Operator wave mechanics

Eigen function In wave mechanics, the Schrodinger equation may be written using the Hamiltonian operator H as... 

Requiring the variation of L to vanish provides a set of equations involving an effective one-electron operator (hKs), similar to the Fock operator in wave mechanics... 

In wave mechanics the electron density is given by the square of the wave function integrated over — 1 electron coordinates, and the wave function is determined by solving the Schrddinger equation. For a system of M nuclei and N electrons, the electronic Hamilton operator contains the following tenns. 

The wave mechanics discussed in Chapter 2 is a linear theory. In order to develop the theory in a more formal manner, we need to discuss the properties of linear operators. An operator 4 is a mathematical entity that transforms a function ip into another function 0... 

To represent observables in n-dimensional space it was concluded before that Hermitian matrices were required to ensure real eigenvalues, and orthogonal eigenvectors associated with distinct eigenvalues. The first condition is essential since only real quantities are physically measurable and the second to provide the convenience of working in a cartesian space. The same arguments dictate the use of Hermitian operators in the wave-mechanical space of infinite dimensions, which constitutes a Sturm-Liouville problem in the interval [a, 6], with differential operator C(x) and eigenvalues A,... 

Because of these properties of Hermitian functions it is accepted as a basic postulate of wave mechanics that operators which represent physical quantities or observables must be Hermitian. The normalized eigenfunctions of a Hermitian operator constitute an orthonormal set, i.e. 

The quantum-mechanical state is represented in abstract Hilbert space on the basis of eigenfunctions of the position operator, by F(q, t). If the eigenvectors of an abstract quantum-mechanical operator are used as a basis, the operator itself is represented by a diagonal square matrix. In wave-mechanical formalism the position and momentum matrices reduce to multiplication by qi and (h/2ni)(d/dqi) respectively. The corresponding expectation values are... 

For systems with classical analogues every observable quantity is an algebraic function of <7, and p . The wave-mechanical operator acting on the -function is... 

Other formulations that are analogous to wave mechanics in an arbitrary basis, are obtained by defining the operator A(-a ) according to... 

The fundamental equivalence between Schrodinger s wave-mechanical and Heisenberg s matrix-mechanical representation of quantum theory implies that H (or Hm>) may be viewed as a differential operator or a matrix. The latter viewpoint is usually more convenient in the... 

Apart from the operational, wave or action-based pictures of quantum mechanics provided by Heisenberg, Schrodinger, or Feynman, respectively, there is an additional, fully trajectory-based picture Bohmian mechanics [20,23]. Within this picture, the standard quantum formalism is understood in terms of trajectories defined... 

In Chapter 1 we have discussed the familiar realization of quantum mechanics in terms of differential operators acting on the space of functions (the Schrodinger wave function formulation, also called wave mechanics ). A different realization can be obtained by means of creation and annihilation operators, leading to an algebraic formulation of quantum mechanics, sometimes called matrix mechanics. For problems with no spin, the formulation is done in terms of boson creation, b (, and annihilation, ba, operators, satisfying the commutation relations... 

The (bilinear) expansion in the products of boson operators b]j serves to ensure the correspondance with quantum mechanics. To see this explicitly, say A, B, and C are operators familiar from wave mechanics and let A,B, and C be their corresponding matrix representations. If [A, 3] = C, then [A, B] = C. Now define... 

Physicist P. A. M. Dirac suggested an inspired notation for the Hilbert space of quantum mechanics [essentially, the Euclidean space of (9.20a, b) for / — oo, which introduces some subtleties not required for the finite-dimensional thermodynamic geometry]. Dirac s notation applies equally well to matrix equations [such as (9.7)-(9.19)] and to differential equations [such as Schrodinger s equation] that relate operators (mathematical objects that change functions or vectors of the space) and wavefunctions in quantum theory. Dirac s notation shows explicitly that the disparate-looking matrix mechanical vs. wave mechanical representations of quantum theory are actually equivalent, by exhibiting them in unified symbols that are free of the extraneous details of a particular mathematical representation. Dirac s notation can also help us to recognize such commonality in alternative mathematical representations of equilibrium thermodynamics. 

Transition Moment.—A wave-mechanical quantity whiph is proportional to the square root of the intensity of a transition, and is given by the integral f wave functions of the initial and final states. The dipole moment vector, M is given by M = Ser where r is the radius vector from the center of gravity of the positive charge to the electron. M is also known as the dipole moment operator. 

The only assumption, in addition to Bohr s conjecture, is that the electron appears as a continuous fluid that carries an indivisible charge. As already shown, Bohr s conjecture, in this case, amounts to the representation of angular momentum by an operator L —> ihd/dp, shown to be equivalent to the fundamental quantum operator of wave mechanics, p —> —ihd/dq, or the difference equation (pq — qp) = —ih(I), the assumption by which the quantum condition enters into matrix mechanics. In view of this parallel, Heisenberg s claim [13] (page 262), quoted below, appears rather extravagent ... 

That is to say that the quantity p from classical mechanics is thus replaced in wave mechanics by an operator. Likewise the potential energy is also replaced by an operator, namely the multiplication of the function of the potential energy V by the wave function 9. 

In the calculation of the total energy W (p. 123) as well as other dynamical quantities which can be written classically as a function of p and q, the method of calculation in wave mechanics is to be found by means of the transformation to the corresponding operators according to the same rules. 

The resonance concept thus removes both difficulties. In fact, if Kekule had arrived on experimental grounds at the hypothesis that the oscillation between the two structures might at the same time be the cause of the stabilization, then the resonance concept would have been anticipated in the same way as the tetrahedral carbon atom of Van 5t Hoff and Le Bel anticipated the wave-mechanical theory of mixed wave functions. Now the resonance concept can, however, like the tetrahedral carbon atom, be operated even without the wave mechanical theories, which lie at the basis of these concepts, being always applied explicitly. 

Until this stage all discussions have been based on classical mechanics. However, in the present formulation the translation to quantum mechanics is quite straightforward, since quantum momenta can be defined from the momentum transformation with only slight modifications. Furthermore, the resulting expressions are general in the sense that they can be derived without considering any particular representation of the momenta as differential operators. They apply equally well in a wave mechanical context. 

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