To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands (“digits”), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation.
To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands (“digits”), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation.
To simulate the quantum systems at classical or quantum computers, it is necessary to reduce continuous observables (e.g. coordinate and momentum or energy and time) to discrete ones. In this work we consider the continuous observables represented in the positional systems as a series of powers of the radix mulitplied over the summands (``digits``), which turn out to be Hermitean operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them and the effects of choice of numeral system on the lattices and representations. Furthermore, during the construction of the digital representation renormalizations of diverging sums naturally occur.
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on quantum digits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to discrete observables. We consider this problem based on expanding quantum observables in series in powers of two and three analogous to the binary and ternary representations of real numbers. The coefficients of the series (“digits”) a re, t herefore, Hermitian o perators. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary and ternary expansions of quantum observables automatically lead to renormalization of some divergent integrals and series (giving them finite values).
To simulate a quantum system with continuous degrees of freedom on a quantum computer based on qubits, it is necessary to reduce continuous observables (primarily coordinates and momenta) to binary observables. We consider this problem based on expanding quantum observables in series in powers of two, analogous to the binary representation of real numbers. The coefficients of the series (“digits”) are therefore orthogonal projectors. We investigate the corresponding quantum mechanical operators and the relations between them and show that the binary expansion of quantum observables automatically leads to renormalization of some divergent integrals and series (giving them finite values).
Quantum mechanics can be formulated either in terms of matrix mechanics or in terms of wave mechanics. This chapter centers on a discussion of the historical origins, mathematical structure, and extension to non-Hermitian quantum theory of matrix mechanics. The development of atomic discrete spectra and the old quantum theory showed that classical mechanics could not account for the discreteness of atomic energy levels and spectral lines, which motivated Heisenberg, Born, Jordan, and others to establish matrix mechanics centered on noncommuting physical quantities. Combining the Bohr atomic model, angular momentum quantization, and radiative transitions, one can see how matrix mechanics describes the energy level structure and quantum transitions of atomic systems, and how the Heisenberg equation of motion gives the fundamental form of quantum dynamics. The example of the harmonic oscillator further shows that the discreteness of the energy spectrum can be obtained directly from the matrix method, thereby establishing a clear connection among transition frequencies, matrix elements, and radiation-induced transitions. The matrix representation corresponds to the operator theory in Hilbert space, where state vectors, inner products, complete bases, and spectral decomposition constitute the fundamental mathematical framework of quantum mechanics. For closed quantum systems, physical observables are generally required to be represented by Hermitian or self-adjoint operators, so as to ensure that their eigenvalues possess the physically acceptable property of being real numbers. The Hermiticity of an operator is also closely related to its domain and boundary conditions, which provides a mathematical possibility for the emergence of non-Hermitian Hamiltonians. Based on these discussions, this chapter further explains how non-Hermitian dynamics is connected with open systems, dissipative processes, and special boundary conditions, thereby demonstrating the natural connection between traditional matrix mechanics and modern non-Hermitian quantum theory.
In this chapter we take up systems whose state spaces form a continuum. These are, naturally, infinite dimensional vector spaces (often called Hilbert spaces). Unfortunately, we could not do any better than treating such systems by formal analogy with their finite dimensional counterparts. A rigorous treatment would require a background in functional analysis and is beyond the scope of the present work. Thus, we essentially lay out a dictionary for translating the mathematical ingredients and formulae, from discrete to continuum descriptions, motivated by plausibility arguments. In particular, we argue why states are now described by complex square integrable functions (called wave functions) and observables by Hermitian differential operators. We then move on to show how to quantize quantum systems with classical analogues, which comprise the bulk of real life physical systems. Here, we explicitly show how the most nontrivial component of the quantization, i.e., the description of the representation of the momentum operator (in position space) can be deduced from both, the de Broglie hypothesis, and the canonical commutation relation. The “particle in a box” problem is treated as an illustration. The parity operator, Ehrenfest's theorem and continuity conditions have been relegated to the problem section.
In this chapter we shall mainly present the basic formalism that was initially developed by Heisenberg, also known as matrix mechanics (see Subsec. 1.5.7). We will first introduce in Sec. 2.1 the concept of quantum observables. Then, the problem of discrete and continuous spectra will be discussed and the basic non-commutability of quantum-mechanical observables will be deduced. While in Sec. 2.1 we discuss observables on a general formal level, in Sec. 2.2 some basic quantum-mechanical observables will be defined, and then different representations discussed and commutation relations derived. In Sec. 2.3 a basic uncertainty relation is derived. In the same section the relationship between uncertainty, superposition, and complementarity will be discussed. Finally, in Sec. 2.4 complete subsets of commuting observables will be shown to be Boolean subalgebras pertaining to a quantum algebra which is not Boolean. BASIC FEATURES OF QUANTUM OBSERVABLES This section is devoted to a general and formal exposition of quantum observables. In Subsec. 2.1.1 we shall learn how one can mathematically represent quantum observables as Hermitian operators. In Subsec. 2.1.2 we shall see how to change a basis, while in Subsec. 2.1.3 we shall find the relationship between eigenvalues of the observables and probabilities and learn how to calculate mean values. In Subsec. 2.1.4 we shall deal with an operator diagonalization. Finally, in Subsec. 2.1.5, the basic non-commutability of quantum observables will be presented by means of an example.