- Magnetic Excitons. - We show that one can associate the excited states of the system with quasi-particles that propagate along the chain.. - 20.1 The Molecule CsFeBr 3. - j = 1 in the basis. - What are the eigenvalues of ˆ J 2 and ˆ J z. - Write the action of the operators ˆ J. - ˆ J x ± i ˆ J y on the states | σ. - In the molecule Cs Fe Br 3 , the ion Fe 2+ has an intrinsic angular mo- mentum, or spin, equal to 1. - The molecule has a plane of symmetry, and the magnetic interaction Hamiltonian of the ion Fe 2+ with the rest of the molecule is. - 20.2 Spin–Spin Interactions in a Chain of Molecules. - We are only interested in the magnetic energy states of the chain, due to the magnetic interactions of the N Fe 2+ ions, each with spin 1.. - 1, to be the orthonormal basis of the states of the system. - it is an eigenbasis of the operators { J ˆ z n } where ˆ J n is the spin operator of the n-th ion (n = 1. - The magnetic Hamiltonian of the system is the sum of two terms ˆ H = H ˆ 0 + ˆ H 1 where. - Show that | σ 1 , σ 2. - σ N is an eigenstate of ˆ H 0 , and give the cor- responding energy value.. - What is the ground state of ˆ H 0 ? Is it a degenerate level?. - What is the energy of the first excited state of ˆ H 0 ? What is the degeneracy d of this level? We shall denote by E 1 the corresponding eigenspace of ˆ H 0 , of dimension d.. - Show that ˆ H 1 can be written as H ˆ 1 = A. - 20.3 Energy Levels of the Chain. - We now work in the subspace E 1 . - Owing to the periodicity of the chain, we define | N + 1. - Show that. - ψ n , where | ψ n is orthogonal to the subspace E 1. - Without giving the complete form of | ψ n , give an example of one of its components, and give the energy of the eigenspace of ˆ H 0 to which | ψ n belongs.. - Write the action of ˆ T and ˆ T † on the states | n. - Check that, in the subspace E 1 , H ˆ 1 and A( ˆ T + ˆ T. - Show that the eigenvalues λ k of ˆ T are the N -th roots of unity (we recall that N is assumed to be even):. - We seek, in E 1 , the 2N eigenvectors | q k. - (b) Show that. - (c) Show that the states | q k. - defined using (20.1) and (20.2) are ortho- normal.. - (d) Show that the vectors | q k. - are also eigenvectors of ˆ T † and ˆ T + ˆ T. - and give the corresponding eigenvalues.. - and write the expan- sion of the states | n. - in the basis | q k. - We limit ourselves to the first excited level of ˆ H 0 , and we want to calcu- late how the perturbation lifts the degeneracy of this level. - We recall that, in the degenerate case, first order perturbation theory consists in diagonaliz- ing the restriction of the perturbing Hamiltonian in the degenerate subspace of the dominant term ˆ H 0. - (a) Explain why the results of questions 3.3 and 3.5 above allow one to solve this problem.. - (c) Draw qualitatively the energies E(q k ) in terms of the variable q k which can be treated as a continuous variable, q k. - What is the degeneracy of each new energy level?. - 20.4 Vibrations of the Chain: Excitons. - We now study the time evolution of the spin chain.. - Suppose that at time t = 0, the system is in the state. - We assume that the initial state is | Ψ (0. - Show that P n (t) is the same for all sites of the chain.. - (b) The molecules of the chain are located at x n = na, where a is the lattice spacing. - Show that the probability amplitude α n (t) is equal to the value at x = x n of a monochromatic plane wave. - where C is a constant, q = q k , and x is the abscissa along the chain.. - (c) Show that Ψ k (x, t) is an eigenstate of the momentum operator ˆ p x. - h/i)∂/∂x along the chain.. - Show that the value of p(q) ensures the periodicity of Ψ k (x, t), i.e. - Ψ k (x, t), where L = N a is the length of the chain.. - In a more complete analysis, one can associate quasi-particles to the magnetic excitations of the chain. - At very low temperatures, T ≈ 1.4 K, the chain is in the ground state of ˆ H 0 . - If low energy neutrons collide with it, they can create excitons whose energy and momentum can be determined by measuring the recoil of the neutrons. - Experimental measurement of the excitation energy E(q) as a function of q between − π and 0. - (b) What do you think of the approximation D A and of the comparison between theory and experiment? How could one improve the agreement between theory and experiment?. - (c) Is it justified to assume that the chain is in its ground state when it is at thermal equilibrium at 1.4 K? We recall the Boltzmann factor:. - We assume that N 1, that the coefficients ϕ k have significant values only in a close vicinity of some value k = k 0 , or, equivalently, q ≈ q 0 , and that, to a good approximation, in this vicinity,. - Show that the probability P n (t) of finding σ n = +1 at time t is the same as the probability P n (t ) of finding σ n = +1 at another time t whose value will be expressed in terms of t and of the distance between the sites n and n. - Interpret the result as the propagation of a spin excitation wave along the chain. - We now assume that the initial state is | Ψ (0. - (b) Calculate the probabilities P 1 (t) and P 2 (t), in the case N = 2, and inter- pret the result.. - (c) Calculate P 1 (t) in the case N = 8. - Is the evolution of P 1 (t) periodic?. - πx 2 cos(x − nπ/2 − π/4) if x >. - 20.5 Solutions. - Section 20.1: The Molecule CsFeBr 3. - The eigenstates are the states | σ . - The state | 0 corresponds to the eigenvalue E = 0, whereas. - Section 20.2: Spin–Spin Interactions in a Chain of Molecules 20.2.1. - The energy is D, and the degeneracy 2N, since there are N possible choices of the non-vanishing σ n , and two values ± 1 of σ n . - A direct calculation leads to the result.. - Section 20.3: Energy Levels of the Chain. - The action of the perturbing Hamiltonian on the basis states is, set- ting. - The definition of ˆ T, ˆ T † and | n. - obviously have the same matrix elements in the subspace E 1. - Conversely, we will see in the following that each N-th root of unity is an eigenvalue.. - N, the eigenvectors are of unit norm and we recover the solution given in the text of the problem.. - are eigenvectors of ˆ T † with the complex conjugate eigenvalues λ ∗ k . - Therefore they are also eigenvectors of ˆ T + ˆ T † with the eigen- value λ k + λ ∗ k = 2 cos q k. - (e) From the definition of the vectors, we have n. - The restriction of ˆ H 1 to the subspace E 1 is identical to A( ˆ T + ˆ T. - In E 1 , the operator A( ˆ T + ˆ T. - is diagonal in the basis | q k. - (20.3) corresponding to the states | q k. - Section 20.4: Vibrations of the Chain: Excitons 20.4.1. - At time t the state of the chain is (cf. - α n | 2 = N 1 , which is the same on each site.. - (b) In the expression. - 20.2, one finds D + 2A eV, and D − 2A eV. - Second order perturbation theory is certainly necessary to account quantitatively for the experimental curve which has a steeper shape than a sinusoid in the vicinity of q. - (q − q 0 )u 0 in the vicinity of q 0 , we obtain. - Since the global phase factor does not contribute to the probability, one has P n (t. - This corresponds to the propagation of a wave along the chain, with a group velocity. - π/2 and a = 0.7 nm, we find v g ∼ 1500 ms − 1 . - These are the usual oscillations of a two- state system, such as the inversion of the ammonia molecule.
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