A quantum mechanical moment problem

Alexei Daletskii

Stanisław Lech Woronowicz

Stanisław Lech Woronowicz

Francis J. Narcowich

Long ago, Moyal [Proc. Cambridge Philos. Soc. 45, 99 (1949)] formulated a moment problem in the context of the Wigner–Weyl phase-space formulation of quantum mechanics. The problem amounts to giving necessary and sufficient conditions for a sequence of numbers to be moments of a Wigner function. In this paper, that problem is solved, and so is a truncated version of it.

Paolo Vanini

Pei-Lin You, Xiang-You Huang

The linear Stark effect for the first excited state of the hydrogen atom shows that, in the unperturbed states, the atom has a permanent electric dipole moment (EDM) of magnitude 3eao (ao is Bohr radius). The EDM is not induce by the external field but is inherent behavior of the atom. But the calculation of quantum mechanics tells us that unperturbed states of hydrogen atom have no EDM! In the effect, two of four states have no energy shift. What are the EDM of the hydrogen atoms in the two states? Quantum mechanics can not answer the problem. The statement that the EDM of the two states is perpendicular to the field only comes from guesses in quantum mechanics. The two problems had puzzled physicists for more than 80 years. By introducing a new parameter this article gives a satisfactory explanation for the effect. Our calculation discovered that, in the unperturbed states, the atoms not only have EDM with 3eao but also can have only three directions of quantization! It is an entirely unexpected discovery. This research is a vital clue that not all is well with quantum mechanics. We prophesy that hydrogen-like atoms, such as K, Rb and Cs atom etc, may have large EDM of the order of magnitude eao in their ground state (see arXiv:0809.4767, arXiv:0810.0770 and arXiv:0810.2026).