A New Spin on Photoemission Spectroscopy Page: 37 of 259
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Technically, with systems of more than two particles, a state can be constructed that is
a symmetric eigenket of P12 and and an antisymmetric eigenket of P23, for example, and
such a state is said to have mixed symmetry. Because the Hamiltonian must be invariant
to permutation, a state's permutation symmetry is fixed for all time.
As far as we know, all systems of identical particles either completely symmetric or anti-
symmetric with respect to permutation - no mixed symmetry ever occurs. This is called the
"Symmetrization Postulate" and is an empirical fact of nature, underivable from the fun-
damental postulates of quantum mechanics.t Bosons are particles which form systems that
are completely symmetric and hence obey Bose-Einstein statistics. Fermions are particles
which form systems that are completely antisymmetric and hence obey Fermi-Dirac statis-
tics. A most amazing result is the connection between permutation symmetry and spin:
integer spin particles are bosons and half-integer spin particles are fermions. This seemingly
arbitrary rule is in fact derivable from relativistic quantum mechanics which proves that
integer spin particles cannot be fermions and half-integer spin particles cannot be bosons.11
This connection is typically referred to as the "spin-statistics theorem" and is one of the
most intriguing aspects of spin. Actually understanding the proof, however, is anything but
elementary and attempts for more intuitive explanations have been revisited. 12
Electrons, with spin s = 1/2, are then fermions, and they must form states which are
completely antisymmetric with respect to permutation. This has immediate far-reaching
consequences as antisymmetric states cannot be formed if two particles are in the same
state. For example, the required antisymmetric state of three electrons in 1a), 13), and ry)
Pay) = (a) )1 )- 13)1a)y) + 2y)Ia)L3) - a)y)B) + /3)K7)Ia) - ly)#)1a)) .
Because Ia3y) = 0 if 1a) = 13), it is clear that a system of many electrons, and other
fermions, cannot exist with more than one electron in a given state with all equal quantum
numbers (this result holds, of course, for system of arbitrary N fermions). This is the cause
tMany quantum physics textbooks suggest that this is derivable from quantum mechanics, but these
treatments ignore the possibility of mixed symmetry states. An illuminating discussion is found in Ballen-
tine's textbook. 10
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Jozwiak, Chris. A New Spin on Photoemission Spectroscopy, thesis or dissertation, December 1, 2008; United States. (https://digital.library.unt.edu/ark:/67531/metadc1014237/m1/37/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.