6 Multi-particle Systems (2)

Transcription

1 6 Multi-particle Systems (2) Exercise 6.1: Four particles in one dimension Four one-dimensional particles of mass m are confined to a length L with periodic boundary conditions. The Hamiltonian Ĥ for one particle reads Ĥ = ˆp2 2m. (1) The particles do not interact. The single-particle states have wavefunctions ψ n (x) = 1 L e 2πinx/L, with n = 0,±1,±2,... The creation operator for a particle in a state with wavefunction ψ n (x) is denoted â n for bosons and â n,σ for spin 1/2 fermions with spin σ =,. (a) What are the ground state energy and the ground state degeneracy if the four particles are distinguishable? (b) What are the ground state energy and the ground state degeneracy if the four particles are indistinguishable spin-0 bosons? Express the ground state(s) in terms of the vacuum state 0 and the creation operators a n. (c) What are the ground state energy and the ground state degeneracy if the four particles are indistinguishable spin-1/2 fermions? Express the ground state(s) in terms of the vacuum state 0 and the creation operators a n,σ. Exercise 6.2: Bosons in a harmonic oscillator potential Consider indistinguishable spin 0 bosons with mass m in the three-dimensional harmonic oscillator potential with frequency ω, V(x,y,z) = 1 2 mω2 (x 2 +y 2 +z 2 ). (2) Thesingle-particlegroundstateisnondegenerateandhasenergy(3/2) ω. Thelowestexcited single-particle state has energy (5/2) ω and is threefold degenerate. 1

2 (a) Letâ 0 betheoperatorthatannihilatesabosoninthesingle-particlegroundstate. Give an expression for â 0 in terms of the boson annihilation operator ˆψ(r) in the position representation. (b) What is the degeneracy of the two-boson ground state? And what is its energy? (c) What are the energy and the degeneracy of the lowest-lying two-boson excited state? Express the lowest-lying two-boson excited state in terms of the creation and annihilation operators â 0 and â 0, and in terms of the creation and annihilation operators â 1,j and â 1,j, j = 1,2,3, of a boson in each of the lowest-lying excited single-particle states. (If the lowest-lying two-boson excited state is degenerate, give expressions for each of the degenerate states.) Exercise 6.3: Hydrogen molecule A simple model for the Hydrogen molecule is given by the Hamiltonian with Ĥ = Ĥ0 +Ĥ1, Ĥ 0 = U 2 (ˆn 1, + ˆn 1, ) 2 + U 2 (ˆn 2, + ˆn 2, ) 2, Ĥ 1 = t(ĉ 1, ĉ2, +ĉ 2, ĉ1, +ĉ 1, ĉ2, +ĉ 2, ĉ1, ), where ˆn j,σ = ĉ j,σĉj,σ. In this Hamiltonian, only one spin-degenerate orbital is taken into account for each one of the two Hydrogen atoms that make up the Hydrogen molecule. The operator ĉ j,σ annihilates an electron at atom j (j = 1,2) and with spin σ. Further, U and t are positive parameters that describe the Coulomb repulsion of two electrons localized at the same atom and the hybridization between the atomic orbitals, respectively. The total number of electrons N = 2. (a) What is the dimension of the two-electron Hilbert space for this Hamiltonian? (b) Show that the two-electron ground state of the Hamiltonian Ĥ0 is fourfold degenerate and give expressions for the four eigenstates. Use the language of second quantization. 2

