# Appendix A (p.242) Dirac bra-ket operator representation of one-particle Hamiltonians

# Appendix A (p.242) Dirac bra-ket operator representation of one-particle Hamiltonians

Throughout this book electronic models of conjugated polymers are developed within the number or second quantization representation. This representation is particularly powerful for treating many-body problems. However, as it is less familiar than a first quantization approach, this appendix explains the equivalence of the two approaches for single particle Hamiltonians. We take two examples: the fermion noninteracting (or Hückel) Hamiltonian and the Frenkel exciton model.

# A.1 The Hückel Hamiltonian

The Hückel Hamiltonian, described in Chapter 3, is

*t*

_{n}=

*t*(1 +

*δ*

_{n}).

As explained in Section 2.4, ${c}_{n\sigma}^{\u2020}$ creates an electron with spin *σ* in the spin-orbital, *χ* _{n}(**r**,*σ*). We define the Dirac ket state as

*n, σ*〉 is formally equivalent to the spin orbital

*χ*

_{n}(

**r**,

*σ*).

The bra state, 〈*n, σ*|, is the conjugate to the ket state. The scalar product of a bra and ket is defined as

If the states form an orthonormal set then,

For an orthonormal set we notice that the operator

*n, σ*〉 onto the state |

*m, σ*〉: (p.243)

Thus, |*m, σ*〉 〈*n, σ*| and ${c}_{m\sigma}^{\u2020}{c}_{n\sigma}$ are equivalent: both have the effect of transferring an electron from the spin-orbital *χ* _{n}(**r**,*σ*) to the spin-orbital *χ* _{m}(**r**, *σ*).

We can therefore express the Hückel Hamiltonian (eqn (A.1)) as

As described in Section 3.4.1, for cyclic undimerized chains this Hamiltonian is diagonalized by the Bloch states,

*N*is the number of sites. To demonstrate this, consider

# A.2 The Frenkel exciton Hamiltonian

The Frenkel exciton Hamiltonian,

*mth*molecule. The ket state

*m*th molecule. (In analogy to eqn (A.2), we may regard $\prod _{n=1}^{N}|\text{GS}{\u3009}_{n}$ as the vacuum state.)

(p.244) Equation (A.11) is equivalent to

For a dimer it is easy to show that the symmetric and antisymmetric states

*H*with energies (Δ +

*J*) and (Δ −

*J*), respectively (with

*J*

_{mn}=

*J*

_{nm}=

*J*).