An uncertainty relation for the orbital angular momentum operator. (English) Zbl 1369.81053
Summary: A common reducible representation space of the Lie algebras \(\operatorname{su}(1,1)\) and \(\operatorname{su}(2)\) is equipped with two different types of scalar products. The representation bases are labeled by the azimuthal and magnetic quantum numbers. The generators of \(\operatorname{su}(2)\) are the \(x\)-, \(y\)- and \(z\)-components of the orbital angular momentum operator. The representation of each of these Lie algebras is unitary with respect to only one of the scalar products. To each positive magnetic quantum number a family of the \(\operatorname{su}(1,1)\)-Barut-Girardello coherent states is associated. The normalization and resolution of the identity condition for the coherent states are realized in two different approaches, i.e. the unitary and the non-unitary approaches. For the coherent states of the non-unitary case we calculate the uncertainty relation for the Hermitian \(x\)- and \(y\)-components of the angular momentum operator. While the unitary case leads to the known uncertainty relation for the Hermitian \(x\)- and \(y\)-components of \(\operatorname{su}(1,1)\) Lie algebra.
MSC:
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 81R30 | Coherent states |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
| 17B81 | Applications of Lie (super)algebras to physics, etc. |
| 62J10 | Analysis of variance and covariance (ANOVA) |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
orbital angular momentum operator; coherent states; uncertainty relation; \(\operatorname{su}(2)\) and \(\operatorname{su}(1,1)\) Lie algebrasReferences:
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