By Shi-Hai Dong
This ebook introduces the factorization process in quantum mechanics at a complicated point, with the purpose of placing mathematical and actual ideas and strategies just like the factorization process, Lie algebras, matrix parts and quantum regulate on the reader’s disposal. For this objective, the textual content presents a complete description of the factorization strategy and its broad purposes in quantum mechanics which enhances the conventional assurance present in quantum mechanics textbooks.
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Extra resources for Factorization Method in Quantum Mechanics
Once the exact solutions which, sometimes, can be expressed by the associated Laguerre functions, confluent hypergeometric functions and other special functions are known, we may construct the creation and annihilation operators by acting the first differential operator d/dξ on the normalized eigenfunctions and then by using the recursion relations among those special functions. Second, we may construct a suitable Lie algebra in terms of the obtained ladder operators so that we may study other properties of quantum system based on the Lie algebra.
10) eiβ0 = 1, where β0 = 2π for SO(3) group and β0 = 4π for SU(2) group. Hence, we find that must be integer for group SO(3) and 2 for group SU(2). This important property will be useful in constructing the quantum mechanical Hamiltonian by using the Lie algebra generators. 5. Properties of non-compact groups SO(2, 1) and SU(1, 1) We now give a brief review of non-compact Lie groups SO(2, 1) and SU(1, 1). 11) where the operator L3 is considered as the generator of the geometrical rotation, while L1 and L2 are the Lorentz transformation.
Abelian group: if f g = gf , we say that the elements f and g commute. If all elements of G commute, then G is a commutative or Abelian group. If G has a finite number of elements, it has finite order n(G), where n(G) is the number of elements. Otherwise, G has infinite order. , f ; h ∈ S −→ f h ∈ S. , G → H. Isomorphism: an isomorphism is a homomorphism which is one-to-one and "onto" . From the viewpoint of the abstract group theory, isomorphic groups can be identified. In particular, isomorphic groups have identical multiplication tables.
Factorization Method in Quantum Mechanics by Shi-Hai Dong