Sheet 5 Due Date: 10:15 23/11/2018 J

Home , Ladder operator

Freie Universit¨at Berlin Tutorials for Advanced Quantum Mechanics Wintersemester 2018/19 Sheet 5 Due date: 10:15 23/11/2018 J. Eisert

1. Representing spins with fermions(4×2 points) 1 Spin- 2 lattice models are ubiquitous in the study of condensed matter physics and 1 quantum statistical mechanics. In the lectures you learned that a single spin- 2 system can be represented in C2 by associating the spin-up and spin-down states 1 0 with vectors | ↑i := and | ↓i := respectively which form a basis 0 1 and that measuring in the {| ↑i, | ↓i} basis is described by the Pauli z matrix1 σz. Measuring whether a particle is spin-up or spin-down is thus described by an z 1 z 1 operator S = 2 σ , i.e. an operator with eigenvalues ± 2 for the spin-up and spin down states respectively. One can also measure√ in the real and imaginary√ spin superposition bases |±i := (| ↑i ± | ↓i)/ 2 and |i±i := (| ↑i ± i| ↓i)/ 2 with x 1 x y 1 y corresponding operators S = 2 σ and S = 2 σ . These operators satisfy the canonical commutation relations i j k [S ,S ] = iijkS (1)

where ijk is the Levi-Civita symbol which is 0 if any two indices are the same and (−1)π(P ) where π(P ) is the parity of any permutation away from the order i, j, k = 1 x, y, z. These commutation relations define the algebra of spin- 2 observables in a similar manner to the bosonic and fermionic case. For spin ladder operators defined as S± = Sx ± iSy (2) (a) Justify the term spin ladder operators by finding the action of S± on the states | ↑i and | ↓i (b) Show that {S+,S−} = 1 (3) and [S+,S−] = 2Sz (4) which is another canonical way of defining the spin algebra. (c) The anti-commutation relations in (3) and the suggestive names might prompt us to propose a representation of the spin system in terms of fermions by associating the state | ↑i with an occupied fermionic particle state f †|0i := |1i and the state | ↓i with the vacuum f|1i := |0i. In this representation the spin raising and lowering operators would identified with fermionic creation + † − z † 1 and annihilation operators via S = f , S = f and S = f f − 2 . Using the fermionic anti-commutation relations2 show that under this definition the spin operators satisfy (4).

1 0 0 1 0 −i 1The Pauli matrices are given by σz = , σx = , σy = 0 −1 1 0 i 0 2 † † † {fj , fk } = δjk, {fj , fk} = {fj , fk } = 0

1 (d) Consider a 1-D chain of spins with sites labelled j = 1, 2, ..., N where the N- NN C2 site states live in the Hilbert space H = j=1 j . A spin ladder operator for just one lattice site, j, is given by the corresponding operator defined in (2) (i.e. the original definition in terms of Pauli matrices) on the Hilbert space, C2 + 1 + 1 j , tensored with the identity on all the others, e.g. S2 = ⊗ S ⊗ ⊗ ...⊗. Given the above results, we might be tempted to represent the spin raising and lowering operators on a site j with with fermionic creation and anni- + † − hilation operators for orbitals j = 1, 2, ..., N via Sj = fj , Sj = fj and z † 1 Sj = fj fj − 2 . Explain why the representation breaks down in this case. + + (Hint: consider the commutator [S1 ,S2 ]) 2. Representing spins with bosons(4 × 2 points) We can also construct representations of the spin algebra in terms of bosons, not 1 only for spin- 2 systems but for arbitrary spin-S systems. The state of a spin- S system is typically written |S, mi where m is the Sz spin component which 1 can take values −S, −(S − 1), ...S − 1,S. For spin- 2 particles the only allowed 1 values are ± 2 , while in general there are 2S + 1 possible values for m. This state must be an eigenstate of Sz and also the total spin operator defined as S2 := (Sx)2 + (Sy)2 + (Sz)2 satisfying,

Sz|S, mi = m|S, mi, S2|S, mi = S(S + 1)|S, mi (5)

(a) Using the bosonic commutation relations3 show that the representation

1/2 † Sˆ− = a† 2S − a†a , Sˆ+ = Sˆ− , Sˆz = S − a†a (6)

where a and a† are bosonic creation and annihilation operators reproduces (4). (Hint: you can prove this without explicitly expanding the square root!) (b) We now investigate a representation where one spin system is represented by two bosonic modes (or orbitals) deﬁned by operators (a, a†) and (b, b†) via † ˆ+ † ˆ− ˆ+ ˆz 1 † † the mapping S = a b, S = S , S = 2 a a − b b . Show via the bosonic commutation relations that,

a†S+m b†S−m |S, mi = |∅i (7) p(S + m)! p(S − m)!

satisfy the deﬁnitions in (5), where |∅i is the bosonic vacuum state. (c) Consider the operators given by the following linear combination of the bosonic operators from the previous question √ √ √ √ a1 = T a + Rb, b1 = T b − Ra (8)

with T,R ∈ R. Find the conditions on T and R such that a1 and b1 also satisfy the bosonic commutation relations.

3 † † † [aj , ak] = δjk,[aj , ak] = [aj , ak] = 0

2 (d) The above transformations describe many physical processes, for example the combination of two spatially distinct beams of light (represented by the modes a and b) being combined (or interfered) on a particular kind of glass cube that transmits or reﬂects each mode in a manner described by the co- eﬃcients T and R. If we apply this transformation and then measure the † † operator a1a1 − b1b1, what does this correspond to in terms of a spin mea- surement? Based upon your answer, how do we interpret the interference transformation in the spin picture?

3. Time evolution of the ﬁeld operators(4 points)

In lectures you saw that the Hamiltonian for interacting particles in a potential in terms of the ﬁeld operators was given by, Z 2∆ H = dξΨ†(ξ) −~ + V (ξ) Ψ(ξ) 2M 1 (9) 1 Z + dξdξ0Ψ† (ξ0)Ψ†(ξ)V (ξ, ξ0) Ψ(ξ)Ψ (ξ0) 2 2 Using the Heisenberg equations of motion which simplify in this case to ∂ i Ψ(ξ, t) = [H, Ψ(ξ, t)] (10) ~∂t and ﬁeld operator commutation relations,

[Ψ(ξ), Ψ(ξ0)] = Ψ†(ξ), Ψ† (ξ0) = 0 (11) Ψ(ξ), Ψ† (ξ0) = δ (ξ − ξ0) (12)

and the identity

[AB, C] = A[B,C] + [A, C]B (13)

show that (10) has the structure of a nonlinear Schr¨odinger equation, namely

∂ 2∆ i Ψ(ξ, t) = −~ + V (ξ) Ψ(ξ, t) ~∂t 2M 1 Z (14) 0 † 0 0 0 + dξ Ψ (ξ , t) V2 (ξ, ξ )Ψ(ξ , t) Ψ(ξ, t)