By Shi-Hai Dong
This paintings introduces the factorization process in quantum mechanics at a complicated point with an target to place mathematical and actual ideas and methods just like the factorization strategy, Lie algebras, matrix parts and quantum keep an eye on on the reader’s disposal. For this function a entire description is equipped of the factorization approach and its vast functions in quantum mechanics which enhances the normal assurance present in the prevailing quantum mechanics textbooks. relating to this vintage strategy are the supersymmetric quantum mechanics, form invariant potentials and staff theoretical techniques. it's no exaggeration to claim that this system has develop into the milestone of those techniques. actually the author’s motive force has been his wish to offer a finished evaluation quantity that comes with a few new and important effects in regards to the factorization strategy in quantum mechanics because the literature is inundated with scattered articles during this box, and to pave the reader’s means into this territory as speedily as attainable. the outcome: transparent and comprehensible derivations with the mandatory mathematical steps integrated in order that the clever reader can be capable of stick to the textual content with relative ease, specifically whilst mathematically tricky fabric is presented.
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Additional resources for Factorization Method in Quantum Mechanics (Fundamental Theories of Physics)
3) k = 1, 2, 3, . . , N. Here j|Lk |i denotes the matrix element of the operator Lk , and |i , |j are two different states of the system. 1). As we know, once the irreducible representations are obtained, then any matrix representation can be constructed by using the irreducible representations as building blocks. In general, for a given matrix representation, we can obtain an identical block diagonal structure by applying a similarity transformation to all the matrices. If so, we say that the representation is reducible since the block matrices can form smaller dimensional matrix representations of the Lie algebra.
In fact, all unitary irreducible representations of the Lie algebra so(3) are finite dimensional . Additionally, all nontrivial unitary irreducible representations of the Lie algebra so(2, 1) are infinite dimensional . On the other hand, we find that the unitary irreducible representations of the Lie algebra so(2, 1) are not as well known as those of the Lie algebra so(3), but as a spectrum generating algebra, the Lie algebra so(2, 1) has played an important role in many simple quantum systems [168, 169, 209].
On the other hand, it often runs parallel to the differential equation approach due to the great scientist Schr¨odinger. Even though Pauli used the algebraic method to treat the hydrogen atom in 1926  and Schr¨odinger also solved the same problem almost at the same time , their fates were quiet different. This is because the standard differential equation approach was more accessible to the physicists than the algebraic method. As a result, the algebraic approach to determine the eigenvalue of the hydrogen atom was largely forgotten and the algebraic techniques went into abeyance for a few decades.