Canonical conjugate

In addition, there could be a mechanical or electromagnetic interaction of a system with an external entity which may do work on an otherwise isolated system. Such a contact with a work source can be represented by the Hamiltonian U p, q, x) where x is the coordinate (for example, the position of a piston in a box containing a gas, or the magnetic moment if an external magnetic field is present, or the electric dipole moment in the presence of an external electric field) describing the interaction between the system and the external work source. Then the force, canonically conjugate to x, which the system exerts on the outside world is... 

The components of the position operator, therefore, commute with one another furthermore they are canonically conjugate to the momentum operators... 

A canonically conjugate momentum may be obtained from (32) in the usual way as the derivative of the Lagrangian,... 

Here pt and, v, are the canonically conjugated integrals of motion corresponding to the ith degree of freedom A(t,ph, v ) is the work depending on the same... 

The quantities h, (p0 and /, v(/o constitute (to within a constant multiplier) two pairs of the canonically conjugate arbitrary constants. Therefore we may choose them as the phase variables while averaging over T in (27). We note that these quantities refer to a local phase space corresponding to any chosen direction of the symmetry axis. Hence, integration performed in the overall phase space dTls corresponding to an isotropic fluid should additionally include averaging over all possible inclinations 0 of the symmetry axis C to the a.c. field vector E. Thus,... 

We represent the phase-volume element dV in the form dT = dhdl dtp0, where h and canonically conjugated variables, (p0 is an initial instant that enters in addition to time (p in the law of motion of a dipole. Another pair of canonical variables is /, 0 we omit differential df>() in dT, since the variables we use do not depend on the azimuthal coordinate < )0. 

Indeed, in this context, position and momentum operators are canonically conjugated operators, through the relations (62). When eq.(61) is used as... 

A new momentum operator Pj a must therefore be introduced, defined in such a way to be canonically conjugated to A.y-/V through the commutation relations... 

With that in mind, we have to suggest a second important correction which concerns the role of the momentum operators, haDjmomentum operator PjjCt is the canonically conjugated partner of Xj>a, we understand that the use of haDj Ct in the representation Phi,h2(g 1532), although legitimate, is improper and subtle. Moreover, since hi and h2 are connected by the relation (68), it should be useful to introduce the new parameter /ieff in the formulation. Both these requirements can be fulfilled if we determine a new unitary irreducible and infinite-dimensional representation of the group D"... 

The observation of a quantum-mechanical system involves the disturbance of the state being observed the Heisenberg6 uncertainty principle [5] dictates that the uncertainty Ax in position x and the uncertainty Apx in momentum px in the x direction (or in y or in z, or the uncertainty in any two "canonically conjugate" variables, e.g. energy E and time f, or angular momentum L and phase (f>, i.e. variables whose... 

The initial and final sets of dynamical variables deciding the classical action, namely (Qi, Ei) and (Q2, (2) in this case, are the quantum observables specifying the initial and final states. Then we should assign real numbers to them. Since t is canonically conjugate to E and cannot be observed quantum mechanically, we can choose any complex number for it and the lapse times s = t2 t may take a complex number. To our knowledge, such prescription for complexifying canonically paired observables was first presented by Miller [2]. 

The variable canonically conjugate to the action I is the angle variable 6. According to (3.1.36) it can be obtained from the generating function... 

Canonically-conjugate observables do not commute. Corresponding to a generalised position coordinate q there is a generalised momentum p. The commutation law is... 

Since atoms are strongly affected by the central potential of the nucleus, an important part in electron—atom collision theory is played by states that are invariant under rotations. From the general dynamical principle that invariance under change of a dynamical variable implies a conservation law for the canonically-conjugate variable we expect rotational invariance to imply conservation of angular momentum. Hence angular momentum... 

The transformation from one pair of canonically conjugate coordinates q and momenta p to another set of coordinates Q = Q(p,q,t) and momenta P = P(p>qT) is called a canonical transformation or point transformation. In this transformation it is required that the new coordinates (P,Q) again satisfy the Hamiltonian equations with a new Hamiltonian H P,Q,t) [35] [43] [52]. 

As an Interlude between sections, it is useful to use Eq. (7) to obtain, at once, the general expressions for the angular momenta canonically conjugate to the Euler angles, as well as their relations with the corresponding cartesian components in the body frame. It is straightforward to obtain... 

The relation thus formulated is capable of immediate generalization. Consider in the first place, as an example with one degree of freedom, the case already treated above (p. 100), that of the rotator. Here the co-ordinate is the azimuth q== (f>y to which belongs, as canonically conjugated quantity, the angular momentum (or, in other words, the moment of momentum) p. In the free rotation p is constant, i.e. independent of the angle turned through. Thus... 

The non-commutativity which presents itself here is not, however, of the most general kind, as the theory shows for the left-hand expression, with a pair of canonically conjugated variables, can take... 

For the dynamic properties of the impurity system the formalism of (3.7)—(3.10) is not convenient, since the total Hamiltonian arises from (2.3), (2.4) as well as (2.5) and in the former two the dynamical variables are the normal mode coordinates of the lattice and their canonical conjugates. When the coefficients 5Vy are written out in terms of these variables, their formal expression up to the second order in Qj(fc) is... 

Let (p, q) be one set of canonically conjugate coordinates and momenta (the old variables) and (P,Q) be another such set (the new variables).13 (P, Q, p and q are IV-dimensional vectors for a system with N degrees of freedom, but for the sake of clarity multidimensional notation will not be used the explicitly multidimensional expressions are in most cases obvious.) In classical mechanics P and Q may be considered as functions of p and q, or inversely, P and Q may be chosen as the independent variables with p and q being functions of them. To carry out the canonical transformation between these two sets of variables, however, one must rather choose one old variable and one new variable as the independent variables, the remaining two variables then being considered as functions of them. The canonical transformation is then carried out with the aid of a generating function, or generator, which is some function of the two independent variables, and two equations which express the dependent variables in terms of the independent variables.13... 

In the time concept of the pre-relativistic mechanics, the observable quantities, time t and energy E, have to be considered as another canonically conjugate pair, as in classical mechanics. The dynamic law (time-dependent energy term) of the Schrodinger equation will then completely disappear [19]. A good occasion for Weyl to introduce the relativistic view would have been his contributions to Dirac s electron theory. His other colleagues developed the method of the so-called second quantization that seemed easier for the entire community of physicists and chemists to accept. 

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