3 (c) Give second-quantized expressions for the operators Ŝz, Ŝ+, and Ŝ (or, alternatively, Ŝ z, Ŝx, and Ŝy) of the total spin. (d) Show that Ĥ0 and Ĥ1 commute with the total spin Ŝ. (e) Using second-order perturbation theory in Ĥ1, determine the energy, degeneracy, and total spin of the ground state of Ĥ. Hint: Arrange the eigenstates you found under (b) such, that they are eigenstates of Ŝ z and Ŝ2. Exercise 6.4: Spin (a) In the Pauli theory, an electron is described as a particle with a spin degree of freedom only. For a single electron, there are only four relevant operators: the identity operator 1 and the three components s x, s y, and s z of the spin operator. In second quantized language, the identity operator 1 becomes ĉ ĉ + ĉ ĉ, where ĉ σ and ĉ σ are creation and annihilation operators for an electron with spin σ. What are the second-quantized expressions for the operators ŝ x, ŝ y, and ŝ z? (b) In the Pauli-Schrödinger theory an electron is described by a wavefunction ψ(r, σ). Express the operators ŝ x, ŝ y, and ŝ z in terms of the creation and annihilation operators ˆψ (r,σ) and ˆψ(r,σ) for an electron with spin σ at position r. Exercise 6.5: Two-particle operators Operators Ĝ that involve a sum over operators ĝ(ij) that act on the degrees of freedom of two different particles i and j are called two-particle operators, N Ĝ N = ĝ (ij) = 1 ĝ (ij). 2 i<j An example of a two-particle operator is the Coulomb interaction for particles of charge q, Û N = 1 q 2 2 4π r i r j. i j i j 3

4 (a) Show that a two-particle operator may be represented as Ĝ N = N(N 1) 2 k 1,k 2,k 1,k 2 k 3,...,k N k 1 k 2 k 3...k N g(k 1,k 2 ;k 1,k 2) k 1k 2k 3...k N, where and k 1 k 2...k N = 1 ( k P1 k P2... k PN ) N! P g(k 1,k 2 ;k 1,k 2) = ( k 1 k 2 )ĝ (12) ( k 1 k 2 ). (b) Show that the corresponding Fock-space operator Ĝ can be represented as Ĝ = 1 â (k 1 )â (k 2 )g(k 1,k 2 ;k 2 1,k 2)â(k 2)â(k 1), k 1,k 1 k 2,k 2 where the operators â (k) and â(k) are creation and annihilation operators, respectively. Exercise 6.6: Occupation number basis If the single-particle basis states { k } are labeled by a discrete quantum number k, one can construct a normalized basis for the N-boson Hilbert space H (S) N using occupation numbers. For a given (symmetrized) basis state k 1 k 2...k N, one defines n k as the number of times the quantum number k appears in the N-tuple k 1, k 2,..., k N. (a) Show that the dot product k 1...k N k 1...k N of two symmetrized basis states is zero, unless the two basis states k 1...k N and k 1...k N have the same set of occupation numbers {n k }. In that case, k 1...k N k 1...k N = 1 n k!. N! (b) One defines the state {n k } as N! {n k } = k n k! k 1...k N, k 4

5 where N = k n k and the quantum number k appears precisely n k times in the N- tuple k 1,..., k N. (The order in which the quantum numbers appear is not relevant.) Show that these states form an orthonormal basis for the Fock space F. (c) Show that (d) Show that â k...n k... = n k +1...n k +1..., â k...n k... = n k...n k â kâk...n k... = n k...n k..., so that the operator ˆn k = â kâk gives the occupation of the single-particle state k. (e) Find an expression for the state {n k } in terms of the creation operators â k vacuum state 0. and the Exercise 6.7: Commutation relations The boson annihilation operators a k and ψ(r) in the continuous momentum representation and in the position representation are related as 1 ˆψ(r) = dkâ (2π) 3/2 k e ik r. (a) Give the inverse of the above relationship and give the corresponding relations between the creation operators. (b) Starting from the commutation relations of the boson creation and annihilation operators in the position representation, derive the commutation relations in the continuous momentum representation. Exercise 6.8: Current density 5

