Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Implementation of Second Quantization for Fermionic systems in Python : An application to the Tight-binding and Hubbard models

Jonuzi, Leon

Published Jan 1, 2025• 0 citations
OpenAlex

Abstract

The Quantum-Mechanical description of subatomic particles is the most accurate theory of these particle’s behavior. The first quantization is the standard method for solving the Schrodinger equation, but the dimension of the partial differential equation increases rapidly, especially when dealing with more than one particle. These differential equations become difficult to solve and, although there exist methods in this formalism, a new method is approached. This thesis starts with an introduction where the first quantization is discussed in multi-particle physics and electronic systems. The second quantization formalism is then introduced and discussed with some definitions and notation. Some applications of the new formalism are then discussed, such as some spin operators and the number operator. Some new models and operators are introduced, such as the tight-binding model, the Hubbard model, and the number operator. Some numerical applications are also discussed for these models and operators. The second quantization formalism is then implemented in Python, and some analysis is performed for the tight-binding and Hubbard model. In other words, this thesis focuses on the implementation of the second quantization formalism for fermions in Python and is then used to analyze two models. It is also a first step into the multi-particle quantum mechanics.

Related papers

Lecture 9: Second Quantization, Fock Space, the Tight Binding Hamiltonian, Two and N Particle Operators, and the Hubbard Model

Travis S. Taylor

In this chapter, we explore the advanced formalism of second quantization, which provides a powerful framework for describing quantum systems with variable particle numbers. We begin with an overview of how operators are second quantized, introducing the essential tools of creation and annihilation operators. These operators form the foundation of Fock space, a versatile representation that accommodates the quantum states of systems with differing particle counts.

CRC Press eBooks•Nov 21, 2025•0 citations

Second quantization

Alexander Altland, Benjamin D. Simons

The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focuses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti) ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.

Cambridge University Press eBooks•Mar 11, 2010•2 citations

The Schrödinger equation written in the second quantization formalism: derivation from first principles

Olavo Leopoldino da Silva Filho, S S João Augusto, Marcello Caio Ramon e Barros Ferreira

Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space $(Q,P)$, obtained by a canonical transformation upon the original phase space $(q,p)$, using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators $(\hat{q},\hat{p})$ for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.

Revista do Professor de Física•Apr 17, 2021•0 citations

Second quantization

Yuli V. Nazarov, Jeroen Danon

In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.

Cambridge University Press eBooks•Jan 3, 2013•0 citations

Second quantization

Alexander Altland, Benjamin D. Simons

The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focusses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti)ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body quasi-particle space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general. In fact, second quantization can be considered the first major cornerstone on which the theoretical framework of quantum field theory was built.

Cambridge University Press eBooks•Jun 1, 2006•6 citations

Numerically exact quantum dynamics for indistinguishable particles: The multilayer multiconfiguration time-dependent Hartree theory in second quantization representation

Haobin Wang, Michael Thoss

A new theory is proposed to accurately simulate quantum dynamics in systems of identical particles. It is based on the second quantization formalism of many-body quantum theory, in which the Fock space is represented by occupation-number states. Within this representation the overall Fock space can be formally decomposed into smaller subspaces, and the wave function can be expressed as a multilayer multiconfiguration Hartree expansion involving subvectors in these subspaces. The theory unifies the multilayer multiconfiguration time-dependent Hartree theory for both distinguishable and indistinguishable particles. Specific formulations are given for systems of identical fermions, bosons, and combinations thereof. Practical implementations are discussed, especially for the case of fermions, to include the operator algebra that enforces the symmetry of identical particles. The theory is illustrated by a numerical example on vibrationally coupled electron transport.

