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Second Quantization of Conserved Particles Electrons, 3He, 4He, etc. And of Non-Conserved Particles Phonons, Magnons, Rotons… We Found for Non-Conserved Bosons • E.g., Phonons that we can describe the system in terms of canonical coordinates • We can then quantize the system • And immediately second quantize via a canonical (preserve algebra) transform • We create our states out of the vacuum • And describe experiments with Green functions • With Creation of (NC) Particles at x • We could Fourier transform our creation and annihilation operators to describe quantized excitations in space poetic license • This allows us to dispense with single particle (and constructed MP) wave functions • We saw, the density goes from • And states are still created from vacuum • These operators can create an N-particle state • With conjugate • Most significantly, they do what we want to! Think <x|p> • That is, they take care of the identical particle statistics for us • I.e., the operators must • And the Slater determinant or permanent is automatically encoded in our algebra Second Quantization of Conserved Particles • For conserved particles, the introduction of single particle creation and annihilation operators is, if anything, natural • In first quantization, • Then to second quantize • The density takes the usual form, so an external potential (i.e. scalar potential in E&M) • And the kinetic energy • The full interacting Hamiltonian is then • It looks familiar, apart from the two ::, they ensure normal ordering so that the interaction acting on the vacuum gives you zero, as it must. There are no particle to interact in the vacuum • Can I do this (i.e. the ::)? p42c4 Transform between different bases • Suppose we have the r and s bases • Where • I can write (typo) • If this is how the 1ps transform then we use if for operators x or k (n) • With algebra transforming as • I.e. the transform is canonical. We can transform between the position and discrete basis • Where is the nth wavefunction. If the corresponding destruction operator is just Is this algebra right? • It does keep count • Since – F [ab,c]=abc-cab + acb-acb =a{b,c}-{a,c}b – B [ab,c]=abc-cab + acb-acb =a[b,c]+[a,c]b Eq. 4.22 – • For Fermions • It also gives the right particle exchange statistics. • Consider Fermions in the 1,3,4 and 6th one particle states, and then exchange 4 <-> 6 • Perfect! • And the Boson state is appropriately symmetric • 3 hand written examples Second Quantized Particle Interactions • The two-particle interaction must be normal ordered so that • Also hw example