6 Show that the current density ĵ(r) in a many-particle system is given by the expression and prove the continuity relation ĵ(r) = 2mi [ˆψ (r) rˆψ(r) ( rˆψ (r))ˆψ(r)] r ĵ(r)+ ˆn(r) = 0, where ˆn(r) is the particle density, as an operator identity in the many-particle system. Exercise 6.9: Wick s theorem Wick s theorem gives a relation between many-particle correlation functions and singleparticle correlation functions for non-interacting particles in thermal equilibrium. (a) Show that, for non-interacting Bosons, ˆψ (r 1 )... ˆψ (r n )ˆψ(r n)... ˆψ(r 1) = P ˆψ (r P1 )ˆψ(r 1)... ˆψ (r Pn )ˆψ(r n) where the sum is over permutations P of the numbers 1, 2,..., n. (b) Extend your proof to the case that the Heisenberg-picture operators ˆψ (r i ) and ˆψ(r i) are evaluated at different times t i and t i, respectively, i = 1,...,n. (c) In the above expression, the creation operators appear to the left of the annihilation operators. One refers to this as the normal order of the operators. Can you find a version of Wick s theorem that is valid if the order of the creation and annihilation operators is arbitrary? (d) Generalize your answers to items a, b, and c to the case of identical spin 1/2 fermions. Exercise 6.10: Tight-binding Hamiltonian In certain crystals the valence electrons are tightly bound to their host ions. In the simplest approximation, in which one considers one bound state (per spin direction) per lattice ion, 6

7 one may take the one-particle Hilbert space to be spanned by states n,σ bound at the nth lattice ion. The one-particle Hamiltonian then takes the form Ĥ = u t n,σ m,σ, (3) n,m n.n. where u is the bound-state energy of the electrons and t is a probability amplitude for tunneling between neighboring lattice sites. In the above equation, n.n. stands for nearest neighbor : the double sum is restricted to neighboring lattice ions. The probability amplitude t is usually referred to as the hopping amplitude. (a) Using second-quantization language, give an expression for the many-electron generalization of the Hamiltonian (3) in terms of creation and annihilation operators ĉ nσ and ĉ nσ for an electron with spin σ in a bound state at lattice ion n. (b) Consider a one-dimensional lattice with N sites and periodic boundary conditions. Use the discrete Fourier transformation σ ĉ n,σ = 1 e ikj ĉ k,σ N to write the second-quantized Hamiltonian takes the diagonal form k H = k,σ ε k c kσ c kσ and calculate the corresponding eigenvalues ε k. (c) Repeat question (b) for a two-dimensional square lattice with N 2 sites and periodic boundary conditions. Now the discrete Fourier transformation involves wavevectors k = (k x,k y ). Plot contours of constant energy ε kx,k y in the k x k y plane. Note: In high-temperature superconductors, the conduction electrons are confined to parallel CuO planes, where the ions form a two-dimensional square lattice. In this case the twodimensional model you considered above serves as a starting point for many theoretical investigations. Exercise 6.11: Super-exchange 7

8 In certain crystals the valence electrons are tightly bound to their host ions. In this exercise, you consider two neighboring ions and the (outer) electrons bound to them. You may use the simplest approximation, in which one considers one bound state (per spin direction) per lattice ion. In that case, the second-quantized Hamiltonian takes the simple form Ĥ = ε g (ˆn 1 + ˆn 2 )+t σ (ĉ 1σĉ 2σ +ĉ 2σĉ 1σ )+ U 2 (ˆn 1(ˆn 1 1)+ ˆn 2 (ˆn 2 1)), (4) where ĉ iσ and ĉ iσ are creation and annihilation operators for an electron on lattice ion i (i = 1,2), ˆn i = ĉ i ĉi +ĉ i ĉi, ε g is the energy of the bound state at each ion, t is a parameter that describes the tunneling rate between electrons bound to the two ions, and U is a parameter that describes the repulsive interaction between electrons bound to the same ion. The interaction term is taken to be proportional to ˆn i (ˆn i 1) in order to ensure that there is no interaction contribution to Ĥ if n i = 1. (a) Show that 1 2ˆn i(ˆn i 1) = ĉ i ĉi ĉ i ĉi. For deep wells, interaction effects typically dominate tunneling, U t. This is the case you are asked to consider in the remainder of this exercise. (b) Show that, if there are two electrons present, the ground state is fourfold degenerate if you neglect tunneling. Give explicit expressions for the ground states using the creation operators ĉ iσ. (c) Again neglecting tunneling arrange your expressions for the four degenerate ground states, such that each state is an eigenstate of the total spin Ŝ2 and its z-component Ŝ z. (d) If you take tunneling into account, will the ground state energies of part (d) change? Does the change depend on spin? If yes, in what way? And to what order in t? Can you calculate the shift of the ground state energies to leading order in t? Exercise 6.12: Hanbury Brown Twiss and noise correlations In this exercise you are asked to look at variations of the Hanbury Brown Twiss experiment for identical particles. First consider two identical spin 0 particles of mass m 0 localized in 8

