Second Quantization -- Fermions Do essentially the same steps as with bosons. Order all states and put in ones and zeros for filled and unfilled states respectively. Example: three particle system

Slater Determinant Slater determinate Makes wave function antisymmetric Keeps track of signs for f ’s. E’ i ’s are ordered. Only the N occupied states go into determinant. Occupation number space

Fermion Anticommutators 1. Two particles cannot be in the same state. 2. 0,1 are eigenvalues of number operator. 3. Going to creation and annihilation operators

Phases for Occupation States Again define occupation state as: where n i =0,1 Need to be careful of phases for raising and lowering operators: Phases become 1 for number operator!

Phase Factor in Hamiltonian Reorder both sides to normal order. (First move the W to where E k should be; then move it to its proper position. Phase factor depends on whether W>E k or <E k Define:

Continued First term always there. Second and third term may be present iff: Remembering: Coupled Equations

Going to Second Quantization

Green Term Continued Likewise for two other kinetic energy terms (see problem sets). Put delta functions in explicitly Go to “after” n’s

Potential Energy Term k level is now occupied. Must include in sum as we move to the l level. This is the term i<j<k<l.

Green Term Likewise for other terms (see problem sets). Sign from extra -1 in phase Reorder to restore + in front

Both Kinetic and Potential Terms Second Quantization Result Note order to preserve sign.