Abstract
The evolution of a N-dimensional quantum mechanical system where N = 1 or 2 is described by the solutions to a time-dependent Schrödinger equation [1, 2, 3],
where the Schrödinger operator is given by
and V(x, τ) is, for the moment, an arbitrary time-dependent potential. The symbol x = (x 1, x 2) refers to the 2-tuple of coordinates in the 2-dimensional configuration space [4]. If N = 1, then we shall write x = x l = x. The symbol ∂ σ represents the partial derivative with respect to x σ . The analytical solution of Eq. (1) is possible for certain choices of the τ-dependent potential, V(x, τ), because in those cases, Eq. (1) admits Lie symmetries. We shall work with space-time or kinematical symmetries of Eq. (1).
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This is a simplification and not a necessary requirement. See Ref. [7].
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Truax, D.R. (1997). Lie Symmetries in Quantum Mechanics. In: Calais, JL., Kryachko, E. (eds) Conceptual Perspectives in Quantum Chemistry. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5572-4_5
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DOI: https://doi.org/10.1007/978-94-011-5572-4_5
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