9 harmonic wells of frequency ω 0 at x = ±d/2 in one dimension. You may neglect interactions between the particles. Initially, the system is described by the two particle wavefunction where ψ(x 1,x 2 ) = φ(x 1 d/2)φ(x 2 +d/2)±φ(x 2 d/2)φ(x 1 +d/2) (5) φ(x) = 1 x 2 (2πσ 2 ) 1/4e 4σ 2, σ = 2m 0 ω 0, is the wavefunction for a single particle localized in harmonic well at x = 0 and the + ( ) sign refers to bosons (fermions) respectively. We assume σ d, so that the wells are well separated and the harmonic oscillator ground state wavefunctions φ(x d/2) and φ(x + d/2) may be taken to be orthogonal. (a) The harmonic trapping potential is suddenly turned off at time t = 0. For t > 0 the two-particle wavefunction evolves under the free Hamiltonian Ĥ = ˆp2 1/2m + ˆp 2 2/2m. What is the probability P(x,t) for measuring a particle at position x as a function of time? Does this probability depend on statistics? In the long time limit, what is shape of the probability distribution? Indicate the time evolution of P(x,t) by making a few representative plots of P(x,t) at different times t. Hint: First show that for t > 0. φ(x,t) = 1 (2πσ 2 ) 1/4 (1+iω 0 t) 1/2e x2 /4σ2 (1+iω 0 t) (b) Now compute the joint probability distribution for the positions of the two particles P(x 1,x 2,t) = ψ(x 1,x 2,t) 2 in the long time limit t. How could one measure this function in principle? Does this function depend on statistics? Could we infer the spacing between the particles at t = 0 from measuring correlations between the locations where we find the identical particles at a time t 0? Next you are asked to consider a one-dimensional lattice of harmonic potential wells with separation d. Each well has one particle. In second quantized language, the initial state G has the form G = m â m 0, (6) where a m is creation operator for a particle in the ground state of a well at x = md. 9

10 (c) Express the creation operator â m and the corresponding annihilation operator â m in terms of the creation and annihilation operators ˆψ (x) and ˆψ(x) that create and annihilate a particle at position x. (d) Again, the confining potential is suddenly switched off at time t = 0. For t < 0, when the lattice of harmonic wells was in place, the harmonic well ground states were (to a very good approximation) eigenstates of the Hamiltonian. Hence, for t < 0, the Heisenberg-picture time dependence of the operators â m and â m was trivial, â m (t) = â m e iω 0t/2, where ω 0 is the frequency of the harmonic well and â m is the annihilation operator at time t = 0. For t > 0, the time dependence is nontrivial, because the harmonic well ground states are no longer eigenstates of the Hamiltonian. Show that â m (t) = dxw m (x,t)ˆψ(x) (7) where w n (x,t) is the time evolved single particle wavefunction for a particle initially localized in the well centered at x = md and ˆψ(x) is the annihilation operator for a particle at position x at time t = 0. (e) Compute the density n(x,t) = G ˆψ (x,t)ˆψ(x,t) G (8) in terms of the single-particle wavefunctions w m (x,t). Does your result depend on the statistics of the particles? Hint: Invert Eq. (7), i.e., write ˆψ(x,t) = iâiw i (x,t) + jˆb j v j (x,t). The annihilation operators ˆb j are for higher lying states of the harmonic oscillators (higher lying bands in condensed matter jargon). Argue that the terms containing the ˆb j s vanish in our expectation value and therefore it is valid to replace ˆψ(x,t) iâiw i (x,t). (f) Compute the density-density correlation function C(x,x,t) = G ˆn(x,t)ˆn(x,t) G, ˆn(x,t) = ˆψ (x,t)ˆψ(x,t), (9) in terms of the single-particle wavefunctions w m (x,t). Does your result depend on the statistics of the particles? 10

