Second quantization ( $\hat{ψ} (x) = \sum_{α} φ_{α} (x) a_{α}$ )

Second quantization describes identical-particle quantum states in Fock space using creation operators and annihilation operators. Relative to an orthonormal basis of one-particle states, a one-body operator becomes

\sum_{α β} h_{α β} a_{α}^{†} a_{β}

. The canonical commutation relation implements bosonic statistics, while the canonical anticommutation relations implement fermionic statistics. Expanding the quantized field in a complete basis gives a delta-function equal-time bracket; restricting the expansion to finitely many modes replaces that delta function by the corresponding projection operator kernel.

Table of contents

Two-mode approximation for a Bose gas ( $\hat{ψ} (r) ≃ φ_{L} (r) a_{L} + φ_{R} (r) a_{R}$ )

 0  0

Project a quantized field onto two low-energy orbitals. A sufficient regime has temperature, hopping and interaction shifts small compared with the energy gap to discarded orbitals. Small interwell contact interaction integrals can then be neglected while retaining a small quantum tunnelling matrix element. The projected field's operator commutator is the two-mode projection operator kernel, rather than a complete spatial Dirac delta function.

Bose-Hubbard model ( $H = - \sum_{ij} t_{ij} a_{i}^{†} a_{j} + \frac{U}{2} \sum_{i} n_{i} (n_{i} - 1) - μ \sum_{i} n_{i}$ )

 1  0

A lattice boson Hamiltonian operator combining hopping and on-site repulsion. The on-site interaction counts distinct particle pairs through

n_{i} (n_{i} - 1) /2

. It is the bosonic counterpart of the Hubbard model; projecting a continuum contact interaction onto localized orbitals gives

U = g \int ∣ φ_{i} ∣^{4}

when intersite overlaps are negligible.

Two-site Bose-Hubbard model ( $H = - t (a_{L}^{†} a_{R} + a_{R}^{†} a_{L}) + \frac{U}{2} \sum_{j = L, R} n_{j} (n_{j} - 1)$ )

 0  0

Two localized orbitals coupled by quantum tunnelling and on-site interactions give the smallest Bose-Hubbard model. A symmetric double-well potential with lowest one-particle energies

E_{0}, E_{1}

has

t = (E_{1} - E_{0}) /2

and common diagonal energy

(E_{0} + E_{1}) /2

. At fixed total particle number, that common diagonal term is constant. A fixed-number two-mode condensate gives a simple variational method for this model, while stronger interactions can require occupation-correlated trial quantum states.

Contact interaction (quantum mechanics, $H_{int} = \frac{g}{2} \int \hat{ψ}^{†} (x)^{2} \hat{ψ} (x)^{2} d x$ )

 0  0

In a one-dimensional identical-particle model, a contact interaction has two-body potential

U (x, x^{'}) = g δ (x - x^{'})

, using the Dirac delta distribution. For bosons, its second quantization is the displayed normally ordered Hamiltonian operator. Projection onto orthonormal modes gives

H_{int} = (g /2) \sum_{ijk l} I_{ijk l} a_{i}^{†} a_{j}^{†} a_{k} a_{l}

, where

I_{ijk l} = \int φ_{i}^{*} φ_{j}^{*} φ_{k} φ_{l} d x

. The overlap integrals determine which mode-changing processes survive. A finite-mode model describes that projection; discarded modes can still matter to the full physical system.

Momentum-space contact-interaction Hamiltonian ( $H_{int} = \frac{u}{2 V} \sum_{k, p, q} a_{k + q}^{†} a_{p - q}^{†} a_{p} a_{k}$ )

 0  0

For a periodic box of volume

V

, expanding

ψ (r) = V^{- 1/2} \sum_{k} e^{i k \cdot r} a_{k}

H_{int} = \frac{u}{2} \int d^{3} r ψ^{†} ψ^{†} ψψ

gives the displayed normal ordering. The spatial integral is

V δ_{k_{1} + k_{2}, k_{3} + k_{4}}

by Fourier orthogonality, enforcing momentum conservation and leaving the factor

V^{- 1}

. The coupling

u

in three dimensions has units of energy times volume.

