Coordinate and momentum operators

Now, consider the case of spinless particles not subject to external electronic and magnetic fields. We may now choose the unitai7 operator U as the unit operator, that is, T = K. For the coordinate and momentum operators, one then obtains... 

The operators J, Q, D, D have the physical meaning of the angular momentum, quadrupole, coordinate, and momentum operators, respectively. 

Transforming to coordinate and momentum operators using Eqs (2.152), the interaction term... 

Here R and P are sets of coordinate and momentum operators, respectively. The unperturbed system is described by the Hamiltonian Ho, which is the sum of the kinetic energy (T(P)) and the potential energy (V(R)) operators. The time-dependent perturbation is denoted as W(R,t). In the examples presented below, the latter is an electric dipole interaction with an external field E(t) ... 

For reasons that will become clear in Section V, the operators and D in (2.95) have basically the same physical meaning as the coordinate and momentum operators. The final result of this replacement is that one can write the second-quantized form of the spherical invariant Hamiltonian operator in terms of U(4) coupled boson operators (up to two-body interactions and with the compact notation h j = b =pI-3> k = 2, 3,... 

Since involves only the coordinate and momentum operators of particle 1, we have [G qi)G2 q2)] = G2 q2)HxGi qx), since, as far as Ax is concerned, Gj is a constant. Using this equation and a similar equation for H, we find that Eq. (6.20) becomes... 

However, another subtle consequence of these relations is that from dynamical quantum equation may be abstracted (beyond the coordinate and momentum operators) the general equation of motion for an arbitrary operator ... 

Although both expressions (28.31) and (28.32) are equivalent, they differ a somewhat in the individual orders of the Taylor expansion, since coordinate and momentum operators do not commute. This difference is not significant here but the latter one is more convenient due to the symmetric properties of the final expressions and it will be discussed hereafter. 

Transforming to coordinate and momentum operators using Eqs (2.152), the interaction term in (9.44) is seen to depend on the momenta. A more standard interaction expressed in terms of the coordinates only, say xj %2 j when transformed into the creation and annihilation operator representation will contain the four products a j 2 1 1 1 Tbe neglect of the last two terms in Eq. (9.44)... 

The projected nature of the second-quantization operators has many ramifications. For exan le, relations that hold for exact operators such as the canonical commutation properties of the coordinate and momentum operators do not necessarily hold for projected operators. Similarly, the projected coordinate operator does not commute with the projected Coulomb repulsion operator. It should be emphasized, however, that these problems are not peculiar to second quantization but arise whenever a finite basis is employed. They also arise in first quantiztttion. but not until the matrix elements are evaluated. 

In order to apply the representation theory of so(2,1) to physical problems we need to obtain realizations of the so(2, 1) generators in either coordinate or momentum space. For our purposes the realizations in three-dimensional coordinate space are more suitable so we shall only consider them (for N-dimensional realizations, see Cizek and Paldus, 1977, and references therein). First we shall show how to build realizations in terms of the radial distance and momentum operators, r, pr. These realizations are sufficiently general to express the radial parts of the Hamiltonians we shall consider linearly in the so(2,1) generators. Then we shall obtain the corresponding realizations of the so(2,1) unirreps which are bounded from below. The basis functions of the representation space are simply related to associated Laguerre polynomials. For finding the eigenvalue spectra it is not essential to obtain these explicit realizations of the basis functions, since all matrix elements can... 

The transformations between coordinate and momentum space are accomplished via the fast Fourier transform (FFT) algorithm. This transforms the coordinate space evaluation of exp( — iFAt/2ft) into momentum space the effect of exp( — iTAt/h) is determined in this space to yield the momentum space function the momentum space function is transformed back to a coordinate space function via another FFT and finally, the new coordinate space function is further evolved by the local operator exp( — jFAt/21i) to yield the coordinate space wavefunction at time t-F At. The analogous technique can be accomplished with the simpler formula of Eq. (4.6a), but this does not reduce the number of FFT evaluations per time step, and thus is of no utility. This is clear when one notes that the values of < exp( —t FAt/2/t) > and

are evaluated at the grid points (in the FFT) just once at the beginning of the calculation and stored. Essentially all the work is involved in the FFT computation. 