11 Hint: First normal order the expression for C, C(x,x,t) = G ˆψ (x,t)ˆψ(x,t)ˆψ (x,t)ˆψ(x,t) G = δ(x x )ˆn(x,t)± G ˆψ (x,t)ˆψ (x,t)ˆψ(x,t)ˆψ(x,t) G, the + ( ) sign refers to bosons (fermions) respectively. Then argue that, as before, you can replace ˆψ(x,t) iâiw i (x,t) in the normal ordered expectation value. (g) Explicitly evaluate the density and the density-density correlation function in the long time limit ω 0 t 1. What difference do you find between fermions and bosons? Take a lattice with N = 2 and compare your result to part (b). Plot C(x,0,t) (without the δ-function at the origin) for N = 2,4,40. If you did everything right, you should have found bunching or antibunching of particles in this free expansion setup. This effect has been demonstrated in experiments with ultra-cold fermions and bosons where one derives C(x,x,t) from the statistics of many images after time-of-flight expansion. The goal of a measurement of the density-density correlation function is to gain information on the many-particle state before the confining potential was turned off. One hopes, e.g., to find unique features that prove that this state has correlations (such as a ferromagnet or antiferromagnet). In a different context, similar effects show up in the temporal current noise of electrical transport through mesoscopic conductors. You can find more information in E. Altman, E. Demler and M. Lukin, Phys. Rev. A 70, (2004) (the original theory article for this setup in the cold-atom context), S. Foelling et al., Nature 434, 481 (2005) (experiment with bosons), T. Rom et al., Nature 444, 733 (2006) (experiment with fermions), and Ya. M. Blanter and M. Büttiker, Phys. Rep. 336, 1 (2000) (mesoscopic conductors). Exercise 6.13: Stability of Bose and Fermi systems A system of Bosons with an attractive interaction will collapse into a volume of the order of the range of the attractive force; the bosons will not occupy all available space uniformly. In this exercise you show that a system of fermions is similarly unstable in the presence of an attractive interaction with a finite range. (a) Consider a system of Bosons subject to an attractive interaction potential u(r) = u 0 if r < a and 0 otherwise. The Bosons are not confined by an external potential. Show that the ground state energy of N Bosons is bounded from above by cn 2, where c is a positive constant, and argue that this implies that the Bose systems collapses into a finite volume. 11

