Separability of the wave equation

The quantum states of samples of solids, liquids or gases, of which we have spoken in the statistical discussion, are obtained in principle by solving equation (12 12). However, the equation can only be solved in actuality under especially simple conditions, particularly where there are separable energy states. The particular significance of separability—and this applies whether we carry out the separation as between different particles (as in the case of a perfect gas) or as between the different kinds of energy (translational, rotational, etc.) of any single molecule—is that it allows equation (12 12) to be broken down into a number of simpler differential equations, each containing a smaller number of variables. 

In order to illustrate this point consider the form of equation (12 12) when it contains only two independent variables x and y. We shall set about the problem of separating this equation into two others, one containing x only and the other containing y only. For simplicity it will be supposed that the physical situation corresponds to one of constant potential energy, i.e. V is not a function of a or y 

Solution of these equations (other than the trivial solution arsO, y s=0) can be obtained only when E has the values 3 or — 1. For any other values of E the equations are inconsistent and thus the two equations could not refer simultaneously to the same physical problem involving tho variables x and y. The eigenvalue E Z has as its solution, or eigenfunction, x y and the eigenvalue E— — has as its eigenfunction y. 

This would be the wave equation of a particle of mass m which moves only in an x-y plane of constant potential energy. Alternatively, it could refer to a system of two particles, of equal mass, one confined to motion along the x axis and the other to motion along the y axis. 

Let it be assumed, for purposes of trial, that an eigeufonction corresponding to a permitted value of Ey may be expressed as a product of a function which depends only on x and a function which depends only on y  

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The two degrees of freedom associated with the ring puckering are, therefore, an ordinary vibration and a type of one-dimensional rotation in which the phase of the puckering moves around the ring the latter is not, however, a true rotation since there in no angular momentum about the axis of rotation, and so is described as a pseudo-rotation . This separation of the wave equation is not exact, but it has been stated that exact separation is possible and, on the assumptions of harmonic oscillations and small amplitudes of vibrations, leads to the same results as those given. 

Because of the interelectronic repulsion term l/ri2, the electronic Hamiltonian is not separable and only approximate solution of the wave equation can be considered. The obvious strategy would be to use Hj wave functions in a variation analysis. Unfortunately, these are not known in functional form and are available only as tables. A successful parameterization, first proposed by James and Coolidge [89] and still the most successful procedure, consists of expressing the Hamiltonian operator in terms of the four elliptical coordinates 1j2 and 771 >2 of the two electrons and the variable p = 2ri2/rab. The elliptical coordinates 4> 1 and 2, as in the case of Hj, do not enter into the ground-state wave function. The starting wave function for the lowest state was therefore taken in the power-series form... 

This equation can be solved by separation of variables, provided the potential is either a constant or a pure radial function, which requires that the Lapla-cian operator be specified in spherical polar coordinates. This transformation and solution of Laplace s equation, V2 / = 0, are well-known mathematical procedures, closely followed in solution of the wave equation. The details will not be repeated here, but serious students of quantum theory should familiarize themselves with the procedures [15]. 

In Chapter 3 we investigated the development in time of a decaying state, expressed in terms of the time-independent eigenfunctions satisfying a system of two coupled differential equations, resulting from the separation of the Schrodinger equation in parabolic coordinates. In this analysis we obtained general expressions for the time-dependent wave function and the probability amplitude. 

Equation 17-9, the second part of the wave equation, is a function of p and

[Pg.107]

Contrary to the mass of the nucleus, its size influences the binding energy considerably in heavy ions (Fig. 10). In studying nuclear size effects nowadays always a spherically symmetric charge distribution of the nucleus is assumed which allows a separation of the Dirac equation and corresponding wave function into an angular part and a radial part similar to the point nucleus case. The radial Dirac equation then reads [45]... 

Similar considerations apply when the wave equation contains more than two independent variables. However, it is to be emphasized that separability is not a necessary property of the wave equation, and it can be achieved only when the potential energy is constant or is a particularly simple function of the co-ordinates. In the present connexion three important tjrpes of separability are as follows ... 

In equation (12 8) the partition function Q was expressed as a product of the partition functions /, of each of the N molecules, together with the factor jN which allows for indistinguishability. As noted in the last section, the justification for this procedure is the separabiUty of the wave equation in the case of a perfect gas. In addition, there is also the separability which is expressed by equations (12 21) and (12-22), and this permits a factorization of / itself. Let <0 - and be the degeneracies of the th translational level and the kth internal level respectively. The degeneracy of the energy state... 

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

This part of our chapter has shown that the use of the two variables, moduli and phases, leads in a direct way to the derivation of the continuity and Hamilton-Jacobi equations for both scalar and spinor wave functions. For the latter case, we show that the differential equations for each spinor component are (in the nearly nomelativistic limit) approximately decoupled. Because of this decoupling (mutual independence) it appears that the reciprocal relations between phases and moduli derived in Section III hold to a good approximation for each spinor component separately, too. For velocities and electromagnetic field strengths that ate nomrally below the relativistic scale, the Berry phase obtained from the Schrddinger equation (for scalar fields) will not be altered by consideration of the Dirac equation. 

In the few two- and three-dimensional cases that pemiit exact solution of the Schroedinger equation, the complete equation is separated into one equation in each dimension and the energy of the system is obtained by solving the separated equations and summing the eigenvalues. The wave function of the system is the product of the wave functions obtained for the separated equations. 

This separation will allow the students to properly assess the measurement process, which plays a special and complex role in QM that is different from its role in any classical theory. Just as Kepler s laws only cover the free-falling part of the trajectories and the course corrections, essential as they may be, require tabulated data, so too in QM, it should be made clear that the Schrbdinger equation governs the dynamics of QM systems only and measurements, for now, must be treated by separate mles. Thus the problem of inaccurate boundaries of applicability can be addressed by clearly separating the two incompatible principles governing the change of the wave function the Schrbdinger equation for smooth evolution as one, and the measurement process with the collapse of the wave function as the other. 

The wave equation is a second-order partial differential equation in three variables. The usual technique for solving such an equation is to use a procedure known as the separation of variables. However, with r expressed as the square root of the sum of the squares of the three variables, it is impossible... 

In order to solve the wave equation for the hydrogen atom, it is necessary to transform the Laplacian into polar coordinates. That transformation allows the distance of the electron from the nucleus to be expressed in terms of r, 9, and (p, which in turn allows the separation of variables technique to be used. Examination of Eq. (2.40) shows that the first and third terms in the Hamiltonian are exactly like the two terms in the operator for the hydrogen atom. Likewise, the second and fourth terms are also equivalent to those for a hydrogen atom. However, the last term, e2/r12, is the troublesome part of the Hamiltonian. In fact, even after polar coordinates are employed, that term prevents the separation of variables from being accomplished. Not being able to separate the variables to obtain three simpler equations prevents an exact solution of Eq. (2.40) from being carried out. 

The wave equation (34) is found to be separable in terms of the product function... 

There are several important conclusions to be drawn from the Hj example. As expected, separation of the electronic wave equation permits exact solution of the one-electron problem. The result provides a benchmark... 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

The basic idea of the EIM is to use a separation ansatz for the x- and the y-dependency of the field, x,y)- x)-constant refractive indices the wave equation thus reads as... 

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