An operator that acts on one particle at a time, like the kinetic energy or an outside potential, is a sum over the particles in the language of wavefunctions: $\hat{F}=\sum_{k}f(k)$. In Fock space it becomes $\hat{F}=\sum_{ij}\bra{i}f\ket{j}\hat{a}^{\dagger}_{i}\hat{a}_{j}$ — take a particle out of state $j$ and put it into state $i$, with the amplitude of the one-particle matrix element. The formula is the same for bosons and fermions, and it does not mention the number of particles: one operator serves every $N$. In the eigenbasis of $f$ it is diagonal, $\sum f_{i}\hat{n}_{i}$, each particle contributing the eigenvalue of its state. An interaction between pairs becomes $\tfrac12\sum\bra{ij}v\ket{kl}\hat{a}^{\dagger}_{i}\hat{a}^{\dagger}_{j}\hat{a}_{l}\hat{a}_{k}$: two out, two back in. Field operators $\hat{\psi}(x)=\sum\phi_{i}(x)\hat{a}_{i}$ remove a particle at a point, and $\hat{\psi}^{\dagger}(x)\hat{\psi}(x)$ counts the particles there. A simple model shows the method at work: particles hopping between neighbouring sites of a ring. Rewritten with waves running round the ring, the hopping is diagonal, with the energies $-2t\cos k$ of a band.

A shop’s price list works for any number of customers: it says what one item costs, and the till adds up whatever is bought. A one-body operator is such a price list: it says what happens to one particle, and the operators apply it to however many there are.

The energies $-2t\cos k$ of particles hopping round a ring, with $t=1$ and $k$ across from $-\pi$ to $$. A ring of eight sites allows the eight waves , the dots; a longer ring fills the dashed band more densely.

An operator $\sum_{k}f(k)$ acting on the particles one at a time, and an interaction $\frac12\sum_{k\ne l}v(k,l)$ between pairs, become in Fock space

• The same formulas hold for fermions with $\hat{c}$ in place of $\hat{a}$, and neither depends on $N$.
• In the eigenbasis of $f$, : free particles have .

Write each site operator as a sum of wave operators, $\hat{a}_{j}=N^{-1/2}\sum_{k}e^{ikj}\hat{a}_{k}$, with so that site is site 1. In each product the sum over sites keeps only equal , because the phases cancel otherwise. A hop one way contributes , the hop back , and together . The same steps hold for fermions.

A unitary change of basis from sites to waves round the ring.

A geometric sum of $N$-th roots of unity.

The sum over $j$ sets $k'=k$.

The hop back, the Hermitian conjugate of the last line.

Each wave is a one-particle eigenstate, which $N$ particles fill as bosons or as fermions.

The product $\hat{a}^{\dagger}_{i}\hat{a}_{j}$ takes a particle out of state and puts it into state . A one-body operator is a sum of such moves weighted by the matrix elements .

In the eigenbasis of $f$ the one-body operator is a sum of eigenvalues times occupations: each particle contributes the eigenvalue of its state.

Each term $\hat{a}^{\dagger}_{i}\hat{a}_{j}$ removes one particle and adds one, so one-body operators commute with the total number operator.

Particles hopping between neighbouring sites of a ring of $N$ sites have one-particle energies $-2t\cos k$, with $k=2\pi m/N$.

The field operator $\hat{\psi}(x)=\sum_{i}\phi_{i}(x)\hat{a}_{i}$ removes a particle at the point , and is the density of particles there.

Great work! You now know how to:

• write one-body and two-body operators with creation and annihilation operators
• see that they keep the number of particles
• diagonalise hopping on a ring into a band