Heisenberg commutator

Starting from the standard Heisenberg commutator [p, q] — —hi, it is readily shown [119] that the coefficients satisfy the following commutation rules... 

The covariant Heisenberg commutation relations in phase space can be written... 

In this manner the mathematical formalism incorporates the important experimental result that not all physical observables can be known simultaneously for a quantum system, the incompatibility of a knowledge of the position and its conjugate momentum for a particle being an example of this behaviour. The non-compatibility of these two observables is enshrined in the Heisenberg commutation rules... 

Again taking Sq, equal to eq, the first of the commutators in the expanded product yields — bP. The resulting operator equation must be valid for any state including one for which Ps T > = 0, which, for arbitrary Sq, yields the last of the Heisenberg commutator relations... 

The multipole electromagnetic held can be quantized in much the same way as plane waves [2]. We have to subject the complex held amplitudes in the expansion (17) to the Weyl-Heisenberg commutation relations of the form... 

We now introduce the ideas of Weyl to distinguish between pure states and mixtures. Pure states were mathematically represented by eigenvectors of observables, which described the properties of a particle or a dynamic state. On the other hand, mixtures were composed of pure states of a certain mixing relationship. These aspects are clearly important to chemists and obviously to the electrochemists too. The canonical variables, G and H [19], have to satisfy the canonical or Heisenberg commutation relation, derived from Equations 3.12 and 3.13 ... 

Here, we used the Heisenberg commutator relation between the conjugate position and momentum operators x, px = ih. The magnetic moment matrix element of the intra-ligand transition with respect to the common origin of the coordinate system is given by ... 

Projecting the Hamiltonian and other quantum-mechanical operators onto a finite basis sets has serious consequences, even if the basis is large. It can be shown that certain quantum-mechanical rules eo ipso cannot be represented in finite basis. This leads to serious inconsistencies inherent in practical quantum chemistry where the basis set is finite in nearly all calculations. One such example is given by the Heisenberg commutation rule between the coordinate operator q and the canonically conjugated momentum operator p ... 

Substituting these expressions into the Heisenberg commutation rule of Eq. (8.19) we obtain ... 

Summarizing, we have noted that the Heisenberg operators Q+(t) obey field free equations i.e., that their time derivatives are given by the commutator of the operator with Ha+(t) = Ho+(0) and that this operator H0+(t) is equal to H(t) = H(0). The eigenstates of H0+ are, therefore, just the eigenstates of H. We can, therefore, identify the states Tn>+ with the previously defined >ln and the operator [Pg.602]

We shall again postulate commutation rules which have the property that the equations of motion of the matter field and of the electromagnetic field are consequences of the Heisenberg equation of motion ... 

Now in quantum theory the description of a physical system in the Heisenberg picture for a given observer O is by means of operators Q, which satisfy certain equations of motion and commutation rules with respect to O s frame of reference (coordinate system x). The above notion of an invariance principle can be stated alternatively as follows If, when we change this coordinate frame of reference (i.e., for observer O ) we are able to find a new set of operators that obeys the same equations of motion and the same commutation rules with respect to the new frame of reference (coordinate system x ) we then say that these observers are equivalent and the theory invariant under the transformation x - x. The observable consequences of theory in the new frame (for observer O ) will then clearly be the same as those in the old frame. 

In the Heisenberg-type description the existence of such a unitary or anti-unitary operator U is inferred from the fact that the set of observable Q and Q satisfy the same commutation rules. 

This general expression relates the uncertainties in the simultaneous measurements of A and B to the commutator of the corresponding operators A and B and is a general statement of the Heisenberg uncertainty principle. 

The fact that quantum observables are represented by matrices immediately suggests problems of non-commutation. For instance, the observables can be measured at the same time only if they have a complete orthogonal set of eigenvectors in common. This happens only when they commute, i.e. XY = YX, or the commutator [X, Y] = XY — YX = 0. This is a central feature of the matrix formulation of quantum theory discovered by Heisenberg, Born and Jordan while trying to explain the observed spectral transitions of the hydrogen atom in a more fundamental way than the quantization... 

From the Heisenberg formalism momentum should be represented by an operator that does not commute with x, i.e. [x,p] = ih. The momentum operator can therefore not also be multiplicative, but can be a differential operator. The representation p <----ih-J gives the correct form when operating... 

Equation (31) is known as Heisenberg s equation of motion and is the quantum-mechanical analogue of the classical equation (17). The commutator of two quantum-mechanical operators multiplied by 2mfh) is the analogue of the classical Poisson bracket. In quantum mechanics a dynamical quantity whose operator commutes with the Hamiltonian, [A, H] = 0, is a constant of the motion. 

It commutes with the action of the Heisenberg algebra 1). Hence (9.23) holds even for m or n = 0. 

The operator a i) in the Heisenberg algebra, of course, corresponds to the operator constructed in Chapter 8. But our commutator relation (8.14) differs from the standard one, we need to modify operators. In fact, it is more natural to change also the sign of the bilinear form. Hence we dehne... 

I use temporarily roman n, m to include zero.) Show that the Heisenberg operators an(t), a (t) obey the same rules provided they are taken at the same f. The commutation relations of two of these operators taken at different times are not simple they involve the solution of the equations of motion. 

Here q and p are Heisenberg operators, y is the usual damping coefficient, and (t) is a random force, which is also an operator. Not only does one have to characterize the stochastic behavior of g(t), but also its commutation relations, in such a way that the canonical commutation relation [q(t), p(t)] = i is preserved at all times and the fluctuation-dissipation theorem is obeyed. ) Moreover it appears impossible to maintain the delta correlation in time in view of the fact that quantum theory necessarily cuts off the high frequencies. ) We conclude that no quantum Langevin equation can be obtained without invoking explicitly the equation of motion of the bath that causes the fluctuations.1 That is the reason why this type of equation has so much less practical use than its classical counterpart. 

Heisenberg representation (matrix mechanics) the position and momentum are represented by matrices which satisfy this commutation relation, and ilr by a constant vector in Hilbert space, the eigenvalues E being the same in two cases,... 

The formulation of quantum mechanics requires a representation of the Heisenberg group on the Hilbert space L2 (R") spanned by the functions tp ( ) where the variable indicates a n—dimensional vector = ( 1 , n) whose elelments have physical units of a length [/]. Let us first introduce the set of operators, generators of the Lie group H", I, Xj, and hDj (j = 1,..., n) satisfying the commutation relation... 

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