Dedicated to the 60th birthday of Prof. Dr. Harry Paul
- Autor(in)
- Referenz
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- Seitenbereich
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0181 - 0197
- Schlagwort(e)
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<KWD>Boson operator
Dual orthogonality and completeness
Eigenvalue problem
Squeezed coherent states
- Zusammenfsg.
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The eigenvalue problem for arbitrary linear combinations <I>k</I>α + μα<sup>‡</sup> of a boson annihilation operator α and a boson creation operator α<sup>‡</sup> is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | <I>n</I> >, (<I>n</I> = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ &tbond; μ/<I>k</I> and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators <I>k</I>α + <I>k</I>* α<sup>‡</sup> to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α<sup>‡</sup> are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ &tbond; μ/<I>k</I> as a parameter.
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