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33 where vac is the vacuum state it is natural to take

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Unformatted text preview: 0 )N |vac p N! (3.3) where |vac is the vacuum state. It is natural to take this state as zeroth order approximation for finite but small interactions. Expectation value of the 3 4CHAPTER 3. BOSE-EINSTEIN CONDENSATION OF WEAKLY INTERACTING ATOMIC GASES Hamiltonian (3.1 in this state is E = −µN0 + U0 2 N 2V 0 (3.4) Minimizing with respect to N0 we find relation between the number of particles and the chemical potential N0 µ = V U0 (3.5) To proceed to the next order in the interaction it is convenient to introduce the idea of broken symmetry. When we consider two point correlation functions for the state (3.3) N0 V (3.6) bp eipr (3.7) ΨN |Ψ† (r2 )Ψ(r1 )|ΨN = 1 Ψ(r) = √ V p they do not depend on the relative distance between the two points. This is the definition of the long range order. Naively one expects property (3.6) to hold when individual expectation values of Ψ(r) and Ψ† (r) are equal to (N0 /V )1/2 . However individual expectation values of these operators vanish since they change the number of particles by one and state |...
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