12 (b) Show that the same result also holds for a system of fermions, and interpret your result. (c) Investigate whether your answers to (a) and (b) depend on dimensionality. Exercise 6.14: Excitations in a non-uniform weakly interacting Bose-Einstein condensate In a weakly interacting Bose-Einstein condensate, the annihilation operator ψ(r) can be written as ˆψ(r) = ψ 0 (r)+δˆψ(r), (10) where ψ 0 (r) is a complex-valued function, which describes the condensate, and δˆψ(r) is an operator which, together with its hermitian conjugate δˆψ (r), satisfies the standard commutation relations for annihilation and creation operators and describes excitations in the Bose-Einstein condensate. Since ψ 0 (r) is the ground state wavefunction, we can choose it to be real. An effective Hamiltonian for these excitations can be found by expanding the full Hamiltonian for the weakly interacting Bose gas to second order in δˆψ(r), Ĥ µ ˆN = E 0 µn 0 + dr [ δˆψ (r)ĥ0δˆψ(r)+ un 0(r) 2 ( δˆψ (r)δˆψ (r)+δˆψ(r)δˆψ(r)) ], (11) where E 0 and N 0 are the energy of the condensate and the number of particles in the condensate, respectively, Ĥ 0 = 2 2m 2 r +v(r)+2u ψ 0 (r) 2 µ, (12) and ˆn 0 (r) = ˆψ 0 (r) 2 is the condensate density. As in the case of the uniform Bose gas, we would like to write this Hamiltonian in the form Ĥ = E 0 µn 0 + ξ nˆb nˆbn, (13) n wheretheˆb n andˆb n arestandardbosoncreationandannihilationoperatorsandallexcitation energies ξ n are positive. Hereto we make the ansatz δˆψ(r) = n [u n (r)ˆb n v n(r)ˆb n], (14) where the u n (r) and v n (r) are functions that still need to be determined. 12

13 (a) The most convenient way to determine the functions u n and v n is to look at the time dependence of ˆψ(r). From the Hamiltonian (11), we have i δ ˆψ(r) = Ĥ 0ˆψ(r)+un0 (r)δˆψ. On the other hand, for the ansatz (13), the time dependence of each operator ˆb n is ˆbn (t) =ˆb n e iξnt/. Show that these equations imply that the functions u n (r) and v n (r) need to satisfy the eigenvalue equations ( ) ( )( ) un (r) Ĥ ξ n = 0 un 0 (r) 2 un (r). (15) v n (r) un 0 (r) 2 Ĥ0 v n (r) (b) Show that eigenvalues ξ n of the eigenvalue problem (15) come in pairs ±ξ n, with eigenfunctions (u n,v n ) and (v n,u n). (c) The eigenvalue problem (15) has a different structure than you are used to, because the matrix-valued operator in Eq. (15) is not a hermitian operator in the usual sense. Instead, it is self-adjoint with respect to the dot product defined as (u n,v n ) (u n,v n ) = dr[u n (r) u n (r) v n (r) v n (r)]. Show that, if there are no eigenfunctions (u n,v n ) of zero norm, the eigenvalues ξ n are real, and that the eigenfunctions with positive ξ n form a complete set with orthonormality conditions dr[u n(r)u m (r) vn(r)v m (r)] = δ nm, dr[u n (r)v m (r) v n (r)u m (r)] = 0. (d) Show that the operators ˆb n may be expressed in terms of the operators δˆψ(r) as, ˆbn = dr[u nδˆψ(r)+v nδˆψ(r) ]. (e) Showthattheabovenormalizationofthefunctionsu n andv n impliesthattheoperators b n and b n satisfy standard commutation relations for Boson annihilation and creation operators. 13

14 (f) Show that the Hamiltonian for the excitations indeed has the form (13), up to a constant term, and calculate this constant. Exercise 6.15: Conduction electrons in a metal In the Sommerfeld model, conduction electrons in a metal are modeled as an ideal Fermi gas. For Cu, the electronic density is /cm 3. (The densities for most other metals are within a factor 10 of this value.) (a) Assuming that the conduction electrons in Cu form an ideal Fermi gas, calculate the Fermi energy ε F and the Fermi wavenumber k F. (b) Are conduction electrons in a metal at room temperature described by a degenerate, or by a non-degenerate Fermi gas? Exercise 6.16: Chemical potential of an ideal Fermi gas Consider an ideal Fermi gas with density n. (a) Calculate its chemical potential µ for low temperatures k B T ε F, keeping correction terms up to order T 2. (b) Calculate the chemical potential µ in the high-temperature limit k B T ε F. (c) Repeat your calculations of items (a) and (b) for ideal Fermi gases in one and two dimensions. Exercise 6.17: Hartree-Fock theory of the uniform Fermi gas In general, the Hartree-Fock equations can not be solved in closed form. An exception is the case of the uniform Fermi gas, a Fermi gas not subject to an external potential. Because of translation invariance, the single-particle eigenfunctions are plane waves. (This can be seen both from the variational approach and from the mean-field approach.) 14

