By Barry G. Adams

This e-book presents an advent to using algebraic equipment and sym bolic computation for easy quantum structures with functions to massive order perturbation idea. it's the first ebook to combine Lie algebras, algebraic perturbation concept and symbolic computation in a kind appropriate for college students and researchers in theoretical and computational chemistry and is with ease divided into elements. the 1st half, Chapters 1 to six, presents a pedagogical creation to the real Lie algebras so(3), so(2,1), so(4) and so(4,2) wanted for the learn of straightforward quantum platforms akin to the D-dimensional hydrogen atom and harmonic oscillator. This fabric is acceptable for complicated undergraduate and starting graduate scholars. Of specific value is using so(2,1) in bankruptcy four as a spectrum producing algebra for a number of very important platforms corresponding to the non-relativistic hydrogen atom and the relativistic Klein-Gordon and Dirac equations. This method offers an attractive and significant substitute to the standard textbook strategy utilizing sequence recommendations of differential equations.

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**Extra resources for Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory**

**Sample text**

These results will be useful later for the representation theory of so( 4). In the final section an alternate derivation of the matrix representation of a vector operator is given using some of the more advanced aspects of angular momentum theory based on the Wigner-Eckart theorem. This section is not essential for the understanding of later chapters. 2) The Casimir operator does not belong to so(3) since it is not a linear combination of the generators2. 3) lThe word "unirrep" refers to representations of the Lie group by unitary matrices or operators but it is common to also use it to refer to representations of the Lie algebra.

4 Realizations of so(2,1) It is not as straightforward to obtain realizations of so(2,1) as it is for so(3). Given the correspondence principle it is clear how to extend the classical orbital angular momentum to its quantum mechanical counterpart in order to obtain a realization in the coordinate representation for the generators of so(3). However there is no similar classical analogue in the coordinate representation for the generators of so(2,1). Since our goal is to use so(2,1) as a spectrum generating algebra for simple quantum systems such as the hydrogen atom and the harmonic oscillator, we look for realizations which can be used to express the hamiltonians for these systems in terms of the generators Tk.

Finally some useful vector identities involving L are r·L = L·r = 0, p·L = L·p= 0, p X L = r l - p(r. 35) P = 2ip. 5. 31) illustrates another useful rule: since r X r = O. This result also follows from the general rule: If A = L,jk ajk8jk is such that ajk is antisymmetric in j and k and 8jk is symmetric in j and k, then A = O. We can conclude that r • L = 0 since fjkl is antisymmetric in j and k but is symmetric in j and k. 1 Prove that if A = L,jk ajk8jk is such that ajk is antisymmetric in j and k and 8jk is symmetric in j and k, then A = O.