Abstract
So far, we have discussed only one-particle problems. We now turn our attention to cases in which more than one particle is present.
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Notes
- 1.
For the last 20 years it has been understood that, although this postulate holds true in our three-dimensional world, there is a whole range of intermediate possibilities – anyons – between bosons and fermions, in two dimensions. In some cases there are surface layers a few atoms thick in which the concept of anyons is realized, as in the fractional quantum Hall effect (Sect. 7.6.2 †).
- 2.
Pauli produced a demonstration of this relation between spin and statistics which involved many complications of quantum field theory. Feynman’s challenge that an elementary proof of the spin-statistics theorem be provided has not yet been answered.
- 3.
To reconcile the successes of the (fermion) quark model with the requirement that the total wave function be antisymmetric, it is necessary to hypothesize that each quark comes into three different species, which are labeled by the colors red, green and blue. Baryon wave functions may thus be antisymmetrized in color subspace.
- 4.
There is an alternative coupling scheme in which the orbital and spin angular momenta are first coupled to yield the angular momentum of each particle: \({\vec{\hat{J}}}_{i} ={ \vec{\hat{L}}}_{i} +{ \vec{\hat{S}}}_{i}\) (i = 1, 2), as in (6.11). Subsequently, the two angular momenta are coupled together: \(\vec{\hat{J}} ={ \vec{\hat{J}}}_{1} +{ \vec{\hat{J}}}_{2}\). The two coupling schemes give rise to two different sets of basis states.
- 5.
The first number is the Coulomb principal quantum number; the orbital angular momentum follows the notation of Table 5.2; the exponent (2) denotes the number of particles with the previous two quantum numbers.
- 6.
See also Sect. 7.8 † .
- 7.
See also Sect. 7.8 † .
- 8.
This linearity also holds in three dimensions (sound waves).
- 9.
The sources [47] have been used for this section.
- 10.
Einstein himself suggested H2 and He4 as possible candidates for B–E condensation. Only in 1938 the He4 superfluid transition at 2.4 K was interpreted as a transition to the B–E condensate. However, this interpretation was marred by the presence of large residual interactions.
- 11.
The main source of this section is [52].
- 12.
The Slater determinant for | M l | + 1 electrons moving in the first Landau level is written in (7.50).
- 13.
Low (high) energy localized states arise around impurity atoms which have an excess (dearth) of positive charge.
- 14.
See [58], p. 417.
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Bes, D.R. (2012). Many-Body Problems. In: Quantum Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20556-9_7
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