By Professor Dr. Franz Schwabl (auth.)
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Additional resources for Advanced Quantum Mechanics
Spin-1/2 Fermions dicted 9 and experimentally observed for electrons on the surface of helium. 10 Its melting curve has also been determined. 11). 095e2/2ao. 6. 3 Modification of Electron Energy Levels due to the Coulomb Interaction The Coulomb interaction modifies the electron energy levels fo(k) = (~~2. We can calculate this effect approximately by considering the equation of motion of the operator akO"(t): akO"(t) = ~ [2: fo(k')at,O",ak'O"', akO"] 2: k',u' = - akO"(t) = ~ k' ,0"' fO (k') [at, 0"" akO" L ak' 0"' "-v-" -~fo(k)akO"(t) .
2b). I:kf(k) F. Schwabl, Advanced Quantum Mechanics © Springer-Verlag Berlin Heidelberg 2004 The volume of k-space per point 34 2. 3) whence it follows that _ 37f2 N _ 3 2 k 3p-y-7fn. 4) Here, k p is the Fermi wave vector, pp = nk p the Fermi momentum 2 , and n = the mean particle density. The Fermi energy is defined by Ep = (fikp) j(2m). For the x-dependence of the ground-state expectation value of the particle density, one obtains y (n(x)) = L (cPo 1'l/Jt (x)'l/Ja (x) IcPo) a 1 = V Lnpa = n. p,a As was to be expected, the density is homogeneous.
Cit. 3 Hartree-Fock Equations for Atoms 51 implies that both the operators am and ak must be present in the set a1 ... 3), otherwise their application to the right on the vacuum state 10) would give zero. 5) - Oms,m sJ 'P: (x)'P; (X')'Pi(X')'Pj (x) }. In the spirit of the Ritz variational principle, the single-particle wave functions 'Pi(X) are now determined so as to minimize the expectation value of H. As subsidiary conditions, one must take account of the normalizations l'Pil 2d3x = 1; this leads to the additional terms -Ei(J d3xl'Pi(X) 12 -1) with Lagrange parameters Ei.
Advanced Quantum Mechanics by Professor Dr. Franz Schwabl (auth.)