Projektpraktikum - TU Graz - Institut für Theoretische Physik ...

More documents

Recommendations

Info

kBTc = 2π2 3/2 N m ζ(3/2)V (1.6) For temperatures lower than Tc the fraction of particles in the ground state or condensed particles can be calculated from 1.5 as: N0 N T = 1 − Tc 3/2 (1.7) This means, that at a temperature lower than Tc a macroscopic amount of particles occupies the ground state and the fraction of condensed particles approaches one as T becomes zero. On the other hand for temperatures higher than Tc the number of condensed particles can be neglected. 7
Chapter 2 Interacting Bosons We now consider a simple model for interacting bosons described by the one dimensional Bose-Hubbard-Hamiltonian: H = L (V (R) − µ) ˆ ψ † R ˆ ψR − t R=1 L R=1 ˆψ † R ˆ ψR+1 + ˆ ψR ˆ ψ † R+1 + U 2 L ( ˆ ψ † R=1 R )2 ( ˆ ψR) 2 . (2.1) Where the first term corresponds to an external potential, which in this work is chosen to be: V (R) = α(R − L 2 )2 . This term also contains the chemical potential, which insures particle conservation. The second term describes the movement of the particles, where just hopping from a lattice point to its nearest neighbors is included. The last part is the interaction potential between 2 particles, where the potential V (R − R ′ ) acts only on particles on the same site: The bosonic field-operators ˆ ψR and ψ † R relations: V (R − R ′ ) = UδRR ′ . (2.2) 8 obey the bosonic commutation
Page 1 and 2: Projektpraktikum Technische Univers
Page 3 and 4: D Calculating the 2L×2L matrix for
Page 5 and 6: Introduction When Einstein develope
Page 7: occupation of the ground state N0
Page 11 and 12: H = (V (R) − µ)|φ(R)| R 2 + U
Page 13 and 14: 2.10 can be written as: H = C + R
Page 15 and 16: ˆ b0|N0, N1, ...〉 = N0|N0 − 1
Page 17 and 18: vk = − 1 2 + (µ + 4t − 2t cos
Page 19 and 20: statistics 1.1 for different temper
Page 21 and 22: and the 2L × 2L dimensional matrix
Page 23 and 24: with a diagonal matrix D. The follo
Page 25 and 26: eigenvalue equation is still valid,
Page 27 and 28: since only the expectation values f
Page 29 and 30: Expectation Value of Fluctuation 2.
Page 31 and 32: with J the jacobi matrix of F . yn
Page 33 and 34: |φ (R)| 4.5 4 3.5 3 2.5 2 1.5 1 0.
Page 35 and 36: The matrix V, on the other hand, ar
Page 37 and 38: equally the next term of the Hamilt
Page 39 and 40: Where a representation of the delta
Page 41 and 42: Calculation now the commutator [ ˆ
Page 43 and 44: B † M B = = L Mmkbk + 2L B m=1
Page 45 and 46: Appendix E Derivation of the condit
Page 47: Bibliography [1] Michael Knap, Enri

Projektpraktikum - TU Graz - Institut für Theoretische Physik ...

Create successful ePaper yourself

Delete template?

Save as template?