10.3 Second quantization of bosonic and fermionic systems

Second quantization is a game-changer for dealing with multi-particle systems in quantum mechanics. It shifts our focus from individual particles to the overall state of the system, making it easier to handle systems with varying particle numbers.

This approach is particularly useful for studying many-body physics and collective phenomena. By using creation and annihilation operators, we can automatically incorporate the symmetry properties of bosons and fermions, simplifying calculations and revealing emergent properties in complex systems.

Second Quantization: Concept and Advantages

Introduction to Second Quantization

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• Second quantization is a formalism in quantum mechanics that describes multi-particle systems using creation and annihilation operators acting on a vacuum state
• In second quantization, the focus is on the states of the system rather than the individual particles, making it easier to handle systems with variable numbers of particles
• The vacuum state represents the state with no particles, and creation operators add particles to the system while annihilation operators remove particles from the system

Advantages of Second Quantization over First Quantization

• Second quantization allows for a more compact and efficient description of multi-particle systems compared to first quantization, especially when dealing with identical particles
• The symmetry properties of bosons and fermions are automatically incorporated into the formalism of second quantization through the commutation and anticommutation relations of the creation and annihilation operators
• Second quantization simplifies the calculation of matrix elements for multi-particle operators and facilitates the study of systems with variable numbers of particles
• The formalism of second quantization is particularly useful in the study of many-body physics, where collective phenomena and emergent properties arise from the interactions between large numbers of particles

Single-Particle Operators in Second Quantization

Creation and Annihilation Operators

• Creation operators (a^† for bosons, c^† for fermions) add a particle to the system in a specific quantum state, while annihilation operators (a for bosons, c for fermions) remove a particle from a specific quantum state
• The action of creation and annihilation operators on the vacuum state and multi-particle states follows specific rules depending on the type of particle (boson or fermion)
• For bosons, multiple particles can occupy the same quantum state, while for fermions, the Pauli exclusion principle prohibits more than one particle from occupying the same quantum state
• The creation and annihilation operators satisfy commutation relations for bosons and anticommutation relations for fermions, which ensure the proper symmetry properties of the multi-particle wave functions

Expressing Single-Particle Operators

• Single-particle operators, such as position, momentum, and energy, can be expressed as combinations of creation and annihilation operators
• The matrix elements of single-particle operators in the second-quantized formalism are related to the matrix elements in the first-quantized formalism through the use of creation and annihilation operators
• For example, the position operator $\hat{x}$ can be expressed as $\hat{x} = \sum_{ij} \langle i | \hat{x} | j \rangle a_i^† a_j$ for bosons and $\hat{x} = \sum_{ij} \langle i | \hat{x} | j \rangle c_i^† c_j$ for fermions, where $|i\rangle$ and $|j\rangle$ are single-particle states
• Similarly, other single-particle operators can be expressed in terms of creation and annihilation operators, allowing for their study within the second-quantized formalism

Second-Quantized Hamiltonian for Bosons and Fermions

Non-Interacting Hamiltonian

• The second-quantized Hamiltonian consists of terms involving creation and annihilation operators, representing the kinetic energy, potential energy, and interactions between particles
• For non-interacting systems, the second-quantized Hamiltonian is a sum of single-particle Hamiltonians, with each term expressed using creation and annihilation operators
• The non-interacting Hamiltonian for bosons is given by $\hat{H}_0 = \sum_i \epsilon_i a_i^† a_i$, where $\epsilon_i$ is the energy of the single-particle state $|i\rangle$
• The non-interacting Hamiltonian for fermions is given by $\hat{H}_0 = \sum_i \epsilon_i c_i^† c_i$, following a similar structure to the bosonic case

Interacting Hamiltonian

• In the presence of interactions, additional terms involving products of creation and annihilation operators are included in the Hamiltonian to represent particle-particle interactions
• The specific form of the interaction terms depends on the nature of the system (e.g., Coulomb interaction for charged particles, contact interaction for neutral atoms)
• For example, the Hamiltonian for a system of interacting bosons can include terms like $\hat{H}_{\text{int}} = \frac{1}{2} \sum_{ijkl} V_{ijkl} a_i^† a_j^† a_k a_l$, where $V_{ijkl}$ represents the interaction strength between particles in states $|i\rangle$, $|j\rangle$, $|k\rangle$, and $|l\rangle$
• Similarly, the Hamiltonian for interacting fermions can include interaction terms with appropriate creation and annihilation operators and interaction strengths

