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5 Atomic Systems
We also need to think about what happens with the electron spins at this stage
of development. This is a little more complicated to deal with unless we utilize
another, more generalized, version of the Pauli Principle that is based on symmetry
arguments. Our discussion begins by noting that all fundamental particles, such
as electrons, protons, and neutrons, are of two basic types, known as bosons and
fermions. These two types of fundamental particle are distinguished by the values
of the intrinsic angular momentum I with which they are associated. The angular
momentum I is referred to as the ‘nuclear spin’ (with magnitude designated by
I ) for want of a better terminology. Fundamental bosons and fermions can also
combine to form what are referred to as composite bosons and fermions, with net
values of I determined by vector addition of the fundamental nuclear spin vectors.
A system of many indistinguishable bosons or fermions can be described by a
multiproduct wavefunction that must be either symmetric or antisymmetric to the
interchange of any pair of particles making up the system. Those particles that are
symmetric to this interchange are called bosons and those that are antisymmetric to
the interchange are called fermions. It turns out that fermions are fundamental or
composite particles that have half-odd-integer values for the magnitudes of their
intrinsic angular momenta (normally, nuclear or electronic spins), while bosons
are fundamental or composite particles that have zero or integer values for the
magnitudes of their intrinsic angular momenta. The best known examples of such
particles are electrons, protons, and neutrons, all of which are fermions with ‘spin’
1
2 , and neutrinos and photons, both of which are bosons, with ‘spins’ 0 and 1,
respectively.
Let us now return to our specific example of the Si atom, and consider how to
represent the wavefunction for the pair-state in which both m l 1 and m l 2 have value
+1, so that the total value for the pair-state will be M L = m l 1 + m l 2 = +2. We have
already said that we may represent a pair-state wavefunction as the product function
for the states making up the pair, so that the wavefunction M L =2 (1, 2) for the pairstate will have the form M L =2 (1, 2) = ψ m l 1 =1 (1)ψ m l 2 =1 (2), which is symmetric
to the interchange of the electron labels 1 and 2. It can also be established, using the
quantum mechanical total electronic orbital angular momentum lowering operator,
that all five M L -states for L = 2 are symmetric to this 1 ↔ 2 interchange (more
generally, all 2L + 1 M L -states for total angular momentum L will have the same
particle interchange symmetry). We can also establish that the three M L pair-states
for L = 1 are antisymmetric to the 1,2 interchange, while the single M L = 0 pairstate corresponding to L = 0 is symmetric to the 1,2 interchange.
In Schrödinger quantum mechanics, the spin dependence is adjoined to the
orbital angular momentum dependence, so that the total wavefunction for the pair
of 3p electrons in the valence shell of the Si atom may be represented by
total (1, 2) = ψ orbital (1, 2)ψ spin (1, 2),
with both ψ orbital (1, 2) and ψ spin (1, 2) having the form of outer product functions.
We know from our construction that the orbital product functions all have specified
interchange symmetries (symmetric if L = 0, 2, and antisymmetric if L = 1), but
5 Atomic Systems
We also need to think about what happens with the electron spins at this stage
of development. This is a little more complicated to deal with unless we utilize
another, more generalized, version of the Pauli Principle that is based on symmetry
arguments. Our discussion begins by noting that all fundamental particles, such
as electrons, protons, and neutrons, are of two basic types, known as bosons and
fermions. These two types of fundamental particle are distinguished by the values
of the intrinsic angular momentum I with which they are associated. The angular
momentum I is referred to as the ‘nuclear spin’ (with magnitude designated by
I ) for want of a better terminology. Fundamental bosons and fermions can also
combine to form what are referred to as composite bosons and fermions, with net
values of I determined by vector addition of the fundamental nuclear spin vectors.
A system of many indistinguishable bosons or fermions can be described by a
multiproduct wavefunction that must be either symmetric or antisymmetric to the
interchange of any pair of particles making up the system. Those particles that are
symmetric to this interchange are called bosons and those that are antisymmetric to
the interchange are called fermions. It turns out that fermions are fundamental or
composite particles that have half-odd-integer values for the magnitudes of their
intrinsic angular momenta (normally, nuclear or electronic spins), while bosons
are fundamental or composite particles that have zero or integer values for the
magnitudes of their intrinsic angular momenta. The best known examples of such
particles are electrons, protons, and neutrons, all of which are fermions with ‘spin’
1
2 , and neutrinos and photons, both of which are bosons, with ‘spins’ 0 and 1,
respectively.
Let us now return to our specific example of the Si atom, and consider how to
represent the wavefunction for the pair-state in which both m l 1 and m l 2 have value
+1, so that the total value for the pair-state will be M L = m l 1 + m l 2 = +2. We have
already said that we may represent a pair-state wavefunction as the product function
for the states making up the pair, so that the wavefunction M L =2 (1, 2) for the pairstate will have the form M L =2 (1, 2) = ψ m l 1 =1 (1)ψ m l 2 =1 (2), which is symmetric
to the interchange of the electron labels 1 and 2. It can also be established, using the
quantum mechanical total electronic orbital angular momentum lowering operator,
that all five M L -states for L = 2 are symmetric to this 1 ↔ 2 interchange (more
generally, all 2L + 1 M L -states for total angular momentum L will have the same
particle interchange symmetry). We can also establish that the three M L pair-states
for L = 1 are antisymmetric to the 1,2 interchange, while the single M L = 0 pairstate corresponding to L = 0 is symmetric to the 1,2 interchange.
In Schrödinger quantum mechanics, the spin dependence is adjoined to the
orbital angular momentum dependence, so that the total wavefunction for the pair
of 3p electrons in the valence shell of the Si atom may be represented by
total (1, 2) = ψ orbital (1, 2)ψ spin (1, 2),
with both ψ orbital (1, 2) and ψ spin (1, 2) having the form of outer product functions.
We know from our construction that the orbital product functions all have specified
interchange symmetries (symmetric if L = 0, 2, and antisymmetric if L = 1), but