Unlike a trajectory in classical mechanics, the wave function depends only on the space-spin coordinates x, jLi of the electrons in the system, and not on their momenta This lack of dependence of the wave function on the momenta reflects Heisenberg s uncertainty principle which in turn arises from the non-commutativity of conjugate pairs of position and momentum operators. 

THEOREM Quantum hermitic operators of coordinate and momentum fulfill the Newtonian laws of motion... 

Finally, worth showing that the ID analysis may be easily generalized to iV-dimensionally space with associate coordinate and momentum representations. The starring point is the generating of the A -unity operator as... 

With these the formalism is open of being applied of whatever quantum systems providing the chosen coordinate and momentum matrices provide through their combination in the Hamiltonian a diagonal matrix that can be read at once for the stationary eigen-values. Moreover, those operators have to fulfill the basic commutation mles above from this reason the choice of coordinate an momentum operators may eventually not being... 

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 ... 

The third postulate asserts that there is a hermitian mathematical operator for each mechanical variable. To write the operator for a given variable (1) write the classical expression for the variable in terms of Cartesian coordinates and momentum components, (2) replace each momentum component by the relation... 

In the beginning of this topic we pointed out some remarks about the cross-platform notation used in this paper. As one can see in the previous chapter we have used expansion coefficients cpq of the one-electron wavefunctions and eigenvectors ia/Pia secular equations for the potential/kinetic oscillator energy defined on the set of complex numbers. In quantum chemistry this is often irrelevant, all the mentioned coefficients may be real, but in solid-state physics the complex number notation is necessary. In a similar way we will use the cross-platform notation for coordinate and momentum oscillator operators, namely Br = br + bt and Br = br-bt. For systems that allow the real number solution of wavefunctions (quantum... 

The solution of time evolution problems for classical systems is facilitated by introducing a classical phase space representation that plays a role in the description of classical systems in a matmer that is formally analogous to the role played by the coordinate and momentum representations in quantum mechanics. The state vectors T ) of this representation enumerate all of the accessible phase points. The phase function /(E ) is given by / (f ) = (f I/), which can be thought to represent a component of the vector f) in the classical phase space representation. The application of the classical Liouville operator (f ) to the phase function /(f ) is defined by (f )/(f ) = (f I/), where is an abstract op-... 

Quantum mechanics can also be developed so that the functions describing the system are functions of momentum coordinates, not position coordinates. This is termed a momentum representation, and in this representation, position and momentum operators take on a different form. The picture of a quantum system, though, is equivalent, and the choice between representations is largely one of convenience. 

The angles 0, (j), and x are the Euler angles needed to specify the orientation of the rigid molecule relative to a laboratory-fixed coordinate system. The corresponding square of the total angular momentum operator fl can be obtained as... 

Using the fact that the quantum mechanical coordinate operators q = x, y, z as well as the conjugate momentum operators (pj = px, Py, Pz are Hermitian, it is possible to show that Lx, Ly, and L are also Hermitian, as they must be if they are to correspond to experimentally measurable quantities. 

In continuum field theory, the field dynamical variable and the coordinates X and t are only parameters. With the conjugate momentum f (T, t), and Hamiltonian operator... 

Thus the point group part of the operation works on the momentum coordinates and the translation part gives rise to a phase factor. We notice that this phase factor reduces to 1 in the diagonal elements, or in general when the difference between the the two arguments of N(p,p ) is a reciprocal lattice vector. 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

For illustration, we consider some examples involving only one variable, namely, the cartesian coordinate x, for which w x) = 1. An operator that results in multiplying by a real function /(x) is hermitian, since in this case fix) = fix) and equation (3.8) is an identity. Likewise, the momentum operator p = (i)/i)(d/dx), which was introduced in Section 2.3, is hermitian since... 

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