Position representation

Suppose we are given an arbitrary system Hamiltonian H(x, p) in terms of the dynamical variables x and p we will be more specific regarding the precise meaning of x and p later. The Hamiltonian is the generator of time evolution for the physical system state, provided there is no coupling to an environment or measurement device. In the classical case, we specify the initial state by a positive phase space distribution function fci(x,p) in the quantum case, by the (position-representation) positive... 

The Bohmian formalism follows from the Schrodinger one in the position representation after considering a change of variables, from the complex wave function field OP, P ) to the real fields (p, S) according to the transformation relation ... 

Applied to the example of a bounded quantum system discussed above, one might choose to examine directly the properties of the stationary basis functions This could, e.g., be done by investigating the coarse grained values of the wave functions in the position representation. Assuming that the quantum system is defined in the two-dimensional x,y space, one investigates the functions... 

Having identified the symmetries of the electronic distortion operators, we now determine the symmetries of the nuclear degrees of freedom. These are defined as the direct product of the positional representation with the symmetry of the translations [12,16]. The n-simplex is situated in a (n — 1)-dimensional space and thus will exhibit (n - 1) translations. The corresponding irrep is denoted as Ft-. One easily realizes that this will correspond to the (n — 1,1) irrep from the center of the simplex one can move in n different directions, but the vectorial sum of all these directions amounts to zero, hence the translational space has one degree of freedom less than the number of sites. The direct product can be decomposed in a standard way as follows ... 

In classical mechanics, positions and momenta are treated on an equal footing in the Hamiltonian picture. In quantum mechanics, they become operators, but it is true that the position r and momentum p of a particle are appropriate conjugate variables that can entirely equivalently describe a state of a system under the commutation relation [r, p] = i (Dirac, 1958). This equivalence is usually demonstrated by the example of the onedimensional harmonic oscillator. The choice of the most appropriate representation depends on convenient description of the phenomenon considered. Generally, the position representation is useful for most bound-state problems such as atomic and molecular electronic structures as well as for many scattering problems. The momentum-space treatment... 

An equally valid but different approach is to work with momentum-space wave functions n(yi,y2> 1 ) which depend upon the momentum-spin coordinates yj = (Pj, o ) jLi of the N electrons in the system but not on their positions. A momentum representation of the wave function does not yield any more or less information than the position representation of the wave function does. However, the momentum representation does provide a different perspective—one from the other end of Heisenberg s eyeglass. 

To interpret resonances and assign quantum numbers, plots of the local density are useful (42). The local density can be obtained by taking the diagonal term in the position representation of Eq. (47) to give... 

The time-evolution operator exp(— ) in the position representation is the Green function... 

The corresponding position representations are obtained by way of Fourier transform. 

This permutational representation is also called the ground representation. It describes the transformation of the coset space. The dimension of this coset space is G / //. In the case of a cluster, where each coset corresponds to a site, it represents the permutation of the positions of the sites. For this reason, it is also called the positional representation. Indeed, Eq. (4.72) may equally well be written as... 

For the X-value, which marks the position of the nonzero element in the /cth column of the matrix P, the product ) k is an element of Ha- We call this the subelement of I in Ha. As an example, for the case of the pyramidal complex, the matrices of the positional representation are listed in Table 4.6. If = gnhg, the diagonal element will be nonzero PicK(g) = 1- The following sum rules will thus hold, as can be verified from Table 4.6 ... 

Table 4.6 Ground or positional representation of the four equatorial ligand sites in a square pyramidal complex the sites are ordered as in Fig. 4.3(a)...

In Chap. 4 we left induction after the proof of the Frobenius reciprocity theorem. In that proof the important concept of the positional representation was introduced. This described the permutation of the sites under the action of the group elements. Further, we defined local functions on the sites which transformed as irreps of the site symmetry. As an example, if we want to describe the displacement of a cluster atom in a polyhedron, two local functions are required a totally-symmetric one for the radial displacement and a twofold-degenerate one for the tangential displacements. In cylindrical symmetry, these are labelled a and tt, respectively. The mechanical representation, i.e. the representation of the cluster displacements, is then the sum of the two induced representations ... 

This is precisely the set of fluorine displacements that we constructed in Sect. 4.8 in order to describe the vibrational modes of UFe. One remarkable result of induction theory is that the mechanical representation can also be obtained as the direct product of the positional representation and the translational representation, T u, this is the representation of the three displacements of the centre of the cluster. 

Theorem 14 Consider a standard fibre, consisting of a function space that is invariant under the action of the group. In a cluster of equivalent sites, we can form a fibre bundle by associating this standard fibre with every site position. The induced representation of the fibre bundle is then the direct product of the irrep of the standard fibre with the positional representation. 

The vertices, being zero-dimensional points, form a set of nodes, (m), which are permuted under the symmetry operations of the polyhedron. The representation of this set is the positional representation, Faiv). The a here refers to the fact that the sites themselves transform as totally-symmetric objects in the site group. If the cluster contains several orbits, the induced representation is of course the sum of the individual positional representations. In Fig. 6.8 the vertex representation is Ai -b T2. In Sect. 4.7 we have already encountered these irreps, when discussing the sp hybridization of carbon. 

Ascent in symmetry tables have been provided by Boyle [4], Fowler and Quinn have listed the irreps that are induced by u-, n-, and S-type orbitals on molecular sites [5], These tables are reproduced below. They are useful for the construction of cluster orbitals. Geg always denotes the regular representation. r corresponds to the positional representation. The mechanical representation is the sum r a + r jt-... 

The similarity in form between the two real equations implied by the single-body spin-0 Schrddinger equation in the position representation (wave mechanics) and the equations of fluid mechanics with potential flow in its Eulerian formulation was first pointed out by Madelung in 1926 [1]. In this analogy, the probability density is proportional to the fluid density, and the phase of the wave function is a velocity potential. A novel feature of the quantum fluid is the appearance of quantum stresses, which are usually represented through the quantum potential. To achieve mathematical equivalence of the models, the hydrodynamic variables have to satisfy... 

In an alternative formulation called the momenrnm representation, the momentum operator is taken to be simply multiplication by the classical momentum and the position operator is i H times the derivative with respect to momentum. The position representation described above is more widely used, but all the predictions concerning observable quantities are the same. 

In the calculation of expectation values it is convenient to introduce the so called bra 4> and ket 4>) notation with the first representing the wavefunction and the second its complex conjugate. In the bra and ket expressions the spatial coordinate r is left deliberately unspecified, so that they can be considered as wavefunctions independent of the representation when the coordinate r is specified, the wavefunctions are considered to be expressed in the position representation . Thus, the expectation value of an operator O in state

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The bra and ket notation can be extended to situations that involve more than one particle, as in the many-body wavefunction relevant to electrons in a solid. For example, such a many-body wavefunction may be denoted by 4 ), and when expressed in the position representation it takes the form... 

The corresponding operator must be defined as i5(r - r ), with the second variable an arbitrary position in space. This choice of the density operator, when we take its matrix elements in the state (p) by inserting a complete set of states in the position representation, gives... 

In such cases we adopt the convention that the order of the single-particle states in the bra or the ket is meaningful, that is, when expressed in a certain representation the nth independent variable of the representation is associated with the nth single-particle state in the order it appears in the many-body wavefunction for example, in the position representation we will have... 

Thus, when expressing matrix elements of the many-body wavefunction in the position representation, the set of variables appearing as arguments in the single-particle states of the bra and the ket must be in exactly the same order for example, in the Hartree theory, Eq. (2.10), the Coulomb repulsion term is represented by... 

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