Sunday, 9 June 2013

Second quantization

Second quantization is a powerful procedure used in quantum field theory for describing the many-particle systems by quantizing the fields using a basis that describes the number of particles occupying each state in a complete set of single-particle states. This differs from the first quantization, which uses the single-particle states as basis.

The starting point of this formalism is the notion of indistinguishability of particles that bring us to use determinants of single-particle states as a basis of the Hilbert space of N-particles states Quantum theory can be formulated in terms of occupation numbers (amount of particles occupying one determined energy state) of these single-particle states. The formalism was introduced in 1927 by Dirac.

The occupation number representation

Consider an ordered and complete single-particle basis $\left\{| \nu_1 \rang, | \nu_2 \rang, | \nu_3 \rang, ...\right\}$ , where $| \nu_i \rang$ is the set of all states $\nu$ available for the $i$ -th particle. In an N-particle system, only the occupied single-particle states play a role. So it is simpler to formulate a representation where one just counts how many particles there are in each orbital $| \nu \rang$ . This simplification is achieved with the occupation number representation. The basis states for an N-particle system in this representation are obtained simply by listing the occupation numbers of each basis state, $|n_{\nu_1}, n_{\nu_2}, n_{\nu_3},\dots \rang$ , where $\sum_j n_{\nu_j} = N$ The notation means that there are $n_{\nu_j}$ particles in the state $\nu_j$ . It is therefore natural to define the occupation number operator $\hat{n}_{\nu_j}$ which obeys