Quantum Mechanics and Spin Systems
With the basic content of this thesis being simulations of quantum spin-1/2 systems, we start introducing the underlying concepts and properties of those and of quantum mechanics in general. Clarifying the basics, we provide a starting point to perform ap
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Quantum Mechanics and Spin Systems
With the basic content of this thesis being simulations of quantum spin-1/2 systems, we start introducing the underlying concepts and properties of those and of quantum mechanics in general. Clarifying the basics, we provide a starting point to perform approximate simulations of these computationally unfeasible problems. An entire summary of the rudiments of quantum mechanics is given in [1–3], on which Sect. 2.1 is based. Section 2.2 builds on [1, 3, 4], where the properties of quantum spin systems are discussed in detail. To introduce quantum entanglement, Sect. 2.3 is based on a detailed summary in [1, 3]. Furthermore, we introduce in this chapter the models considered later in this thesis for benchmarking simulation methods. In particular these are the Bell and Greenberger–Horne–Zeilinger states, as well as the transverse-field Ising model with and without an additional longitudinal field. We also introduce the commonly used simulation method exact diagonalization and the approximate time-dependent density-matrix renormalization group method. Both are well understood and suitable for benchmarking new approximation schemes.
2.1 Concepts of Quantum Mechanics In quantum mechanics we consider systems of particles with different properties being represented as vectors of a complex Hilbert space H. This is a complete vector space under the norm induced by its scalar product. Thus, a pure state of an N -particle quantum system can be described via a vector | = (1 , . . . , N )T which is called the state vector and is an element of the Hilbert space H. We here use the Dirac (bra-ket) notation, where the “ket” | denotes a column vector and the “bra” | © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 S. Czischek, Neural-Network Simulation of Strongly Correlated Quantum Systems, Springer Theses, https://doi.org/10.1007/978-3-030-52715-0_2
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2 Quantum Mechanics and Spin Systems
is its conjugate transpose. It is hence the row vector | = (1∗ , . . . , N∗ ) with the star denoting complex conjugation. The state vectors of quantum systems can be chosen normalized, | =
|
= 1,
(2.1)
and they can be expanded in the basis states |vi of the Hilbert space, | =
dH
cvi |vi ,
(2.2)
i=1
with complex coefficients cvi ∈ C and Hilbert space dimension dH . The state vector is thus a superposition of basis states. Furthermore, we introduce the wave function (v) as the overlap of the state vector with a basis vector v|, (v) = v | ,
(2.3)
which provides a complex-valued amplitude for each basis state. A probability amplitude P(v) for the corresponding basis state is given by the squared wave function, P (v) = | (v)|2 ,
(2.4)
emphasizing the choice of normalized wave functions, since then also P(v) describes a normalized probability. Another possibility to describe the state of a quantum system is to introduce the density matrix as ρˆ = ||,
(2.5)
with the hat denoting that ρˆ
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