Contact-interaction vacuum-energy renormalization

 0  0

A low-energy contact coupling is defined through a physical scattering length. Ultraviolet pieces of zero-point energy cannot be interpreted independently of the matching or counterterms fixing that coupling.

Two-mode Bose gas on a ring ( $H = E N_{1} + (U /4 π) [N (N - 1) + 2 N_{0} N_{1}]$ )

 0  0

Project a bosonic field on a ring onto angular-momentum modes zero and one. The kinetic level spacing is

E = ℏ^{2} / (2 m R^{2})

. The contact-interaction integral enforces angular momentum conservation, leaving two same-mode terms and four mixed terms. The displayed projected Quantum Hamiltonian follows by normal ordering. Couplings to discarded modes belong to the full model and can limit the two-mode approximation.

Contact interaction between two box modes ( $I_{1111} = I_{2222} = 3/ (2 L), I_{1122} = 1/ L$ )

 0  0

For distinct box modes

φ_{1} = 2/ L sin (p π x / L)

and

φ_{2} = 2/ L sin (q π x / L)

, the displayed contact interaction overlaps produce self terms

3 g \sum_{j} n_{j} (n_{j} - 1) / (4 L)

, a density term

2 g n_{1} n_{2} / L

, and pair conversion

g (a_{1}^{† 2} a_{2}^{2} + a_{2}^{† 2} a_{1}^{2}) / (2 L)

. Pair conversion preserves total particle number but prevents diagonalization by individual occupation numbers. The additional overlaps are

I_{1112} = - δ_{q, 3 p} / (2 L)

and

I_{1222} = - δ_{p, 3 q} / (2 L)

, following from

sin^{3} u = (3 sin u - sin 3 u) /4

. Third-harmonic mode pairs therefore also permit single-particle transfer assisted by the other particles. For the first two box modes these additional terms vanish.

Contact interaction between two ring modes ( $H_{int} = \frac{g}{2 L} [n_{1} (n_{1} - 1) + n_{2} (n_{2} - 1) + 4 n_{1} n_{2}]$ )

 0  0

Two distinct normalized plane waves on a periodic one-dimensional ring of length

L

give the displayed projected contact interaction. Momentum conservation excludes pair conversion between these two modes. Four mixed operator orderings give the coefficient of

n_{1} n_{2}

. At fixed particle number

N

, the energy is

g [N (N - 1) + 2 n_{1} (N - n_{1})] / (2 L)

. For repulsive coupling, its minima are the two endpoint Fock states with all particles in one mode; a one-particle energy difference selects the lower mode. This ground-state statement concerns the two-mode projection.

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Second quantization by

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Second quantization is a formalism used in quantum mechanics and quantum field theory to describe and manipulate systems with varying particle numbers. It is particularly useful for dealing with many-body systems, where traditional first quantization methods become cumbersome. In the first quantization approach, particles are described by wave functions, and the focus is on the states of individual particles. However, this approach struggles to accommodate phenomena like particle creation and annihilation, which are crucial in fields like quantum field theory.

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Second quantization by

Ciro Santilli  40 Updated 2025-07-16

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www.quora.com/How-are-quantum-fields-quantized-to-describe-particles

Second quantization also appears to be useful not only for relativistic quantum mechanics, but also for condensed matter physics. The reason is that the basis idea is to use the number occupation basis. This basis is:

convenient for quantum field theory because of particle creation and annihilation changes the number of particles all the time
convenient for condensed matter physics because there you have a gazillion particles occupying entire energy bands

Bibliography:

www.youtube.com/watch?v=MVqOfEYzwFY "How to Visualize Quantum Field Theory" by ZAP Physics (2020). Has 1D simulations on a circle. Starts towards the right direction, but is a bit lacking unfortunately, could go deeper.

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