Coordinate and Momentum Representations

For ID motion, the eigen-value problem of the coordinate operator x 

Therefore, since the eigen-value a may take any all values in the range in a eontinuous way, one coneludes the coordinate spectrum is continuous, thus carrying the basic continuous closure and scalar product relationships 

Moreover, when considering the Fourier expansion of an arbitrary state vector in other discrete ortho-normalized base ( ),  

Other interesting consequence is the coordinate representation of a function of coordinate operator f acting on a state vector m)  

Quantum Nanochemistry-Volume I Quantum Theory and Observability 

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Examples of representations in common use in atomic reaction theory are the coordinate and momentum representations where, if the system under study is a single electron, the basis states are the eigenstates (r and (pi of the position and momentum of the electron respectively. Examples of discrete representations are also important. They will be left until later. 

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... 

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-... 

We recall some basic results of quantum dynamics [3], First, the state of the system and the time evolution can be expressed in a generalized (Dirac) notation, which is often very convenient. The state at time t is specified by x(t)) with the representations x(-Rjf) = (R x t)) and x P,t) = (P x(t)) in coordinate and momentum space, respectively. Probability is a concept that is inherent in quantum mechanics. (R x(t)) 2 is the probability density in coordinate space, and (-P x(f) 2 is H e same quantity in momentum space. The time evolution (in the Schrodinger picture) can be expressed as... 

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 corresponding Sr(Sp) plot for the hydride-exchange Sn2 process again exhibits the maximum (minimum) at TS, with two additional minima (maxima) in its vicinity, where the bond breaking is supposed to occur. These additional, pre-and post-US features are symmetrically placed in the entrance and exit valleys, relative to TS structure, but now they appear at roughly the same values of the Intrinsic Reaction Coordinate (IRC) in both the position and momentum representations. This is in contrast to the two-stage abstraction mechanism, where in the entrance valley the p-space maximum of the Shannon entropy has preceded the associated minimum observed in the r-space. This simultaneous r-localization (p-delocalization) may be indicative of the single-St p mechanism in which the approach of the nucleophile is perfectly synchronized with the concomitant... 

Physically, why does a temi like the Darling-Dennison couplmg arise We have said that the spectroscopic Hamiltonian is an abstract representation of the more concrete, physical Hamiltonian fomied by letting the nuclei in the molecule move with specified initial conditions of displacement and momentum on the PES, with a given total kinetic plus potential energy. This is the sense in which the spectroscopic Hamiltonian is an effective Hamiltonian, in the nomenclature used above. The concrete Hamiltonian that it mimics is expressed in temis of particle momenta and displacements, in the representation given by the nomial coordinates. Then, in general, it may contain temis proportional to all the powers of the products of the... 

The potential energy part is diagonal in the coordinate representation, and we drop the hat indicating an operator henceforth. The kinetic energy part may be evaluated by transfonning to the momentum representation and carrying out a Fourier transform. The result is... 

The Coordinate-Momentum Transformation.—We shall first derive the matrix expressions for the operators Pk on the j-represen-tation, and for the operators Qk on the p-representation. From these we shall then be able to derive the transformation matrices connecting the q- and the -representations. We start with the evident relationships ... 

Traditional hydrogenic orbitals used in atomic and molecular physics as expansion bases belong to the nlm) representation, which in configuration space corresponds to separation in polar coordinates, and in momentum space to a separation in spherical coordinates on the (Fock s) hypersphere [1], The tilm) basis will be called spherical in the following. Stark states npm) have also been used for atoms in fields and correspond to separation in parabolic coordinates an ordinary space and in cylindrical coordinates on (for their use for expanding molecular orbitals see ref. [2]). A third basis, to be termed Zeeman states and denoted nXm) has been introduced more recently by Labarthe [3] and has found increasing applications [4]. 

This commutation relation is easily verified in the coordinate representation leaving x untouched (x = x ) and using the above definition for p. In the momentum representation... 

Generally speaking, the representation in terms of occupation numbers is considered to be an independent quantum-mechanical representation, distinct from the coordinate (or momentum) one. In that case, the occupation numbers for one-particle states are dynamic variables, and operators are the quantities that act on functions of these variables. In this section, second-quantization representation is directly related to coordinate representation in order that in what follows we may have a one-to-one correspondence between quantities derived in each of these representations. 

After about 50 fs (t) has reached the asymptotic region where the torque is essentially zero and the distribution does not alter any further, i.e., the dissociation is over. Figure 10.6 illustrates rotational excitation in the angular momentum representation, whereas Figure 10.2(b) manifests rotational excitation in the coordinate picture. 

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... 

Thus, we obtain a special class of so-called diagonal representations of so(4) characterized by j0 = 0 (or jl = j2). This is analogous to the situation in angular momentum theory where only integral values of orbital angular momentum are possible. The half-odd-integral values are ruled out because of the particular realization of L in terms of coordinates and momenta. 

We keep the quantum number / in the notation for the state because it is necessary to keep track of the parity, which is a property of coordinate or momentum space (represented here by the orbital eigenstate) and has nothing to do with spin space. The coordinate-spin representation of (3.91) may be called a jj coupling function because of its use in states for systems of several electrons where the total angular momentum is obtained by vector addition of the angular momenta J of each electron. 

Throughout this section, the canonical density matrix and the Feynman propagator can be used interchangeably, the transformation P = it taking C into the propagator K, with t the time. While most frequently we shall use the coordinate representation r and r, it will be convenient in this section to work in k or momentum representation, by taking a double Fourier transform with respect to r and r. ... 

Note, that the interaction in the momentum representation depends only on the difference between the momenta p and p. Therefore, we can introduce the coordinate P = p —p. Then, the transition probability can be written as... 

In the Fourier representation, the transformation matrix G = Uf, ((Uf)Jk = (lf /Ng e,2vik > ) is unitary and supplies the means to transform the wave function from coordinate to momentum space. Moreover, the expansion functions gk(q) = e 2 nk ir are eigenvalues of the derivative matrix. Therefore, dkk = (i2Trk/L)hkk which leads to the Fourier derivative formula for order n ... 

Now specify a cartesian coordinate system with orthogonal unit vectors satisfying a 6y = in the laboratory reference frame. It is convenient to introduce a momentum representation for the fields through fourier transformation between x and k in physical applications the momentum vector k can often be identified with the wavevector of a photon. We choose the z-axis to be along the direction of k (i.e. k = k ej. A vector field U(k) may be decomposed into its longitudinal and transverse components... 

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