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Oxford University Press eBooks

Making second quantization work

Tom Lancaster, Stephen J. Blundell

Published Apr 17, 2014• 0 citations
Publisher

Abstract

This chapter considers how to build single-particle operators out of creation and annihilation operators, and this already gives enough information to discuss the tight-binding model of solid state physics and the Hubbard model.
DOI
10.1093/acprof:oso/9780199699322.003.0005

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The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second-quantized, Heisenberg representation, particle-creation, and annihilation operators. The creation operator, ψ †( r , t), when acting to the right on a state of the system, adds a particle to the state at the space-time point r , t; the annihilation operator ψ ( r , t), the adjoint of the creation operator, acting to the right, removes a particle from the state at the point r , t.
The properties of a quantum mechanical system composed of many identical particles are most conveniently described in terms of the second‐quantized, Heisenberg representation, particle‐creation, and annihilation operators. The creation operator ψ ^ † ( r , t ) $ \hat{\psi }^{\dag } ({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_1.tif"/> , when acting to the right on a state of the system, adds a particle to the state at the space‐time point r , $ {\mathbf{r}}, $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_2.tif"/> t ; the annihilation operator ψ ^ ( r , t ) $ \hat{\psi }({\mathbf{r}}, t) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315196596/a397f31e-8107-49bb-8694-b978d38168d6/content/inline-math2_3.tif"/> , the adjoint of the creation operator acting to the right, removes a particle from the state at the point r , t .
The Quantum-Mechanical description of subatomic particles is the most accurate theory of these particle’s behavior. The first quantization is the standard method for solving the Schrodinger equation, but the dimension of the partial differential equation increases rapidly, especially when dealing with more than one particle. These differential equations become difficult to solve and, although there exist methods in this formalism, a new method is approached. This thesis starts with an introduction where the first quantization is discussed in multi-particle physics and electronic systems. The second quantization formalism is then introduced and discussed with some definitions and notation. Some applications of the new formalism are then discussed, such as some spin operators and the number operator. Some new models and operators are introduced, such as the tight-binding model, the Hubbard model, and the number operator. Some numerical applications are also discussed for these models and operators. The second quantization formalism is then implemented in Python, and some analysis is performed for the tight-binding and Hubbard model. In other words, this thesis focuses on the implementation of the second quantization formalism for fermions in Python and is then used to analyze two models. It is also a first step into the multi-particle quantum mechanics.