By Professor Dr. Franz Schwabl (auth.)
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Extra resources for Advanced Quantum Mechanics
This leads to the relation a! al = L (iIA) a! 3) where 'Pi(X) is the single-particle wave function in the co ordinate representation. The creation and annihilation operators corresponding to the eigenstates of position are called field operators. 2 Field Operators The field operators are defined by L 'Pi (x)ai 1j; t (x) = L 'P: (x)al . , at the position x. 5c) i,j ij i,j where the upper sign applies to fermions and the lower one to bosons. We shall now express a few important operators in terms of the field operators.
28 1. 12) the corresponding form applying in the moment um representation. For spin-! fermions, the two possible spin quantum numbers for the z component of S are ± ~. 13b) where (T uu' are the matrix elements of the Pauli matrices. 1 Show that the fully symmetrized (antisymmetrized) basis functions S±'Pil (XI)'Pi2 (X2) ... 'PiN (XN) are complete in the space of the symmetrie (antisymmetrie ) wave functions 'l/Js/a(XI,X2, ... ,XN). Hint: Assurne that the product states 'Pil (Xl) '" 'PiN (XN), composed of the single-particle wave functions 'Pi(X), form a complete basis set, and express 'l/Js/a .
X - x') = 1 - 6"(smx - xcosx)2 . 0 2. - .. -7r---~ 47r- kF lx - x I I Fig. 4. The pair distribution function 9CT<7(X - x') . 16) plotted in Fig. 4. If a fermion is removed at x, the particle density in the vicinity of this point is strongly reduced. In other words, the probability of finding two fermions with the same spin at separations ;S k F1 is smalI. The reduction of 9",,(X - x') at such separations is referred to as an exchange, or correlation hole. It should be emphasized on ce again that this effective repulsion sterns solely from the antisymmetric nature of the state and not from any genuine inter action.