China
Africa
Explore chapters and articles related to this topic
Quantum Nonlinear Optics
Published in Peter E. Powers, Joseph W. Haus, Fundamentals of Nonlinear Optics, 2017
Peter E. Powers, Joseph W. Haus
Two complementary formalisms are applied to solve problems in quantum mechanics; they are called the Schrödinger and Heisenberg pictures. In the former picture, the evolution of the wave functions is described, and averages of measureable quantities are calculated using the time-dependent wave functions. This formalism is commonly taught in books on quantum mechanics. In the latter picture, the evolution of operator variables, which may be decomposed into a set of elementary operators, is followed. The measurable quantities in this case are averages with respect to the initial wave functions. In this chapter, quantum mechanics is developed from the point of view of the Heisenberg picture. Its application connects more simply with the reader because the equations are familiar from our classical treatment of nonlinear optics. The essential complication of quantum mechanics replaces the dynamical variables of classical mechanics by quantum operators.
Electron–Photon Interaction
Published in Joachim Piprek, Handbook of Optoelectronic Device Modeling and Simulation, 2017
The time evolution of the system in the Heisenberg picture is not governed by the Schrödinger equation any more, but by the Heisenberg equation of motion O^˙=iℏ[H^,O^] where [.] denotes the commutator, [H^,O^]=H^O^−O^H^, and O^ is any operator.
Quantum States of Manifested Tribosystem
Published in Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk, Quantum Effects in Tribology, 2017
Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk
From the Heisenberg picture, the state vector ψ, dependent on time, is related to its original value ψ(0)by unitary transformation: ψ(t) = Q(t)ψ(O). That is, the dependence of a physical quantity on time is linked with the basis of Hilbert space. The components of the wave vector ψ are equal to ψ(0)hence, they do not depend on time; that is, they seem to be defined in the space of Schröodinger picture. It is said that, in the Heisenberg picture, the state vector does not depend on time. The Heisenberg picture of quantum mechanics is the closest to classical ways of describing the process of motion.
Locally acting mirror Hamiltonians
Published in Journal of Modern Optics, 2021
Jake Southall, Daniel Hodgson, Robert Purdy, Almut Beige
In order to quantize the EM field in position space, we assume that the basic building blocks of wave packets of light are highly localized field excitations. For simplicity, we only consider light propagating in one direction, i.e. along the x-axis. Using the Heisenberg picture, we denote the annihilation operator of a highly localized field excitation at position x and at a time t by . Here and refer to horizontally and to vertically polarized light, and to excitations propagating in the positive and the negative x-direction, respectively. As in classical electrodynamics, we demand that the expectation values of wave packets travel with the speed of light, c. In this paper, this is taken into account by assuming that for any state of the quantized EM field in the Heisenberg picture. Hence This equation provides a fundamental equation of motion of the quantized EM field in free space.
Statistical quasi-particle theory for open quantum systems
Published in Molecular Physics, 2018
Hou-Dao Zhang, Rui-Xue Xu, Xiao Zheng, YiJing Yan
We treat the issue of L-space truncation, not only from the aforementioned practical aspect, but, more fundamentally, also from the invariance principle of quantum mechanics prescriptions [73]. It demands that a proper truncation scheme in the Schrödinger picture be transferable to a Heisenberg-picture equivalent to Equation (3.13), without further approximations. We will see later that the standard {ρ(n > L)n = 0}-based scheme [25,30–32,111,112] fails in this formal requirement. Considered below is the so-called derivative-resum scheme [73,110].
Rome teleportation experiment analysed in the Wigner representation: the role of the zeropoint fluctuations in complete one-photon polarization-momentum Bell-state analysis
Published in Journal of Modern Optics, 2018
A. Casado, S. Guerra, J. Plácido
In the Heisenberg picture, the state of the electromagnetic field is represented by a time-independent density operator, whereas all the dynamics is contained in the electric field operator, through the time-dependent annihilation and creation operators and . The Wigner representation in the Heisenberg picture establishes a correspondence between the electric field operator and a time-dependent complex amplitude of the field (), through the substitution , . The Wigner function is time independent and corresponds to the Wigner distribution of the initial state. In the context of PDC, the initial state is the vacuum, which is represented in the Wigner representation by a stochastic field , where is the normalization volume, is a unit polarization vector and represents the zeropoint amplitude corresponding to the mode . The Wigner distribution for the vacuum field amplitudes is the Gaussian: where represents the set of zeropoint amplitudes. From the above equation, the following correlation properties hold: where means an average using the Wigner function of the vacuum state as a quasi-probability density.