Second Quantization for Fermionic Operators

Recommended resources

The freely available resources this lesson will direct you to.

Hello! Welcome to the first lesson of our module on many-body formalism.

In the previous module, we explored the optical properties of solids, culminating in nonlinear phenomena like Second Harmonic Generation. While powerful, those semi-classical models have their limits. To truly understand the collective behavior of electrons that governs phenomena like screening, magnetism, and superconductivity, we need a more fundamental quantum mechanical framework.

This lesson introduces that framework. Our learning goal is to express single-particle and two-particle operators in the second quantization formalism for fermions. This is the language of modern condensed matter theory. Given your background in Quantum Field Theory, you'll find the concepts of creation and annihilation operators familiar. Here, we'll apply them not to fundamental fields in spacetime, but to the many-electron system within a crystal.

1. Why Second Quantization?

The "first quantization" approach you learned in introductory quantum mechanics, which describes a system of $N$ particles with a wavefunction $Ψ (r_{1}, σ_{1}, \dots, r_{N}, σ_{N})$ , becomes incredibly cumbersome for condensed matter systems where $N \sim 1 0^{23}$ . There are three main problems:

Indistinguishability: For identical particles (fermions), the wavefunction must be antisymmetric under the exchange of any two particles. This means representing even a simple state requires a Slater determinant, which involves $N!$ terms—an impossibly large number.
Useless Information: The wavefunction keeps track of which particle is at which position. But since the particles are indistinguishable, this is information we don't need and can't measure. All that matters is which single-particle states are occupied.
Variable Particle Number: Many theoretical frameworks (like the grand canonical ensemble) and physical situations (like particle-hole excitations) involve states where the number of particles is not fixed. The first quantization formalism is inherently built for a fixed number of particles.

Second quantization elegantly solves all these issues by working in a space—the Fock space—where the fundamental entities are not the particles themselves, but the occupations of single-particle states.

For a more detailed discussion on the limitations of the first quantization approach, you can read the first part of the following resource.

Why we need 2nd quantization for systems with N >> 1 particles

These notes from the University of British Columbia clearly lay out the challenges of the N-particle wavefunction approach.

Please read Section 2, titled 'Why we need 2nd quantization for systems with N >> 1 particles' (pages 3-5).

2. The Language: Fock Space and Fermionic Operators

Let's quickly review the building blocks. We consider a complete, orthonormal basis of single-particle states, labeled by an index $α$ . For electrons in a solid, $α$ could represent $(k, σ)$ , the momentum and spin.

Fock Space: The state of the many-body system is represented by a vector $∣ n_{1}, n_{2}, \dots, n_{α}, \dots ⟩$ in Fock space, where $n_{α}$ is the number of particles occupying the single-particle state $α$ .
Fermionic Operators: For fermions, the occupation number $n_{α}$ can only be 0 or 1 (Pauli exclusion principle). We define two operators for each state $α$ :
- $c_{α}^{†}$ : The creation operator that adds a particle to state $α$ .
- $c_{α}$ : The annihilation operator that removes a particle from state $α$ .
Anticommutation Relations: The fermionic nature of electrons is entirely captured by their algebra: ${c_{α}, c_{β}^{†}} \equiv c_{α} c_{β}^{†} + c_{β}^{†} c_{α} = δ_{α β}$ ${c_{α}, c_{β}} = 0 and {c_{α}^{†}, c_{β}^{†}} = 0$ The last relation immediately implies that $(c_{α}^{†})^{2} = 0$ , which is the mathematical statement of the Pauli principle: you cannot create two fermions in the same state.
Number Operator: The operator $n^_{α} = c_{α}^{†} c_{α}$ counts the number of particles in state $α$ . Its eigenvalues are 0 and 1.
States: Any N-particle state can be built by acting on the vacuum state $∣0 ⟩$ (the state with no particles) with creation operators. For example, a state with particles in states $α$ , $β$ , and $γ$ is written as $c_{γ}^{†} c_{β}^{†} c_{α}^{†} ∣0 ⟩$ . The order matters, as swapping two operators introduces a minus sign.

3. Single-Particle Operators

A single-particle operator is one that can be written as a sum of identical operators, each acting on a single particle. Examples include the total kinetic energy $T^= \sum_{i = 1}^{N} t^_{i}$ or the potential energy from an external field $U^= \sum_{i = 1}^{N} u^(r_{i})$ .

In second quantization, any such operator $O^^{(1)} = \sum_{i} o^_{i}$ is expressed as:

O^^{(1)} = α, β \sum ⟨ α ∣ o^∣ β ⟩ c_{α}^{†} c_{β}

where $⟨ α ∣ o^∣ β ⟩$ is the matrix element of the single-particle operator $o^$ between the basis states $α$ and $β$ .