15 (a) Argue that interactions do not change the relation n = 3π 2 kf 3 density n and the Fermi wavenumber k F. between the particle (b) Calculate the Hartree-Fock energy levels ξ k,σ for a uniform Fermi gas. Express your answer in terms of the Fourier transform u q of the interaction potential and k F. (c) Find an expression for the ground state energy of the uniform interacting Fermi gas in the Hartree-Fock approximation. (d) Show that your answer agrees with what one expects from first-order perturbation theory. Exercise 6.18: Hard-core bosons in one dimension Motivation of this exercise: In one dimension, it is not possible to interchange particles with a strong repulsive interaction. But if one can not physically interchange particles, there should be no difference between bosons and fermions. In this exercise you consider identical spin 0 bosons with a strong repulsive interaction. We use a description in which the bosons are confined to discrete positions on a one-dimensional chain. An example of such a chain a one-dimensional sequence of harmonic traps, where for each trap only the ground state can be occupied, see the figure below. (The bosons do not have enough energy do occupy higher-lying states.) We denote the operator that creates a boson on site j (i.e., in the ground state of the jth trap for the example mentioned above) with ˆb j and the corresponding annihilation operator with ˆb j N There is a strong repulsive interaction between the bosons that prohibits double occupation of each trap. In principle, this repulsive interaction should be modeled by an appropriate interaction potential in the Hamiltonian. Here, we choose a different approach: In order to model the repulsive interaction, we require that the creation and annihilation operators ˆb j and ˆb i satisfy anticommutation relations if i = j. For i j, the operators still satisfy... 15

16 commutation relations. Hence, we require [ˆb i,ˆb j ] = [ˆb i,ˆb j ] = [ˆb i,ˆb j ] = 0, if i j, {ˆb j,ˆb j } = {ˆb j,ˆb j } = 0, {ˆb j,ˆb j } = ˆ1. (16) Particles described by such creation and annihilation operators are usually referred to as hard-core bosons. (a) Show that the (anti)commutation relations (16) ensure that each trap is occupied by at most one particle. We consider a finite chain, with sites j = 1,2,...,N and define operators ĉ j = e iˆφ jˆbj, where j 1 ˆφ j = π ˆb iˆb i. i=0 (b) Show that ˆφ j /π is the number of particles to the left of trap no. j. (c) Show that ˆφ j commutes with ˆb j, so that ĉ j = e iˆφ jˆbj =ˆb j e iˆφ j. (d) Show that the operators ĉ j obey anticommutation relations on the same site, {ĉ j,ĉ j } = {ĉ j,ĉ j } = 0, {ĉ j,ĉ j } = ˆ1. (e) Show that the operators ĉ j obey anticommutation relations on different sites, too, {ĉ i,ĉ j } = {ĉ i,ĉ j } = {ĉ i,ĉ j } = 0, if i j. Hence, the operators ĉ j, j = 1,...,N, describe fermions. (f) The Hamiltonian of the hard-core bosons reads N 1 Ĥ = t (ˆb i+1ˆb i +ˆb iˆb i+1 ). i=1 This Hamiltonian describes the finite transition amplitudes t between the ground states of neighboring traps, see the figure for a schematic picture. Rewrite this Hamiltonian in terms of the operators ĉ i and ĉ i. 16

