Second Quantization of Single-Particle Operators

Hello! Welcome to our fourth lesson. In our previous work, we established a robust framework for representing many-fermion states in Fock space. We saw how applying a sequence of creation operators to the vacuum, such as $c_{ν_{N}}^{†} \dots c_{ν_{1}}^{†} ∣0 ⟩$ , provides a compact and powerful equivalent to the Slater determinants of first quantization.

Having learned how to represent the states of a many-body system, we now turn our attention to representing the physical observables that act upon them. This lesson's goal is to learn how to express single-particle operators—quantities like kinetic energy or potential energy that are sums of contributions from each particle individually—in the language of second quantization. This is a critical step towards our ultimate goal of constructing and understanding the Hubbard Hamiltonian.

From First to Second Quantization: The General Rule

In first quantization, a single-particle operator for an N-particle system is written as a sum over all particles:

O^= i = 1 \sum N o^_{i}

Here, $o^_{i}$ is the operator acting only on the coordinates of the i-th particle. Examples include the total kinetic energy, $T^= \sum_{i} \frac{p ^ _{i}^{2}}{2 m}$ , or the total potential energy in an external field, $V^= \sum_{i} V (r^_{i})$ .

Working with this form in first quantization can be cumbersome due to the need to explicitly maintain the wavefunction's antisymmetry. The beauty of second quantization is that it provides a universal representation for such operators that is independent of the particle number $N$ and has the symmetry properties built in.

The central result for this lesson is the "dictionary" that translates any single-particle operator into its second-quantized form. Given a complete orthonormal single-particle basis ${∣ α ⟩}$ , the operator $O^$ is expressed as:

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

Let's dissect this expression. The term $c_{α}^{†} c_{β}$ is an operator that first annihilates a particle in the single-particle state $∣ β ⟩$ and then creates a particle in the state $∣ α ⟩$ . In essence, it describes a process that moves a particle from state $∣ β ⟩$ to $∣ α ⟩$ . The coefficient $⟨ α ∣ o^∣ β ⟩$ is the single-particle matrix element of the operator $o^$ . It serves as the amplitude for this transition process.

The following video provides an excellent conceptual overview and a detailed mathematical derivation of this fundamental formula.

One body operators in second quantization

This video from "Professor M does Science" explains why a new representation is needed for one-body operators in many-particle systems and then derives the second-quantized form step-by-step.

First, watch the introduction here to see how a symmetric one-body operator is defined in first quantization. Next, watch the core derivation from the main result. This section introduces the second-quantized formula and gives a clear conceptual interpretation of what the terms c^\dagger_k c_l do. Finally, see what happens when the operator is expressed in the basis of its own eigenstates, from this segment. This shows how the general formula simplifies to a very intuitive form involving number operators.

The key insight from the video is that the operator $O^$ can be viewed as a sum of all possible single-particle transitions $∣ β ⟩ \to ∣ α ⟩$ , each weighted by its corresponding quantum mechanical amplitude.

A More Formal Derivation

Given your background in physics, you'll appreciate a more formal derivation of this central result. A particularly elegant way to prove the formula is to start with the operator expressed in the basis of its own eigenstates and then perform a basis transformation.

The following reading from Henrik Johannesson's lecture notes walks through this proof.

[PDF] “Second quantization” (the occupation-number representation)

This section of the lecture notes provides a rigorous proof for the second-quantized representation of single-particle operators.

Please read the section titled "Proof of the second-quantized representation for single-particle operators" (p. 13-14). The logic of the proof is as follows: First, consider the operator \hat{O} in the basis of its own eigenstates, \{|\tilde{\alpha}\rangle\}, where \hat{o}|\tilde{\alpha}\rangle = o_{\tilde{\alpha}}|\tilde{\alpha}\rangle. In this specific basis, the operator intuitively takes the form \hat{O} = \sum_{\tilde{\alpha}} o_{\tilde{\alpha}} \hat{n}_{\tilde{\alpha}}, where \hat{n}_{\tilde{\alpha}} is the number operator for that eigenstate. This is proven in Eq. (78). Then, the proof transforms this diagonal expression into an arbitrary basis \{|\alpha\rangle, |\beta\rangle, \dots\} using the basis transformation relations we've seen previously. As shown in Eq. (79), this transformation naturally recovers the general formula: \hat{O} = \sum_{\alpha, \beta} \langle \alpha | \hat{o} | \beta \rangle a^\dagger_\alpha a_\beta.

This method is powerful because it starts from a physically transparent, diagonal representation and uses the general machinery of quantum mechanics (basis transformations) to arrive at the universal off-diagonal form.

