Second quantization
The Second quantization (often called occupation number representation, in quantum field theory and field quantization ) is a method for quantum mechanical treatment of Vielteilchenproblemen, especially the processes in which particles are formed or destroyed. It was (see first quantization) developed shortly after the discovery of quantum mechanics in order to describe photons and whose creation and annihilation of quantum mechanics can. The photons appear in the second quantization as the field quanta of the quantized electromagnetic field, which led to the third given name. When it was discovered in the 1930s, that "material " particles can be created and destroyed, the scope of the method was extended to all particles. Thus, in the physics of vivid contrast between particles and waves was abolished in his previous fundamental importance.
The Second quantization is applied in the field of solid state physics, quantum field theory and other many-body theories. It is often the most appropriate framework to treat physical problems theoretically.
The method comes from Paul Dirac (1927 ).
- 4.1 Concrete examples 4.1.1 single-particle operators
- 4.1.2 Coulomb interaction
- 4.1.3 superconductivity
Basic concepts
Following is a brief compilation of some of the main new concepts and their immediate consequences. As expected, thereby, the term "particle" something other than the identical concept in classical mechanics or everyday language. He is here, as well as other seemingly familiar terms - " identity ", " transition ", " destruction " - more metaphorical uses: goal is to be able to operate on catchy way one of the mathematical definitions is when a quantum mechanical explanation of macroscopically observable phenomena in demand, such as staining, while the increase in the current in a semiconductor or the directional distribution of radiation.
- The state of the system under consideration is given as in ordinary quantum mechanics by a normalized vector in a Hilbert space, designed as a so-called Fock space containing the states with different particle numbers.
- There is a state without any particles, the absolute vacuum symbol. The vacuum state is normalized, so should not be confused with the zero vector.
- There is a creation operator for each particle, which puts it in a defined state in the world, icon ( for another particle, etc.). Of 1 -particle with a particle in the state p is then given by. 2- particulate state to a second particle of the same kind, but in the state k, is then added by again applying the generator. For more particles corresponding to more creation operators.
- Since the "a" particles are identical among themselves, must come out no other condition at an interchange in the order of generation. At best, the sign may change. This is ensured by the conditions
- The operator for the destruction of a particle in state p. An application example: Here can be the destruction of an existing particle in vacuum, the empty vacuum back. The destroyer is the producer of the Hermitian adjoint operator. That this is so right, you can see for example when calculating the norm of, ie the scalar product with its adjoint vector:
- The transition from the state of a particle P to K is performed by the operator. The particles were destroyed in p and generates a new one in k With this description, a procedure is misleading questions out of the way, arising out of everyday experience with macroscopic particles:
- Annihilator k with generators p interchangeable, except they refer to the same state. Then:
- The operator, which indicates the number of particles present in the state of P as a proper value, the particle number. It is the same for fermions and bosons. ( For fermions it has no eigenvalues other than 0 and 1 )
- The relationship of a 1- Teilchenzustands with his " old" wave function is obtained by a localized locally generated particles ( state) and, together with the scalar product, which yes, the amplitude of a state indicating in the other:
Mathematical construction
The crucial work, configuration space and second quantization, is derived from the Russian physicist Vladimir Fock in the year 1932.
Be an orthonormal single-particle basis of a quantum-mechanical system ( that is, a set of wave functions, according to which any Einteilchenwellenfunktion can develop ). Then it is known that each fermionic (or bosonic ) many-body wave function that is so inherently anti-symmetric (or balanced), according to determinants (or permanent ) can be developed regarding this Einteilchenbasis: Be antisymmetric ( such B. spatial and spin coordinates of an electron ). Then there are complex numbers (i.e., at all " configuration ", in which indexes are Einteilchenbasis, there are N complex coefficients ) with
So you can each many-body wave function represented as a linear combination of such determinants - states ( or corresponding permanent states in the bosonic case). These determinants can be represented states are also of great physical significance, since the ground-state wave functions of non-interacting systems as determinants pure states (or permanent status ) in addition to the purely mathematical sense as a development base frequently.
