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We propose proper quantization rule, ∫xAxBk(x)dx−∫x0Ax0Bk0(x)dx=nπ, where k(x)=2M[E−V(x)]/ℏ. The xA and xB are two turning points determined by E=V(x), and n is the number of the nodes of wave function ψ(x). We carry out the exact solutions of solvable quantum systems by this rule and find that the energy spectra of solvable systems can be determined only from its ground state energy. The previous complicated and tedious integral calculations involved in exact quantization rule are greatly simplified. The beauty and simplicity of the rule come from its meaning—whenever the number of the nodes of ϕ(x) or the number of the nodes of the wave function ψ(x) increases by 1, the momentum integral ∫xAxBk(x)dx will increase by π. We apply this proper quantization rule to carry out solvable quantum systems such as the one-dimensional harmonic oscillator, the Morse potential and its generalization, the Hulthén potential, the Scarf II potential, the asymmetric trigonometric Rosen–Morse potential, the Pöschl–Teller type potentials, the Rosen–Morse potential, the Eckart potential, the harmonic oscillator in three dimensions, the hydrogen atom, and the Manning–Rosen potential in D dimensions.

Objective: Obesity-related disease risks may vary depending on whether the subject has metabolically healthy obesity (MHO) or metabolically unhealthy obesity (MUO). At least 5 definitions/criteria of obesity and metabolic disorders have been documented in the literature, yielding uncertainties in a reliable international comparison of obesity phenotype prevalence. This report aims to compare differences in MHO and MUO prevalence according to the 5 most frequently used definitions. Methods: A random sample of 4,757 adults aged 35 years and older (male 51.1%) was enrolled. Obesity was defined either according to body mass index or waist circumference, and the definitions of metabolic abnormalities were derived from 5 different criteria. Results: In MHO, the highest prevalence was obtained when using the homeostasis model assessment (HOMA) criteria (13.6%), followed by the Chinese Diabetes Society (11.4%), Adult Treatment Panel III (10.3%), Wildman (5.2%), and Karelis (4.2%) criteria; however, the MUO prevalence had an opposite trend to MHO prevalence. The magnitude of differences in the age-specific prevalence of MHO and MUO varied greatly and ranked in different orders. The proportion of insulin resistance for MHO and MUO individuals differed significantly regardless of which metabolic criterion was used. Conclusion: The prevalence of MHO and MUO in the Chinese population varies according to different definitions of obesity and metabolic disorders.

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