The Journal of Chemical Physics•Jul 13, 2009•182 citations
The Quantum-Mechanical description of subatomic particles is the most accurate theory of these particle’s behavior. The first quantization is the standard method for solving the Schrodinger equation, but the dimension of the partial differential equation increases rapidly, especially when dealing with more than one particle. These differential equations become difficult to solve and, although there exist methods in this formalism, a new method is approached. This thesis starts with an introduction where the first quantization is discussed in multi-particle physics and electronic systems. The second quantization formalism is then introduced and discussed with some definitions and notation. Some applications of the new formalism are then discussed, such as some spin operators and the number operator. Some new models and operators are introduced, such as the tight-binding model, the Hubbard model, and the number operator. Some numerical applications are also discussed for these models and operators. The second quantization formalism is then implemented in Python, and some analysis is performed for the tight-binding and Hubbard model. In other words, this thesis focuses on the implementation of the second quantization formalism for fermions in Python and is then used to analyze two models. It is also a first step into the multi-particle quantum mechanics.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focuses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti) ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general.
Courses about Quantum Mechanics are generally developed at the level of the mathematical formalism, its syntactic level, seldom treating the interpretation of the theory, its semantic level. In the majority of cases, this first formal approach is not followed by any other course addressing the numerous interpretations of the theory. It has become apparent that this strategy leads to a deficit in the professional formation of the students, exceedingly driving them to technical areas of application of Quantum Mechanics. However, applications such as those on quantum computing and entanglement, as they become more and more encrusted in the deepest foundations of the theory, are asking more and more for the development of this kind of ability. In this paper we develop an approach to deal with second quantization in the realm of Quantum Mechanics. We show that the second quantization Schrodinger equation can be mathematically derived from a classical Hamiltonian written in the phase space (Q,P)(Q,P)(Q,P), obtained by a canonical transformation upon the original phase space (q,p)(q,p)(q,p), using an axiomatic quantization procedure developed elsewhere. We apply this quantization process to a bosonic system, modelled by the harmonic oscillator problem. We then make reverse engineering to show that Schwinger's second quantization approach to fermionic systems furnishes the path to derive a Schrodinger equation explicitly written in terms of the usual momenta and positions operators (q^,p^)(\hat{q},\hat{p})(q^​,p^​) for such systems. Finally, we use these mathematical developments to address the commonly accepted statement that 'the spin has no classical analog'.
In this chapter we describe a technique to deal with identical particles that is called second quantization . Despite being a technique, second quantization helps a lot in understanding physics. One can learn and endlessly repeat newspaper-style statements particles are fields , fields are particles without grasping their meaning. Second quantization brings these concepts to an operational level. Since the procedure of second quantization is slightly different for fermions and bosons, we have to treat the two cases separately. We start with considering a system of identical bosons, and introduce creation and annihilation operators (CAPs) that change the number of particles in a given many-particle state, thereby connecting different states in Fock space. We show how to express occupation numbers in terms of CAPs, and then get to the heart of the second quantization: the commutation relations for the bosonic CAPs. As a next step, we construct field operators out of the CAPs and spend quite some time (and energy) presenting the physics of the particles in terms of these field operators. Only afterward, do we explain why all this activity has been called second quantization. In Section 3.4 we then present the same procedure for fermions, and we mainly focus on what is different as compared to bosons. The formalism of second quantization is summarized in Table 3.1.
The purpose of this chapter is to introduce and apply the method of second quantization, a technique that underpins the formulation of quantum many-particle theories. The first part of the chapter focusses on methodology and notation, while the remainder is devoted to the development of applications designed to engender familiarity with and fluency in the technique. Specifically, we will investigate the physics of the interacting electron gas, charge density wave propagation in one-dimensional quantum wires, and spin waves in a quantum Heisenberg (anti)ferromagnet. Indeed, many of these examples and their descendants will reappear as applications in our discussion of methods of quantum field theory in subsequent chapters. In the previous chapter we encountered two field theories that could conveniently be represented in the language of “second quantization,” i.e. a formulation based on the algebra of certain ladder operators â k . There were two remarkable facts about this formulation: firstly, second quantization provides a compact way of representing the many-body quasi-particle space of excitations; secondly, the properties of the ladder operators were encoded in a simple set of commutation relations (cf. Eq. (1.33)) rather than in some explicit Hilbert space representation. Apart from a certain aesthetic appeal, these observations would not be of much relevance if it were not for the fact that the formulation can be generalized to a comprehensive and highly efficient formulation of many-body quantum mechanics in general. In fact, second quantization can be considered the first major cornerstone on which the theoretical framework of quantum field theory was built.
A new theory is proposed to accurately simulate quantum dynamics in systems of identical particles. It is based on the second quantization formalism of many-body quantum theory, in which the Fock space is represented by occupation-number states. Within this representation the overall Fock space can be formally decomposed into smaller subspaces, and the wave function can be expressed as a multilayer multiconfiguration Hartree expansion involving subvectors in these subspaces. The theory unifies the multilayer multiconfiguration time-dependent Hartree theory for both distinguishable and indistinguishable particles. Specific formulations are given for systems of identical fermions, bosons, and combinations thereof. Practical implementations are discussed, especially for the case of fermions, to include the operator algebra that enforces the symmetry of identical particles. The theory is illustrated by a numerical example on vibrationally coupled electron transport.