally into two parts. The left beam is blocked and the right beam is now
directed through a magnetic field gradient parallel to the first one, i.e., oriented
in the positive x-direction. The resulting pattern on the detection plate might be
expected to be a single spot, corresponding to the magnetic moments of all
atoms being aligned in the positive x-direction. What is observed in this case,
however, are two spots situated on a vertical axis and corresponding to atomic
magnetic moments aligned in equal numbers in both the positive and negative
28 The wave function
x-directions. The passage of the atoms through the second magnet apparently
realigned their magnetic moments parallel and antiparallel to the positive y-
axis and thereby destroyed the previous information regarding their alignment
by the first magnet.
The original Stern\u2013Gerlach experiment has also been carried out with the
same results using sodium, potassium, copper, gold, thallium, and hydrogen
atoms in place of silver atoms. Each of these atoms, including silver, has a
single unpaired electron among the valence electrons surrounding its nucleus
and core electrons. In hydrogen, of course, there is only one electron about the
nucleus. The magnetic moment of such an atom is due to the intrinsic angular
momentum, called spin, of this odd electron. The quantization of the magnetic
moment by the inhomogeneous magnetic field is then the quantization of this
electron spin angular momentum. The spin of the electron and of other
particles is discussed in Chapter 7.
Since the splitting of the atomic beam in the Stern\u2013Gerlach experiment is
due to the spin of an unpaired electron, one might wonder why a beam of
electrons is not used directly rather than having the electrons attached to atoms.
In order for a particle to pass between the poles of a magnet and be deflected
by a distance proportional to the force acting on it, the trajectory of the particle
must be essentially a classical path. As discussed in Section 1.4, such a particle
is described by a wave packet and wave packets disperse with time\u2013the lighter
the particle, the faster the dispersion and the greater the uncertainty in the
position of the particle. The application of Heisenberg\u2019s uncertainty principle
to an electron beam shows that, because of the small mass of the electron, it is
meaningless to assign a magnetic moment to a free electron. As a result, the
pattern on the detection plate from an electron beam would be sufficiently
diffuse from interference effects that no conclusions could be drawn.2 How-
ever, when the electron is bound unpaired in an atom, then the atom, having a
sufficiently larger mass, has a magnetic moment and an essentially classical
path through the Stern\u2013Gerlach apparatus.
1.8 Physical interpretation of the wave function
Young\u2019s double-slit experiment and the Stern\u2013Gerlach experiment, as de-
scribed in the two previous sections, lead to a physical interpretation of the
wave function associated with the motion of a particle. Basic to the concept of
the wave function is the postulate that the wave function contains all the
2 This point is discussed in more detail in N. F. Mott and H. S. W. Massey (1965) The Theory of Atomic
Collisions, 3rd edition, p. 215\u201316, (Oxford University Press, Oxford).
1.8 Physical interpretation of the wave function 29
information that can be known about the particle that it represents. The wave
function is a complete description of the quantum behavior of the particle. For
this reason, the wave function is often also called the state of the system.
In the double-slit experiment, the patterns observed on the detection screen
are slowly built up from many individual particle impacts, whether these
particles are photons or electrons. The position of the impact of any single
particle cannot be predicted; only the cumulative effect of many impacts is
predetermined. Accordingly, a theoretical interpretation of the experiment must
involve probability distributions rather than specific particle trajectories. The
probability that a particle will strike the detection screen between some point x
and a neighboring point x‡ dx is P(x) dx and is proportional to the range dx.
The larger the range dx, the greater the probability for a given particle to strike
the detection screen in that range. The proportionality factor P(x) is called the
probability density and is a function of the position x. For example, the
probability density P(x) for the curve IA in Figure 1.9(a) has a maximum at
the point A and decreases symmetrically on each side of A.
If the motion of a particle in the double-slit experiment is to be represented
by a wave function, then that wave function must determine the probability
density P(x). For mechanical waves in matter and for electromagnetic waves,
the intensity of a wave is proportional to the square of its amplitude. By