Heisenberg Operator Approach: Nano-Mechanical Oscillator (Part 2) and the Quantum LC Circuit

With the knowledge of the new design rules in Chapter 7, we use this new insight to find the eigenvectors for the nano-vibrator problem, and then we use the same approach to examine the quantum LC circuit. While the usual approach is to use Kirchhoff’s laws to analyze a simple circuit classically, we first see that Hamilton’s equations can in fact be used, giving the same classical result. But then, using the new design rules and the knowledge of the total energy in the circuit, we identify a canonical coordinate and a conjugate momentum that have nothing to do with real space and motion of a particle of mass m. At the same time, consistent with the Schrödinger picture, we continue to see that the time evolution of an observable such as position, x(t), or current, i(t), is not part of the solution. Given that Hamilton’s equations give the same result as Kirchhoff’s law but the quantum solution does not, reinforces the idea that the quantum description is showing features that cannot be imagined with a viewpoint based on classical (i.e. non-quantum) analysis.

The Density Matrix: Bloch Equations

One of the most powerful tools for calculating quantum device performance is based on the density matrix operator. The operator is unique because it is time dependent in the Schrödinger picture. The approach is quite general, but in the systems of interest here, the Hilbert space of the operator includes both the quantum system such as a nano-vibrator or two-level system and the quantized vacuum radiation field. The equation of motion follows from the time dependent Schrödinger equation. It is possible, as we show, to include the generation of spontaneous emission in this system and then, because observables of interest do not depend on the vacuum field, trace over the vacuum field to create a new density matrix called the reduced density matrix. The resulting equations of motion are the Bloch equations. We use these equations to analyze several problems involving two- and three-level systems.

Current Algebra, a U(1) Gauge Theory and the Wess-Zumino-Witten Model

In this note we realise current algebra with anomalous terms in terms of a U(1) gauge theory, in the space of maps M, from S1 into a compact Lie group corresponding to the current algebra. The Wilson loop around a closed curve in M is shown to be the Wess-Zumino-Witten term. This discussion enables a simple understanding of the non-Abelian anomaly in the Schrödinger picture.

Everettian relative states in the Heisenberg picture

Everett's relative-state construction in quantum theory has never been satisfactorily expressed in the Heisenberg picture. What one might have expected to be a straightforward process was impeded by conceptual and technical problems that we solve here. The result is a construction which, unlike Everett's one in the Schrödinger picture, makes manifest the locality of Everettian multiplicity, its inherently approximative nature and its origin in certain kinds of entanglement and locally inaccessible information. (By Everettian , we are referring not only to Everett's own work, but also to versions of quantum theory that elaborate and refine his. The notion of relative states first appeared in Everett (Everett 1973 In The many worlds interpretation of quantum mechanics (eds BS DeWitt, N Graham)). We are proposing a formalism for relative states that is more detailed and more illuminating than Everett's.) Our construction also allows us to give a more precise definition of an Everett ‘universe’, under which it is fully quantum, not quasi-classical, and we compare the Everettian decomposition of a quantum state with the foliation of a space–time.

Normal ordering normal modes

In a soliton sector of a quantum field theory, it is often convenient to expand the quantum fields in terms of normal modes. Normal mode creation and annihilation operators can be normal ordered, and their normal ordered products have vanishing expectation values in the one-loop soliton ground state. The Hamiltonian of the theory, however, is usually normal ordered in the basis of operators which create plane waves. In this paper we find the Wick map between the two normal orderings. For concreteness, we restrict our attention to Schrodinger picture scalar fields in 1+1 dimensions, although we expect that our results readily generalize beyond this case. We find that plane wave ordered n-point functions of fields are sums of terms which factorize into j-point functions of zero modes, breather and continuum normal modes. We find a recursion formula in j and, for products of fields at the same point, we solve the recursion formula at all j.

Picture-change correction in relativistic density functional theory

Relativistic quantum chemical calculations are performed based on one of two physical pictures, namely the Dirac picture and the Schrödinger picture. With regard to the latter, the so-called picture-change effect...