Example 1: The Kinetic Energy Operator

Let's make this concrete with the most important single-particle operator for our purposes: the kinetic energy, $T^= \sum_{i} p^_{i}^{2} /2 m$ . To write this in second-quantized form, we need the matrix elements $⟨ α ∣ \frac{p ^ ^{2}}{2 m} ∣ β ⟩$ .

The natural basis to use here is the momentum eigenbasis, where the single-particle states are plane waves labeled by momentum $k$ and spin $σ$ , i.e., $∣ α ⟩ = ∣ k, σ ⟩$ . In this basis, the kinetic energy operator is diagonal:

⟨ 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 our general formula:

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 cause the sums over $k^{'}$ and $σ^{'}$ to collapse, leaving a single sum:

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

where $ϵ (k) = \frac{ℏ ^{2} k ^{2}}{2 m}$ is the energy of a particle with momentum $k$ , and $n^_{k, σ} = c_{k, σ}^{†} c_{k, σ}$ is the number operator for that state.

This final form is incredibly intuitive: the total kinetic energy is just the sum of the energies of each single-particle state, weighted by the number of particles occupying that state. This diagonal structure will be the foundation for finding the band structure in the tight-binding model later on.

The following reading confirms this derivation and introduces another key operator.

1 Quantum mechanics for one particle - review of notation

These sections from Mona Berciu's lecture notes provide the general theorem and then apply it to the kinetic energy operator.

First, quickly look at Eq. (8) in the section The Theorem (p. 8). This is the general formula we have been discussing. Next, read the subsection on kinetic energy (p. 8-9). This walks through the derivation we just performed, leading to Eq. (12). Note the insightful comment that the total number operator, \hat{N} = \sum_i 1, becomes \hat{N} = \sum_\alpha \hat{n}_\alpha, which is the sum of all occupation numbers.

A Different Perspective: Field Operators

So far, we have worked in an abstract basis $∣ α ⟩$ . A particularly powerful choice of basis is the position basis $∣ r ⟩$ . The creation and annihilation operators associated with this basis are called field operators.

The fermionic field operator $Ψ^_{σ}^{†} (r)$ creates a particle with spin $σ$ at position $r$ . Similarly, $Ψ^_{σ} (r)$ annihilates a particle at that position. These operators are related to our original operators $c_{k, σ}^{†}$ via a Fourier transform.

In terms of field operators, the kinetic energy operator takes the form:

T^= σ \sum \int d^{3} r Ψ^_{σ}^{†} (r) (- \frac{ℏ ^{2} \nabla ^{2}}{2 m}) Ψ^_{σ} (r)

This form is especially useful in quantum field theory and advanced condensed matter theory. It emphasizes the local nature of the kinetic energy operator—it involves derivatives at a point $r$ —and it closely resembles the expression for kinetic energy in classical field theory, which is why second quantization is sometimes described as "quantizing the wave function itself."

The following brief reading introduces this concept formally.

1 Quantum mechanics for one particle - review of notation

This supplementary section introduces field operators and shows how to express operators like kinetic energy and density in the position basis.

Please read the first part of this section, from the introduction (p. 19-20). Focus on understanding the definition of the field operators in Eq. (19) and seeing how the kinetic energy and density operators are expressed using them in Eq. (22) and the subsequent examples.

Conclusion

In this lesson, we have established the essential "dictionary" for translating single-particle operators into the language of second quantization. This bridges the gap between the operators of single-particle quantum mechanics and the many-body formalism we are developing.

Key takeaways from our discussion include:

A single-particle operator $O^= \sum_{i} o^_{i}$ is written in second quantization as $O^= \sum_{α, β} ⟨ α ∣ o^∣ β ⟩ c_{α}^{†} c_{β}$ .
This expression represents a sum over all possible processes that move a particle from state $β$ to state $α$ , weighted by the single-particle matrix element for that process.
In the basis of its own eigenstates, the operator simplifies to the diagonal form $O^= \sum_{α} ϵ_{α} n^_{α}$ , which has a direct physical interpretation: summing the eigenvalues weighted by the occupation of each eigenstate.
The kinetic energy operator, a cornerstone of any Hamiltonian, takes the intuitive diagonal form $T^= \sum_{k, σ} ϵ (k) n^_{k, σ}$ in the momentum basis.

We have now covered single-particle operators. However, the most interesting physics arises from interactions between particles. In our next lesson, we will tackle the representation of two-particle operators, such as the Coulomb repulsion. This will provide us with the final component needed to assemble our first interacting many-body Hamiltonian.

Recommended resources

From First to Second Quantization: The General Rule

A More Formal Derivation

Example 1: The Kinetic Energy Operator

A Different Perspective: Field Operators

Conclusion