Properties of Second-Quantized Operators

Commutation and Anticommutation Relations

• Bosonic creation and annihilation operators satisfy the commutation relations: $[a_i, a_j^†] = \delta_{ij}$, $[a_i, a_j] = [a_i^†, a_j^†] = 0$, where $\delta_{ij}$ is the Kronecker delta
• Fermionic creation and annihilation operators satisfy the anticommutation relations: $\{c_i, c_j^†\} = \delta_{ij}$, $\{c_i, c_j\} = \{c_i^†, c_j^†\} = 0$, where $\{A, B\} = AB + BA$ is the anticommutator
• The commutation and anticommutation relations ensure the proper symmetry properties of bosonic and fermionic wave functions, respectively
• These relations also lead to the Pauli exclusion principle for fermions, which states that no two identical fermions can occupy the same quantum state

Number Operator and Particle Conservation

• The number operator, defined as $\hat{N} = \sum_i a_i^† a_i$ for bosons and $\hat{N} = \sum_i c_i^† c_i$ for fermions, counts the total number of particles in the system
• The number operator commutes with the Hamiltonian in systems where the total number of particles is conserved, such as in closed systems without particle creation or annihilation processes
• The eigenvalues of the number operator correspond to the number of particles in each quantum state, and the total number of particles in the system is given by the sum of these eigenvalues
• In systems where particle number is not conserved, such as in quantum field theories or open systems with particle exchange, the number operator may not commute with the Hamiltonian

Applying Second Quantization to Multi-Particle Systems

Diagonalization of Non-Interacting Hamiltonians

• The second-quantized formalism allows for the study of various multi-particle quantum systems, such as harmonic oscillators, lattice models, and quantum field theories
• For non-interacting systems, the eigenstates and eigenvalues of the Hamiltonian can be obtained by diagonalizing the single-particle Hamiltonian matrix
• The diagonalization procedure involves finding a unitary transformation that transforms the creation and annihilation operators into a new basis where the Hamiltonian is diagonal
• In the diagonal basis, the eigenstates of the system are given by the occupation numbers of the single-particle states, and the eigenvalues correspond to the energies of these states

Perturbation Theory and Interacting Systems

• Perturbation theory can be applied in the second-quantized formalism to calculate corrections to the eigenstates and eigenvalues due to weak interactions
• The perturbation Hamiltonian, which represents the interaction terms, is treated as a small correction to the non-interacting Hamiltonian
• The corrections to the eigenstates and eigenvalues are calculated order by order in the perturbation strength using techniques such as Rayleigh-Schrödinger perturbation theory or the Dyson series
• Perturbative calculations in second quantization involve evaluating matrix elements of the perturbation Hamiltonian between the non-interacting eigenstates and summing over the relevant contributions

Applications in Many-Body Physics

• Second quantization techniques are particularly useful in the study of many-body physics, where collective phenomena and emergent properties arise from the interactions between large numbers of particles
• Examples of systems studied using second quantization include:
  • The Bose-Hubbard model for bosonic atoms in optical lattices, which exhibits phenomena such as superfluidity and Mott insulator transitions
  • The Fermi-Hubbard model for electrons in solids, which captures the interplay between kinetic energy and Coulomb interactions and is relevant for understanding metal-insulator transitions and magnetism
  • The BCS theory of superconductivity, which describes the formation of Cooper pairs and the emergence of a superconducting state in materials
  • Quantum field theories, such as quantum electrodynamics (QED) and quantum chromodynamics (QCD), which describe the fundamental interactions between particles and fields
• Second quantization provides a powerful framework for studying these systems, allowing for the calculation of observables, correlation functions, and response functions that characterize the properties and behavior of many-body quantum systems