Abstract
Group action is extremely important in quantum mechanics. Suppose the Hamiltonian of a quantum system is invariant under a symmetry transformation of its independent parameters such as position, momentum, and time. This invariance will show up as certain properties of the solutions of the Schrödinger equation.
Moreover, the very act of labeling quantum-mechanical states often involves groups and their actions. For example, labeling atomic states by eigenvalues of angular momentum assumes invariance of the Hamiltonian under the action of the rotation group (see Chap. 29) on the Hilbert space of the quantum-mechanical system under consideration.
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Notes
- 1.
It will become clear shortly that the appropriate direction for the action is from the right.
- 2.
We have already encountered the notion of representation in the context of algebras. Groups are much more widely used in physics than algebras, and group representations have a wider application in physics than their algebraic counterparts. Since some readers may have skipped the section on the representation of algebras, we’ll reintroduce the ideas here at the risk of being redundant.
- 3.
Chapter 29 will make explicit the connection between groups and their generators.
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Hassani, S. (2013). Representation of Groups. In: Mathematical Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-01195-0_24
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DOI: https://doi.org/10.1007/978-3-319-01195-0_24
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01194-3
Online ISBN: 978-3-319-01195-0
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