A New Spin on Photoemission Spectroscopy Page: 36 of 259
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Figure 1.4. Indistinguishable scattering events of identical particles. It cannot be known,
even in principle, which of (a) or (b) occurred.
13), and yet have the same exact set of eigenvalues for any possible measurement. This
exchange degeneracy presents quantum mechanics with a tricky situation as a complete set
of eigenvalues does not completely determine the state ket as it does for a single particle.
The way nature apparently avoids this issue has fascinating consequences.
From the definition of what it means for a particle to be identical, it can be shown
that any physical observable, including any physical Hamiltonian, must be invariant to the
permutation of two of the identical particles, requiring the eigenstates of any observable to
be simultaneous eigenstates of permutation. The permutation operator's, P12, effect on a
system can be written as
P12Ia)1) = 1)1a) (1.30)
or perhaps more clearly as
P12'IP(x1, x2)) = I'(x1, x2)). (1.31)
It should be clear that P12 = P21, meaning its eigenvalues are 1. Two particle eigenstates
of permutation are said to be symmetric with an eigenvalue of +1 and antisymmetric with
an eigenvalue of -1. For the two particle example above, the possible states are then the
(l a)3) + 13)a)) (1.32)
and the antisymmetric state
(I a)1) - 13)a)) . (1.33)
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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/m1/36/: accessed April 24, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.