Operator Schrodinger

Here of course H is the Pock-space operator, whereas in Eq. (8-230) it is the Schrodinger operator. 

Foldy, L, L., 497,498,536,539 Foldy-Wouthuysen representation, 537 "polarization operator in, 538 position operator in, 537 Ford, L. R., 259 Four-color problem, 256 Fourier transforms of Schrodinger operators, 564... 

For wave functions like = exp[if x,t)], the squared operator would mask the phase information, since = <3> 2, and to avoid this, a linear Schrodinger operator would be preferred. This has the immediate advantage of a wave equation which is linear in both space and time derivatives. The most general equation with the required form is... 

In the Heisenberg picture the operators themselves depend explicitly on the time and the time evolution of the system is determined by a differential equation for the operators. The time-dependent Heisenberg operator AH(t) is obtained from the corresponding Schrodinger operator As by the unitary transformation... 

The domain of the Schrodinger operator on the graph is the L2 space of differentiable functions which are continuous at the vertices. The operator is constructed in the following way. On the bonds, it is identified as the one dimensional Laplacian — It is supplemented by boundary conditions on the vertices which ensure that the resulting operator is self adjoint. We shall consider in this paper the Neumann boundary conditions ... 

Vol. 1498 R. Lang, Spectral Theory of Random Schrodinger Operators. X, 125 pages. 1991. 

When we include in our model an explicit formula for the energy of the system, we can make stronger predictions. The energy observable for the hydrogen atom is completely described by the Schrodinger operator. 

Why should the spectral data for the alkali atoms resemble the spectral data for hydrogen Our model of the hydrogen atom, along with the Pauli exclusion principle (Section 1.2) and some other assumptions, provides an answer. For example, consider lithium, the third element in the periodic table. Its nucleus has a positive charge of three and it tends to attract three electrons. The Schrodinger operator for the behavior of a single electron in the presence of a lithium nucleus is... 

At long last, it is time to appeal to the Schrodinger operator... 

The Schrodinger operator can be used to make predictions about measurements of the energy of the electron in a hydrogen atom. For example, suppose (j) e satisfies the Schrodinger eigenvalue equation... 

Proposition 8.14 Each negative eigenvalue E of the Schrodinger operator has a finite number of linearly independen t eigenfunctions. 

Fix an eigenvalue E. Suppose we have a solution to the eigenvalue equation for the Schrodinger operator in the given form. I.e, suppose we have a function a 1 and a spherical harmonic function such that... 

It follows that at any critical point ro, i.e., any point such that a (ro) = 0, the real numbers a (to) and Q "(ro) must have the same sign. Hence there are no local maxima of a at points where the value of a is positive. Near the origin (r = 0) we have afi) r, so there must be a point ri such that Q (ri) > 0 and Q (ri) > 0. Hence for all r > ri, we have oi r) > a(ri)-, otherwise there would have to be a local maximum between ri and r, in a region where a, is positive. So Integral 8.18 cannot converge at the upper limit. In other words, a does not yield an -eigen I unction of the Schrodinger operator either. 

Because of the spherical symmetry of physical space, any realistic physical operator (such as the Schrodinger operator) must commute with the angular momentum operators. In other words, for any g e SO(3) and any f in the domain of the Schrodinger operator H we must have H o p(g ] = pig) o H, where p denotes the natural representation of 80(3 on L2(] 3 Exercise 8.15 we invite the reader to check that H does indeed commute with rotation. The commutation of H and the angular momentum operators is the infinitesimal version of the commutation with rotation i.e., we can obtain the former by differentiating the latter. More explicitly, we differentiate the equation... 

We can put these representations (one for each eigenvalue of the Schrodinger operator) together to form a representation of su(2) on the vector space of bound states of the hydrogen atom. We will see in Section 8.6 that there is a physically natural representation of the larger Lie algebra 5o(4) = 5m(2) 5m(2) on the set of bound states of the hydrogen atom. 

We can use the representation theory of the Lie algebra 50(4) along with the stunning fact that there is a representation of 5o(4) on the space of bound states of the Schrodinger operator with the Coulomb potential to make a satisfying prediction about the dimensions of the shells of the hydrogen atom and the energy levels of these shells. 

We consider one eigenspace of the Schrodinger operator at a time. Fix an eigenvalue < 0 of the Schrodinger operator. Let Ve denote the eigenspace corresponding to E. From Proposition 8.5 we know that there is a representation of 5m(2) on the eigenspace Ve- We will extend this to a representation of 5m(2) 5m(2). To this end we introduce three more operators on Ve- Define... 

If we restrict our attention to one energy level of the Schrodinger operator, then the Schrodinger eigenvalue equation (Equation 8.16 ) holds so that... 

En the n-th energy eigenvalue of the Schrodinger operator for the electron in the hydrogen atom, 12... 

Ve eigenspace of the Schrodinger operator corresponding to energy level E, 267... 

In this section the symbol ft is used for the molecular Schrodinger operator. 

E. Balslev, J. Combes, Spectral properties of many-body Schrodinger operators with dilation-analytic interactions, Commun. Math. Phys. 22 (1971) 280. 

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