Second Quantization In Quantum Mechanics

[Lane Frisch]

Secondquantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as the wave functions of matter) are thought of as field operators, in a manner similar to how the physical quantities (position, momentum, etc.) are thought of as operators in first quantization. The key ideas of this method were introduced in 1927 by Paul Dirac,[1] and were later developed, most notably, by Pascual Jordan[2] and Vladimir Fock.[3][4]In this approach, the quantum many-body states are represented in the Fock state basis, which are constructed by filling up each single-particle state with a certain number of identical particles.[5] The second quantization formalism introduces the creation and annihilation operators to construct and handle the Fock states, providing useful tools to the study of the quantum many-body theory.

This exchange symmetry property imposes a constraint on the many-body wave function. Each time a particle is added or removed from the many-body system, the wave function must be properly symmetrized or anti-symmetrized to satisfy the symmetry constraint. In the first quantization formalism, this constraint is guaranteed by representing the wave function as linear combination of permanents (for bosons) or determinants (for fermions) of single-particle states. In the second quantization formalism, the issue of symmetrization is automatically taken care of by the creation and annihilation operators, such that its notation can be much simpler.

In the second quantization language, instead of asking "each particle on which state", one asks "How many particles are there in each state?". Because this description does not refer to the labeling of particles, it contains no redundant information, and hence leads to a precise and simpler description of the quantum many-body state. In this approach, the many-body state is represented in the occupation number basis, and the basis state is labeled by the set of occupation numbers, denoted

Note that besides providing a more efficient language, Fock space allows for a variable number of particles. As a Hilbert space, it is isomorphic to the sum of the n-particle bosonic or fermionic tensor spaces described in the previous section, including a one-dimensional zero-particle space C.

The creation and annihilation operators are introduced to add or remove a particle from the many-body system. These operators lie at the core of the second quantization formalism, bridging the gap between the first- and the second-quantized states. Applying the creation (annihilation) operator to a first-quantized many-body wave function will insert (delete) a single-particle state from the wave function in a symmetrized way depending on the particle statistics. On the other hand, all the second-quantized Fock states can be constructed by applying the creation operators to the vacuum state repeatedly.

The creation and annihilation operators (for bosons) are originally constructed in the context of the quantum harmonic oscillator as the raising and lowering operators, which are then generalized to the field operators in the quantum field theory.[7] They are fundamental to the quantum many-body theory, in the sense that every many-body operator (including the Hamiltonian of the many-body system and all the physical observables) can be expressed in terms of them.

These two equations can be considered as the defining properties of boson creation and annihilation operators in the second-quantization formalism. The complicated symmetrization of the underlying first-quantized wave function is automatically taken care of by the creation and annihilation operators (when acting on the first-quantized wave function), so that the complexity is not revealed on the second-quantized level, and the second-quantization formulae are simple and neat.

These commutation relations can be considered as the algebraic definition of the boson creation and annihilation operators. The fact that the boson many-body wave function is symmetric under particle exchange is also manifested by the commutation of the boson operators.

The raising and lowering operators of the quantum harmonic oscillator also satisfy the same set of commutation relations, implying that the bosons can be interpreted as the energy quanta (phonons) of an oscillator. The position and momentum operators of a Harmonic oscillator (or a collection of Harmonic oscillating modes) are given by Hermitian combinations of phonon creation and annihilation operators,

This idea is generalized in the quantum field theory, which considers each mode of the matter field as an oscillator subject to quantum fluctuations, and the bosons are treated as the excitations (or energy quanta) of the field.

It is particularly instructive to view the results of creation and annihilation operators on states of two (or more) fermions, because they demonstrate the effects of exchange. A few illustrative operations are given in the example below. The complete algebra for creation and annihilation operators on a two-fermion state can be found in Quantum Photonics.[8]

These anti-commutation relations can be considered as the algebraic definition of the fermion creation and annihilation operators. The fact that the fermion many-body wave function is anti-symmetric under particle exchange is also manifested by the anti-commutation of the fermion operators.

where i , j \displaystyle i,j labels any Majorana fermion operators on equal footing (regardless their origin from Re or Im combination of complex fermion operators c α \displaystyle c_\alpha ). The anticommutation relation indicates that Majorana fermion operators generates a Clifford algebra, which can be systematically represented as Pauli operators in the many-body Hilbert space.

The term "second quantization", introduced by Jordan,[11] is a misnomer that has persisted for historical reasons. At the origin of quantum field theory, it was inappositely thought that the Dirac equation described a relativistic wavefunction (hence the obsolete "Dirac sea" interpretation), rather than a classical spinor field which, when quantized (like the scalar field), yielded a fermionic quantum field (vs. a bosonic quantum field).

One is not quantizing "again", as the term "second" might suggest; the field that is being quantized is not a Schrdinger wave function that was produced as the result of quantizing a particle, but is a classical field (such as the electromagnetic field or Dirac spinor field), essentially an assembly of coupled oscillators, that was not previously quantized. One is merely quantizing each oscillator in this assembly, shifting from a semiclassical treatment of the system to a fully quantum-mechanical one.

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There is very little description about the origin of this formalism and how it connects to other problems in physics. For example, I know that I have solved the harmonic oscillator problem in QM using ladder operators, but I do not understand why the mathematics of the quantum harmonic oscillator should be the same as the mathematics I use to describe many body systems.

Are there fundamental symmetries? Are all objects in the universe oscillators in some sense? I was hoping that someone could recommend a good text for an introduction to this second quantization formalism. The other problem is that I am an undergraduate student without a knowledge of QFT, thus I seek a text which does not rely on a detailed knowledge of this area.

Moreover, since second quantization (apparently) has a broader relevance than QFT, particularly in condensed matter physics, cold atom theory, and quantum simulation, it would be good to get an introduction that is focused on where undergraduate courses drop off and relevant to the various different fields that use the formalism.

it starts from the description of $N$ distinguishable particles on $\mathcalH^\otimes N$, goes on to the description of $N$ bosonic/fermionic undistinguishable particles on the symmetric/alternating subspace $\mathcalH^\otimes N, \textsym/alt$, then justifies why an arbitrary number of particles should be described on the $\bigoplus_N \geq 0$ of those;

the occupation number basis and the ladder operators are constructed and illustrated on simple examples (compensating for the not always completely clear explanations of the required combinatorial factors); I would give it extra credit for not fixing once and for all a 1-particle ONB, so that change of basis/representation and the creation/annihilation of arbitrary 1-particle states are naturally covered; formulas are written in a unified way for bosons/fermions when possible;

My only gripe with it is in the first few sentences, where the author goes out of his way to insist that second quantization would have nothing to do with quantizing a second time. This is a widely held attitude: although the second-quantization formalism can in fact be obtained by "quantizing" the wave-equation of the first-quantized theory (see this answer), it seems to have been "forgotten", and texts which acknowledge it tend to be much more advanced and/or older (hence a much harder read, as they use old-fashioned notations and concepts).

These will deal of course with the nonrelativistic electromagnetic field. The justification for all this (at least for the nonrelativistic EM field) is not as lofty as any of your suggestions: one simply recognizes that solutions to Maxwell's equations are superpositions of harmonic oscillators (e.g. superpositions of plane waves with time-harmonic dependence), so one replaces each mode of the EM field with a quantum harmonic oscillator. Quantum harmonic oscillators are easy systems to do many body problems for; if we have a many body Hamiltonian comprising non-interacting QHOs:

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