Interpretation: This form is very intuitive. The term $c_{α}^{†} c_{β}$ represents a process where a particle is annihilated in state $β$ and created in state $α$ . The "amplitude" for this process is given by the corresponding matrix element. The total operator is a sum over all possible such transitions.

The following video provides a rigorous and clear derivation of this fundamental result. Given your preference for formal derivations, it's worth watching in full.

One body operators in second quantization

This video, 'One body operators in second quantization' from the 'Professor M does Science' channel, provides a complete derivation showing how the first-quantized form is equivalent to the second-quantized form.

Watch the video from the beginning (00:02 to 20:02). The first few minutes motivate the problem, the main part (04:02 - 18:13) contains the mathematical proof, and the final segment provides a conceptual summary.

Example: Kinetic Energy

Let's apply this to the kinetic energy operator, $T^$ . We choose the basis of plane waves, so our single-particle states are $∣ k, σ ⟩$ . The single-particle operator is $t^= p^^{2} /2 m$ .

The matrix elements are diagonal in this basis:

⟨ k, σ ∣ \frac{p ^ ^{2}}{2 m} ∣ k^{'}, σ^{'} ⟩ = \frac{ℏ ^{2} k ^{2}}{2 m} δ_{k, k^{'}} δ_{σ, σ^{'}}

Substituting this into the general formula for a single-particle operator:

T^= k σ, k^{'} σ^{'} \sum (\frac{ℏ ^{2} k ^{2}}{2 m} δ_{k, k^{'}} δ_{σ, σ^{'}}) c_{k σ}^{†} c_{k^{'} σ^{'}}

The Kronecker deltas collapse the double sum into a single sum:

T^= k, σ \sum \frac{ℏ ^{2} k ^{2}}{2 m} c_{k σ}^{†} c_{k σ} = k, σ \sum \frac{ℏ ^{2} k ^{2}}{2 m} n^_{k σ}

This final form is wonderfully simple: the total kinetic energy is just the sum of the kinetic energy of each state, $\frac{ℏ ^{2} k ^{2}}{2 m}$ , multiplied by the number of particles in that state, $n^_{k σ}$ .

4. Two-Particle Operators

Two-particle operators describe interactions between pairs of particles. The most important example is the Coulomb interaction, $V^= \frac{1}{2} \sum_{i \neq = j} V (r_{i}, r_{j})$ .

The general second-quantized form for a two-particle operator is:

O^^{(2)} = \frac{1}{2} α, β, γ, δ \sum ⟨ α β ∣ o^∣ δ γ ⟩ c_{α}^{†} c_{β}^{†} c_{γ} c_{δ}

where $⟨ α β ∣ o^∣ δ γ ⟩$ is the two-particle matrix element.

Important Note on Ordering: Pay close attention to the order of the annihilation operators: $c_{γ} c_{δ}$ . This corresponds to the ket $∣ δ γ ⟩$ . The bra $⟨ α β ∣$ corresponds to the creation operators $c_{α}^{†} c_{β}^{†}$ . Getting this order wrong is a very common mistake.

Interpretation: The operator $c_{α}^{†} c_{β}^{†} c_{γ} c_{δ}$ describes a scattering event: two particles are annihilated in states $γ$ and $δ$ , and two particles are subsequently created in states $α$ and $β$ .

Example: Coulomb Interaction

Let's derive the form of the Coulomb interaction in the plane-wave basis. This is one of the most important results in many-body theory. The derivation involves calculating the two-particle matrix element and using the Fourier transform of the potential.

2ndQ.pdf

The notes '2ndQ.pdf' provide a clear derivation for the second-quantized form of a two-particle interaction. We will follow its logic to obtain the result for the Coulomb potential.

Please study Section 4, specifically subsection (c) on page 10, 'Two-particle operator'. Follow the calculation that starts with the matrix element (\langle \mathbf{k}_1, \sigma_1; \mathbf{k}_2, \sigma_2 | u | \mathbf{k}_3, \sigma_3; \mathbf{k}_4, \sigma_4 angle) and ends with Equation (13). I'll summarize the result and its meaning below.

As shown in the resource, the calculation yields the following expression for a general two-body interaction $u (r_{i} - r_{j})$ that is independent of spin:

V^= \frac{1}{2 V} k, k^{'}, q \sum σ, σ^{'} \sum u (q) c_{k + q, σ}^{†} c_{k^{'} - q, σ^{'}}^{†} c_{k^{'}, σ^{'}} c_{k, σ}

Here, $V$ is the volume of the system and $u (q) = \int d^{3} r e^{- i q \cdot r} u (r)$ is the Fourier transform of the interaction potential.

A diagram representing the two-particle scattering process described by the Coulomb interaction term. Two electrons with initial momenta $k$ and $k^{'}$ interact, exchanging a momentum $q$ . They emerge with final momenta $k + q$ and $k^{'} - q$ . Total momentum is conserved, and since the Coulomb interaction is spin-independent, the spins $σ$ and $σ^{'}$ are also conserved.

