Commutator momentum-position

To demonstrate this, set F = P or Q. The two basic postulates of matrix mechanics follow directly from this, in terms of the momentum-position commutator,... 

In Sections V-VII we considered the construction of realizations of so(2, 1), so(4), and so(4, 2), respectively. In order to evaluate the commutators among the generators of these Lie algebras to verify that we did indeed obtain a realization, it is necessary to evaluate some rather complex commutators involving position and momentum coordinates. Thus, we shall collect here a number of useful commutators to facilitate this task. 

It is thus often said that measurements for operators that do not commute interfere with one another. The simultaneous measurement of the position and momentum along the... 

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

Equation (9-392) together with (9-394) and (9-395) are the proofs of the assertions that x is the position operator in the Foldy-Wouthuysen representation.16 (Note also that x commutes with /J the sign of the energy.) We further note that in the FTP-representation the operators x x p and Z commute with SFW separately and, hence, are constants of the motion. In the F W-representation the orbital and spin angular momentum operators are thus separately constants of the motion. The fact that... 

Hence U commutes with both position and momentum operators, and must, therefore, depend only on the spin operators. If s is a spin operator then since 8 is similar to an angular momentum operator... 

Under the action of the parity operator P, the position and momentum commutator [Q,L] = ih, becomes... 

All the components of the position r, and momentum p, vectors of the particle i commute with all those pertaining to particle j provided that i j, so that the fundamental commutation relations are... 

They obey the commutation relation [x, p ] = ih. We consider the transformed position and momentum operators given by... 

These commutation relations are taken to be the basic property of all angular momentum operators, including the spin operators which cannot be expressed in terms of the position coordinates xyz. In addition to the components of A/, we must deal with the operator M2 ... 

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,... 

In eq. (12), R(o) is the function operator that corresponds to the (2-D) configuration-space symmetry operator R(o). In eq. (13), /3 is the infinitesimal generator of rotations about z (eq. (8)) exp(i /3) is the operator [/do)] in accordance with the general prescription eq. (3.5.7). Notice that a positive sign inside the exponential in eq. (2) would also satisfy the commutation relations (CRs), but the sign was chosen to be negative in order that /3 could be identified with the angular momentum about z, eq. (6). 

As in section 2, we introduce the identity, position and momentum operators, labelling the two subsystems with a = 1,2, Ia,Xj,aJlaDjta (as before, j = 1,..., n is a vector index in Rn), whose commutation relations are... 

The second aspect is connected to the mathematical properties at the basis of the theory. The commutation relations holding for position and momentum operators, as illustrated in section 2, are... 

We can summarize the procedure followed in the present section to achieve the result obtained above. In Ref. [15] the new momentum operator was correctly introduced, together with the new commutation relations of eq.(70), but Pjt(X was not used in the formal construction of the theory. The expressions for the quantum-classical variables (position and momentum) are those shown in eqs.(67), insted of eq.(72), because XjiCC and are represented using Phi,h2 9i, 92) in which the operators haDjia, no more generators of the corresponding Lie group, are present11. 

Once a Lie algebra has been defined in the abstract sense via the defining commutation relations Eq. (3), it is of practical interest to find physical realizations of the generators in terms of position and momentum operators, which also satisfy these defining commutation relations. We shall call such a set of concrete operators a realization of the Lie algebra. In practice, as we shall see, we often work backwards by starting with a set of concrete operators... 

The basic commutation relations involving the position (xf), momentum... 

It may be that the wave functions are eigenfunctions of two non-commuting operators corresponding to physical quantities such as p (momentum) and q (position) respectively. Then, by measuring either A or B in system I, it becomes possible to predict with certainty and without disturbing the second system, either the value of Pk or qr. In the first case p is an element of reality and in the second case q is an element of reality. But ipk and commuting operators cannot have simultaneous reality. It was inferred that quantum theory is incomplete. 

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

The best known instance of Eq (4.26) involves the position and momentum operators, X and px. Their commutator is given by... 

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... 

It will be recalled (eqn (5.22)) that the first of Ehrenfest s relationships, obtained by taking the commutator of H with the position vector f, states that the momentum operator p is given by mdf/dt. If one uses the Hamiltonian H° -t- H in the commutator with f, then one correctly determines that the momentum is now given by w in eqn (8.208), that is, mdf/dt = — ( /c)A. 

Ehrenfest s first relation, the definition of the velocity and its associated momentum, is obtained using the commutator of H and the position vector r. This commutator, using the above Hamiltonian and multiplied by m, does indeed yield k as the momentum of an electron in an electromagnetic field... 

One pair of incompatible observables is the position of a particle in a given coordinate (x) and its momentum in the same direction pf). The corresponding operators do not commute. This is in agreement with a non-classical form for the momentum operator p. In fact, the classical commutator x(mvx) — (mvx)x is zero. 

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