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Файл:Baranov.Introduction to Physics Of Ultra Cold Gases.pdf
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- •General aspects
- •Introduction
- •Single particle
- •General aspects
- •Traps
- •Many particles
- •Basics of second quantization
- •Bosons
- •Fermions
- •Single particle operator
- •Two particle operator
- •Bosons
- •Free Bose gas
- •General properties
- •BEC in lower dimensions
- •Trapped Bose gas
- •Parabolic trap
- •Weakly interacting Bose gas
- •BEC in an isotr. harmonic trap at T=0
- •Comparison of terms in GP
- •Thomas-Fermi-Regime
- •Fermions
- •Free Fermions
- •General properties
- •Pressure of degenerated Fermi gas
- •Excitations of Fermions at T=0
- •Trapped non-interacting Fermi gas at T=0
- •Weakly interacting Fermi gas
- •Ground state
- •Decay of excitations
- •Landau-Fermi-Liquid
- •Zero Sound
- •Bardeen-Cooper-Shieffer-Theory
- •General treatment
- •BCS Hamiltonian
- •General energy-momentum relation
- •Calculation for section 3.3.1
- •Lifetime and Fermis Golden Rule
- •Bibliography
2.1. FREE BOSE GAS |
21 |
~ #"~
Obviously the total energy decreases (dissipation occurs) if p V , i.e.
ep pV 0 , ep pV: |
(2.29) |
This is possible for V VC with
Vc = min |
ep |
= min |
p |
= 0: |
(2.30) |
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p p p m |
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For the free BOSE gas dissipation of a flow at T = 0 occurs for arbitrary small velocities.
2.1.3BEC in lower dimensions
In 2D
d3 p |
! |
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d2 p |
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(2.31) |
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(2p~)3 |
(2p~)2 |
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n = Z |
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d2 p |
1 |
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: |
(2.32) |
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(2p~)2 |
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exp |
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ep |
1 |
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T |
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This expression diverges at p & 0 as 1p therefore m 6= 0 always. BEC occurs only at T = 0, when all particles are in the ground state, i.e. TC = 0.
In 1D no BEC occurs (even at T = 0).
However superfluidity in 2D is possible (if we switch on interaction, see below).
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