17 t t t N Exercise 6.19: Bogoliubov transformation for fermions In the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity, fermionic excitations in a superconductor are described by a Hamiltonian of the form with Ĥ = Ĥξ +Ĥ, Ĥ ξ = ξ(ĉ ĉ +ĉ ĉ ), Ĥ = (ĉ ĉ +ĉ ĉ ), where ĉ σ and ĉ σ denote fermionic creation and annihilation operators, respectively, σ =,, and ξ and are real numbers with the dimension of energy. (a) Express the number of fermions N in terms of the operators ĉ σ and ĉ σ. (b) Does the Hamiltonian Ĥ conserve the fermion number N? If not, give an expression for dn/dt in terms of the operators ĉ σ and ĉ σ. In order to bring the Hamiltonian Ĥ into the standard diagonal form, we introduce a new set of operators ˆd = uĉ vĉ, ˆd = vĉ +uĉ, where u and v are real numbers. (c) The operators ˆd σ and their hermitian conjugates satisfy the anticommutation relations for fermion creation and annihilation operators if and only if u 2 +v 2 = 1. Verify this statement for the anticommutator {ˆd, ˆd }. (d) When written in terms of the operators ˆd σ and their hermitian conjugates, Ĥ ξ reads Ĥ ξ = (u 2 v 2 )ξ(ˆd ˆd + ˆd ˆd )+2uvξ(ˆd ˆd + ˆd ˆd )+2v 2 ξ. 17

18 Express Ĥ in terms of the operators ˆd σ and ˆd σ and use your answer to find the choice of u and v for which Ĥ = Ĥξ +Ĥ has the diagonal form, Ĥ = E 0 + σ ε σˆd σˆdσ. The creation and annihilation operators ˆd σ and ˆd σ, with the choice of the parameters u and v you found in part (d), describe fermions. One usually refers to these fermions as quasiparticles to distinguish them from the original fermionic particles that were described by the creation and annihilation operators ĉ σ and ĉ σ. (e) What is the spectrum of Ĥ (i.e., what are the possible eigenvalues of Ĥ)? You may express your answer in terms of the numbers ε σ introduced in part (d). Note: You do not need to have solved parts (c) and (d) in order to answer this question. (f) Express the number of quasiparticles ˆN qp in terms of the operators ˆd σ and ˆd σ. Is ˆN qp conserved? If not, give an expression for d ˆN qp /dt in terms of the operators ˆd σ and ˆd σ. Note: You do not need to have solved parts (c), (d), and (e) in order to answer this question. Exercise 6.20: Bose-Einstein condensation in different dimensions In the lecture you have seen, that an ideal Bose gas in three dimensions acquires a macroscopic occupation of the single-particle ground state at sufficiently low temperatures. In this exercise, you ll investigate whether the same is true for a (hypothetical) ideal Bose gas in one or two dimensions. The ideal Bose gas in two dimensions is described by the Hamiltonian Ĥ = k ε k â kâk, ε k = 2 k 2 2m, where k is a two-dimensional vector. We consider the two-dimensional ideal Bose gas on a finite area L 2 with periodic boundary conditions, so that k x = 2πj x /L and k y = 2πj y /L, with j x and j y integers. The average number of particles in a state with wavevector k is given by the Bose-Einstein distribution function, 1 ˆn k = e (ε k µ)/k BT 1. Here T is the temperature, k B the Boltzmann constant, and µ the chemical potential. 18

19 (a) Calculatetheaverageparticledensity n(µ) = N(µ) /L 2 fortheidealtwo-dimensional Bose gas. What is the maximal density you obtain without a macroscopic occupation of the k = 0 state? What does this imply for the possibility of Bose-Einstein condensation in a two-dimensional ideal Bose gas? (b) Now consider an ideal Bose gas in one dimension, on a line of length L and with periodic boundary conditions. In one dimension, the single-particle states are labeled by the wavenumber k, which takes the values k = 2πj/L, with j an integer. Calculate the average particle density n(µ) = N(µ) /L. What is the maximal density you obtain without a macroscopic occupation of the k = 0 state? What does this imply for the possibility of Bose-Einstein condensation in a one-dimensional ideal Bose gas? 19