By Henk T. C. Stoof

Ultracold Quantum Fields offers a self-contained advent to quantum box thought for many-particle platforms, utilizing practical tools all through. the overall concentration is at the behaviour of so-called quantum fluids, i.e., quantum gases and drinks, yet trapped atomic gases are continually used for instance. either equilibrium and non-equilibrium phenomena are thought of. first of all, within the equilibrium case, the best Hartree-Fock conception for the houses of a quantum fluid within the common part is derived. the focal point then turns to the homes within the superfluid section, and the authors current a microscopic derivation of the Bogoliubov concept of Bose-Einstein condensation and the Bardeen-Cooper-Schrieffer conception of superconductivity. the previous is acceptable to trapped bosonic gases comparable to rubidium, lithium, sodium and hydrogen, and the latter specifically to the fermionic isotope of atomic lithium. within the non-equilibrium case, a couple of themes are mentioned for which a field-theoretical process is principally ideal. Examples are the macroscopic quantum tunnelling of a Bose-Einstein condensate, the section dynamics of bosonic and fermionic superfluids, and their collisionless collective modes.

The ebook relies upon the notes for a lecture direction within the masters programme in Theoretical Physics at Utrecht.

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8. e. ∑ V j, j V j−1, j j = ∑ G−1 0; j, j G0; j ,j = δ j, j . 76) j Note that we cannot calculate the integral exactly, because it is not Gaussian, due to the quartic term in the exponential. However, we are going to perform a trick to transform the quartic term away. 77) , where η is a real variable. Note that there is no longer a quartic term, since we have transformed it away. This is the essence of the Hubbard-Stratonovich transformation, which we use many times when treating interacting quantum gases.

In that case, we have ν |ν = δν ,ν , where δν ,ν is the Kronecker delta that equals one when ν = ν and zero otherwise. An orthonormal set of vectors |ν is called complete if it satisfies the completeness relation ∑ |ν ˆ ν | = 1. 1) ν The completeness relation ensures that an arbitrary vector can be expressed uniquely in terms of the orthonormal basis as |ψ = ∑ |ν ν |ψ ≡ ∑ cν |ν . 3) ν such that |cν |2 can be interpreted as the probability for the system to be in state |ν . e. 4) |µ = ∑ |ν ν |µ .

11 Scattering Theory 53 ˆ ˆ G(E) = Gˆ 0 (E) + Gˆ 0 (E)Vˆ G(E) = Gˆ 0 (E) + Gˆ 0 (E)Vˆ Gˆ 0 (E) + ... 111) to obtain |ν (0) ν (0) |Vˆ |ν (0) ν (0) | |ν (0) ν (0) | ˆ + + ... 109), we find the first-order correction to the eigenenergy (1) Eν = ν (0) |Vˆ |ν (0) . 112) gives rise to the first-order correction |ν (1) to the eigenstates. 109). 114) and the corresponding expression for the hermitian conjugate. In a similar way, we can derive higher-order corrections by looking at higher-order terms in the perturbative expansion.