This form beautifully captures the physics: two particles scatter, exchanging momentum $q$ while conserving the total momentum of the pair. The strength of the scattering for a given momentum transfer $q$ is determined by $u (q)$ . For the Coulomb potential $u (r) = e^{2} / (4 π ϵ_{0} r)$ , the Fourier transform is $u (q) = e^{2} / (ϵ_{0} q^{2})$ .

Test your understanding!

Consider a one-dimensional tight-binding model where electrons can hop between nearest-neighbor sites on a chain. The single-particle hopping Hamiltonian in first quantization would be $H^= - t \sum_{i, σ} (∣ site i + 1 ⟩ ⟨ site i ∣ + ∣ site i ⟩ ⟨ site i + 1∣)$ . Using the general formula for a single-particle operator, what is the second-quantized form of this Hamiltonian? Let $c_{iσ}^{†}$ and $c_{iσ}$ be the creation and annihilation operators for an electron at site $i$ with spin $σ$ .

Show answer

The basis states are the localized site states $∣ i, σ ⟩$ . The operator is $H^= \sum_{α, β} ⟨ α ∣ h^∣ β ⟩ c_{α}^{†} c_{β}$ .
The only non-zero matrix elements of the single-particle Hamiltonian $h^$ are $⟨ i + 1, σ ∣ h^∣ i, σ^{'} ⟩ = - t δ_{σ σ^{'}}$ and $⟨ i, σ ∣ h^∣ i + 1, σ^{'} ⟩ = - t δ_{σ σ^{'}}$ .
Plugging these into the formula gives:

H^= i, σ \sum (- t c_{i + 1, σ}^{†} c_{i, σ} - t c_{i, σ}^{†} c_{i + 1, σ}) = - t i, σ \sum (c_{i + 1, σ}^{†} c_{i, σ} + h.c.)

where h.c. stands for Hermitian conjugate. This describes electrons hopping between adjacent sites.

5. Connection to Quantum Field Theory: Field Operators

Your QFT background makes it natural to connect this to field operators. We can define fermionic field operators that create or annihilate a particle at a specific position $r$ :

Ψ_{σ}^{†} (r) = \frac{1}{V} k \sum e^{- i k \cdot r} c_{k σ}^{†} and Ψ_{σ} (r) = \frac{1}{V} k \sum e^{i k \cdot r} c_{k σ}

These operators are the non-relativistic analogues of the Dirac field operators. They satisfy the canonical equal-time anticommutation relation ${Ψ_{σ} (r), Ψ_{σ^{'}}^{†} (r^{'})} = δ_{σ σ^{'}} δ (r - r^{'})$ .

Using these operators, we can write the full Hamiltonian for an interacting electron gas in a very compact and familiar form:

H^= σ \sum \int d^{3} r Ψ_{σ}^{†} (r) (- \frac{ℏ ^{2} \nabla ^{2}}{2 m}) Ψ_{σ} (r) + \frac{1}{2} σ, σ^{'} \sum \int d^{3} r \int d^{3} r^{'} Ψ_{σ}^{†} (r) Ψ_{σ^{'}}^{†} (r^{'}) u (r - r^{'}) Ψ_{σ^{'}} (r^{'}) Ψ_{σ} (r)

This elegant expression is the starting point for many advanced many-body techniques.

Conclusion

This lesson is a crucial first step into the formal world of many-body physics. You have learned the "grammar" needed to write down the quantum mechanical laws governing interacting electrons.

Key Takeaways:

Second quantization provides a natural and efficient language for many-particle systems, automatically handling particle indistinguishability and allowing for variable particle numbers.
Single-particle operators take the general form $O^^{(1)} = \sum_{α, β} ⟨ α ∣ o^∣ β ⟩ c_{α}^{†} c_{β}$ , representing transitions between single-particle states.
Two-particle operators take the form $O^^{(2)} = \frac{1}{2} \sum_{α, β, γ, δ} ⟨ α β ∣ o^∣ δ γ ⟩ c_{α}^{†} c_{β}^{†} c_{γ} c_{δ}$ , representing scattering events between pairs of particles.
The Hamiltonian for the interacting electron gas can be expressed compactly using either momentum-space operators ( $c_{k σ}$ ) or real-space field operators ( $Ψ_{σ} (r)$ ).

Preview of the Next Lesson

Now that we can write down the full Hamiltonian for an interacting many-electron system, the next question is: how do we solve it? Exact solutions are impossible, so we must turn to approximations. In the next lesson, we will use the tools developed today to tackle the first and most fundamental approximation scheme in many-body physics. You will learn to derive the Hartree-Fock equations from a variational principle applied to a many-electron Hamiltonian. This will be our first concrete application of the second quantization formalism to find an approximate ground state of an interacting system.