The determinant / permanent for configuration you can now renamed
Assign, with number of occurrences of the value of in, number of occurrences of the value in, .... The values are called occupation numbers of the corresponding basis states. The occupation numbers can only be 1 or 0 for fermions, since otherwise the determinant would vanish ( two equal columns).
In this notation, so is the general form of an N-particle Vielteilchenzustands:
The occupation number representation. The antisymmetric and symmetric N-particle Hilbert space is thus spanned by these states with. It seems obvious now introduce a more general space called Fock space, which is spanned by the states with arbitrary finite number of particles:
.
Since operators can be displayed regardless of the specific particle number (see below), this design makes sense. In this room conditions indefinite number of particles are included ( linear combination of states of different particle numbers of certain ). In many-body theory it is normally operated.
Individual determinants states that as I said, for example, special states of a noninteracting system might be, one can clearly specify in the form, if you say to that on which Einteilchenbasis referring.
See also: Slater determinant
Generation, annihilation and Teilchenzahloperatoren
To the above- presented annoying ( anti-) symmetrizing for generating Fockzuständen no longer having to perform, generate the Fockzustände now instead of the vacuum state. These leads are new operators, a " generate " the particles in the basis state or " destroy ", that is, increase or decrease the corresponding occupation number, the whole symmetry problem now lies in the commutation relations:
Definition ( on the basis of the state space to the rest of linear extension )
- In the bosonic case
- In the fermionic case
The prefactors ensure in each case for the non-occurrence of impossible states (eg with occupation numbers less than 0 or greater than 1 for fermions ), for the Wegkapseln the antisymmetry with fermions in other terms, and that the occupation number operators in both cases as
Result. Recalculation shows that these operators reproduce the occupation numbers at determinants states:
.
Commutation relations
For the constructed operators are in the fermionic case, the anticommutation
Where the anticommutator means.
In the bosonic case, the commutation relations are
In the commutator.
A and Zweiteilchenoperatoren
It can be shown that allows all linear operators on the Fock space represented as a linear combination of polynomials in the production / annihilation operators. This is a critical aspect of importance. Particularly important are the so-called single-particle and two-particle operators, which according to either observable individual particles represent their name (eg kinetic energy, position, spin) or interactions between two particles (eg Coulomb interaction between two electrons).
This results in doing simple expressions ( is the number of particles ): Let
A single-particle operator ( i.e., each acts only on the coordinates of the - th particle, are structurally the s but all the same ), we obtain ( by inserting ones and use the valid for bosons and fermions relation, where the index a single-particle state in the Hilbert space of the - th particle features ):
Where the matrix element of Einteilchenoperators is, from which the result formed the basis states, with respect to which has been quantized. For Zweiteilchenoperatoren results analogous:
.
In the expressions is true equality of the operators, as long as they are based on a fixed number of particles. One sees, however, that the zweitquantisierte form of the operators no longer contains the number of particles explicitly. The zweitquantisierten operators so take in systems of different particle number at each the same shape.
Concrete examples
Single-particle operators
Particle density in Zweitquantisierung regarding pulse basis (discrete pulse basis, finite volume with periodic boundary conditions):
Coulomb interaction
In Zweitquantisierung respect to ( discrete ) impulse basis.
Superconductivity
The Second quantization allows the Fock representation and the explicit consideration of states that are not eigenstates of the number operator pond. Such conditions play an important role in the theory of superconductivity.
Transformation between Einteilchenbasen
Creation and annihilation operators with respect to a given Einteilchenbasis can be expressed by corresponding operators with respect to another Einteilchenbasis:
Through these relationships, it is possible to perform a change of basis in the Fock space and to better transform thus given expressions for the currently applied situation suitable shapes. Similarly also field operators with respect to continuous local or pulse bases are generated from the Erzeugungs-/Vernichtungs-Operatoren for discrete Einteilchenbasen how they are used mainly in the quantum field theories.
Generalization: Relativistic Quantum Field Theories
As a generalization arise, as indicated in the footnote, instead of the non-relativistic many-body theory of relativistic quantum field theories.