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Chapter 3
Results
3.1 Free Particles in 1 Dimension
The average value of the number operators ψ ˆ† R ˆ
ψR , which were first introduced
in 2.1 and were split in a condensate contribution φ(R) and a operator
ˆbR in equation 2.4, is given by:
〈 ˆ ψ †
R ˆ ψR〉 = |φ(R)| 2 + 〈 ˆ b †
R ˆ bR〉 (3.1)
Since the external potential V (R) is zero this values are place indepen-
dent and can be calculated for any index R. For convenience R = 0 is chosen
and the place will not be denoted any more.
The simple expression for the condensate density φ 2 reads from 2.8:
φ 2 =
2t + µ
U
The expectation value of the fluctuation operators 〈 ˆ b †ˆ b〉 has to be calculated
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