Quantised variable

The variable q is called a quantised variable and Eq. (2.29) a quantised state system (QSS). equations 2.29a-2.29b constitute a dynamic subsystem with a piecewise constant input trajectory dx (t) and a piecewise constant output trajectory q(t). The third Eq. 2.29c is a static relation. The quantised state system (2.29) can be displayed by the block diagram in Fig. 2.23. 

Kofman proposed a second-order accurate QSS2 method [55], For this method, the quantisation that does not need hysteresis produces piecewise linear output trajectories having discontinuities whenever the absolute value of the difference between a state variable and its quantised variable reaches the quantum. If the QSS2 method is applied to LTI systems then state variables have piecewise parabolic trajectories provided input trajectories are piecewise constant. 

An attractive feature of a discrete event simulation of hybrid systems is that the simulation time advances from discrete event to discrete event. Eor the QSS method, discontinuities in the inputs and the quantised variables dictate the time advance. No iteration is necessary to locate the time point of a discontinuity allowing for an efficient processing of models with discontinuities. Discrete event simulation using the quantised-based integrations needs much less simulation steps than a numerical integration method of comparable accuracy based on time-discretisation. Accordingly, computational costs are saved. Nevertheless, there are still some problems to be tackled with the QSS approach as detailed in [1, Chap. 12.11]. 

In the Born-Oppenheimer approximation the basis set for 3Q,i would consist of products of electronic space and spin functions. Transformation to the gyrating axis system may involve transformation of both space and spin variables, leading to a Hamiltonian in which the spin is quantised in the molecule-fixed axis system (as, for example, in a Hund s case (a) coupling scheme) or transformation of spatial variables only, in which case spatially quantised spin is implied (for example, Hund s case (b)). We will deal in detail with the former transformation and subsequently summarise the results appropriate to spatially quantised spin. 

Hero also the radiating electron moves in a processing ellipse the pure ellipse occurs as the form of the orbit in a pure Coulomb field only, any deviation from which, such as that determined by the variability of mass in the hydrogen atom, implies precession. We quantise this motion as before, and so obtain two quantum numbers n and k. By general agreement tlio t(irms are denoted by a number and a letter. The number indicates the principal quantum number n. For the specifi(jation of the azimuthal quantum number k the following notation has established itself ... 

Ehrenfest has proved that the action variables are adiabatic invariants, i.e. that the quantities J can be quantised. We now postulate the quantum conditions... 

So far, we have not specified the energy, and so v is not yet fixed at any specific value, but is a running variable. In order to quantise the energy, we must as usual impose boundary conditions appropriate for a bound state. These will turn out to quantise u, as we now demonstrate. 

The variables p and q are canonically conjugate, so that the Bohr-Sommerfeld quantisation condition yields ... 

If all the variables are separable, then quantisation poses no problem and is achieved through the Bohr-Sommerfeld rule ... 

From the theoretical point of view, as stated at the outset, there is no problem of quantisation if the variables of the problem are separable. In fact, the problem of H in a strong magnetic field, taking even the simplest possible Hamiltonian with only Coulomb and diamagnetic terms... 

In words —In every system for which the potential energy is invariant with respect to a rotation about an axis fixed in space, the component of the angular momentum about the axis multiplied by 2tt is an action variable. If the energy depends essentially on this quantity, it is to be quantised. 

For an arbitrary external field, on the other hand, the resultant angular momentum p is not in general an integral of the equations of motion and cannot therefore be quantised, but it may happen, in special cases, that p is constant and is an action variable. The relations (8) and (9) will then be true at the same time but pt is the projection of p in the direction of the field and, if a denotes the angle between the angular momentum and the direction of the field, wo have... 

The stationary motions in a weak electric field are, however, essentially different from those in a spherically symmetrical field differing only slightly from a Coulomb field. In the latter (for which the separation variables are polar co-ordinates) the path is plane it is an ellipse with a slow rotation of the perihelion. In the former (separable in parabolic co-ordinates) it is likewise approximately an ellipse, but this ellipse performs a complicated motion in space. If then, in the limiting case of a pure Coulomb field, k or nf be introduced as second quantum number, altogether different motions would be obtained in the two cases. The degenerate action variable has therefore no significance for the quantisation. 

The expansions of the cartesian co-ordinates as functions of the angle variables (to be calculated from (26), 22) must now be introduced, to provide a starting-point for the calculation of the perturbations. In this connection, however, there is one point to be borne in mind. In the unperturbed Kepler motion (without taking account of the variation in mass) only Jx is fixed by the quantum theory, whilst J2, i.e. the eccentricity, remains arbitrary in the relativistic Kepler motion, J2 is also to be quantised and, for a one-quantum orbit, J2=J1=A. We shall not take account quantitatively of the relativistic variation of mass, but we shall assume that the initial orbit of each electron is circular with limiting degeneration J1=A,... 

The derivation of the Hamiltonian resembles the standard procedure the classical Lagrange function is constructed first, then it is used to express the classical Hamilton function and then quantisation is applied by substituting the canonical variables for corresponding quantum-mechanical operators. There are two additional requirements the Hamiltonian should be symmetric with respect to the interchange of two electrons, and it should be Hermitian. 

Instead of discretising the time and using a BDF-based method for the numerical computation of a continuous-time model one may think of quantising the state variables. That is, instead of using a multistep method to compute an approximation of the value x(f +i) of a state variable x at time tjc+i, the question then is at what time the state x will deviate from its current value x (tk) by more than a given quantum A Q. In other words, the task is to find the smallest time step h so that... 

It is sufficient to quantise the static relation between theenergy variable of a storage element and its output power variable. Consider, e.g. the linear 1-port C element in integral causality in Fig.2.24 and let qq(t) = floor(g(f)) the quantisation of its displacement q. The continuous constitutive equations of a C element with the capacitance C... 

The second and the third column of Table B.l list the effort and flow variables in the various energy domains. The variables in the fourth column of Table B.l are the time integral of the efforts and the variables in the fifth column are the time integral of the flows. They are called energy variables because they quantise the amount of energy in the energy storage elements of a model. 

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