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arXiv:2307.14640 · Quantum Information Processing

Euclidean time method in generalized eigenvalue equation

Mi-Ra Hwang, Eylee Jung, MuSeong Kim, DaeKil Park

Published Feb 15, 2024• 1 citation
arXivHTMLPDF
DOI
10.1007/s11128-024-04275-3

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Throughout our discussion of the general principles of quantum mechanics we have emphasized the importance of the eigenvalues and eigenfunctions of the operators representing physical quantities. However, we have seen that solving the basic eigenvalue equations determining these is often not straightforward. In the early chapters discussing the energy eigenvalue equation (or time-independent Schro¨dinger equation) for example, we found that this could often not be solved exactly and that, even when a solution was possible, it frequently required considerable mathematical analysis. Because of this, methods have been developed to obtain approximate solutions to eigenvalue equations. One of the most important of such techniques is known as time-independent perturbation theory and will be the first method to be discussed in the present chapter. Another, known as the variational principle, will be discussed later. We shall confine our discussion to the timeindependent Schro¨dinger equation, but generalization to other eigenvalue equations is quite straightforward. We shall mainly use the wave function representation, but we shall restate some of the important results in Dirac notation so that they can be readily transformed to a matrix representation if desired.
As is known, the hydrogen atom energy levels depend on the principal quantum number n only and are independent of the azimuthal quantum number l. One can say (by using the common but not very suitable term) that there is degeneracy (i.e., the multiplicity of the level) with respect to the azimuthal quantum number. On the other hand, there is a general rule according to which the multiplicity of the Schro¨dinger equation eigenvalues relates to the invariance of the equation under a definite group of transformations. So, for example, the invariance under a common rotation group (spherical symmetry) leads to the energy level independence on the magnetic quantum number m. Therefore, it should be expected that the energy level independence of the azimuthal quantum number is explained by the existence of a certain transformation group that is more general than the three-dimensional rotation group. Up to now, this group of transformations for the Schro¨dinger equation has not been found. In the present work, we shall show that this group is equivalent to the rotation group in the four-dimensional Euclidean space.