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Appendix B Performing the Fourier transformation for the free particles Starting point is the Hamiltonian: H =C + R A ˆ b † R ˆ bR H1 + B( ˆb 2 R + ˆ (b † R )2 ) H2 −t ˆ† b R ˆbR+1 + ˆbR ˆb † R+1 H3 A Fourrier transformation given in equations 2.13 is performed. Again the Hamiltonian is split up into 3 parts. For the first term called H1: H1 = R A 1 L = Aˆb † k,k ′ = k,k ′ k,k ′ k ′ˆ bk e i(k−k′ )Rˆ b † k ′ˆ bk 1 L A ˆ b † k ′ˆ bkδk,k 37 R e i(k−k′ )R ′ = k A ˆ b † k ˆ bk . . (B.1)
Where a representation of the delta distribution (the Kronecker delta in 1 that case) δk,k ′ = L R ei(k−k′ )R was used in the second line. Fourier transforming the second term leads to: H2 = B R = B k,k ′ ˆ b 2 R + ( ˆ b † R )2 1 N R = B ( ˆbk ˆbk ′ + ˆb † k,k ′ = B Finally the last term becomes: k H3 = −t R = −t k,k ′ = −t k e i(k′ +k)Rˆ bk ˆ bk ′ + e−i(k′ +k)Rˆ b † k ˆ b † k ′ k ˆb † k ′)δk,−k ′ ˆ bk ˆ b−k + ˆ b † k ˆ b † −k ˆ b † R ˆ bR+1 + ˆ bR ˆ b † R+1 (e ik′ ˆ b † k ˆbk ′ + e−ikˆbk ˆb † 1 k ′) N e ikˆ b † k ˆ bk + e −ik (1 + ˆ b † k ˆ bk) = −t e ik − 2t k k ˆ b † k ˆ bk cos k e i(k−k′ )R R δk,k ′ Where in the third line the bosonic commutation relations were used. That the set operators ˆ bk and ˆ b † k really obey them is shown in Appendix C. 38
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 and 8: occupation of the ground state N0
Page 9 and 10: Chapter 2 Interacting Bosons We now
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: equally the next term of the Hamilt
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

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