Using quantum theory and MATLAB software, the basic properties of two-dimensional linear harmonic oscillators in quantum mechanics are systematically studied in polar coordinate system, and obtain the visualized results. The results show that, in polar coordinate system, with the exception of special case nr
=0, |m|=0, the degeneracy of two-dimensional linear harmonic oscillator is 2nr
+|m|+1, and the corresponding energy eigenvalues is ħω (2nr
+|m|+1). The number of intersection line between wave function and the plane with ψ=0 is 2nr
+m. In the case of nr
=0, the maximum number of probability density distributions is 2|m|. The results of this visualization are in complete agreement with the theoretical results. The visualization results in different coordinate systems can be verified with each other, which opens up a new research idea and also provides an idea for other quantum theoretical models to be studied.
Understanding human mobility patterns is of great importance for a wide range of applications from social networks to transportation planning. Toward this end, the spatial-temporal information of a large-scale dataset of taxi trips was collected via GPS, from March 10 to 23, 2014, in Beijing. The data contain trips generated by a great portion of taxi vehicles citywide. We revealed that the geographic displacement of those trips follows the power law distribution and the corresponding travel time follows a mixture of the exponential and power law distribution. To identify human mobility patterns, a topic model with the latent Dirichlet allocation (LDA) algorithm was proposed to infer the sixty-five key topics. By measuring the variation of trip displacement over time, we find that the travel distance in the morning rush hour is much shorter than that in the other time. As for daily patterns, it shows that taxi mobility presents weekly regularity both on weekdays and on weekends. Among different days in the same week, mobility patterns on Tuesday and Wednesday are quite similar. By quantifying the trip distance along time, we find that Topic 44 exhibits dominant patterns, which means distance less than 10 km is predominant no matter what time in a day. The findings could be references for travelers to arrange trips and policymakers to formulate sound traffic management policies.
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