Gauge invariant hydrogen atom Hamiltonian

For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this puzzle is recently provided by us in [2]. Based on the separation of the electromagnetic potential into pure gauge and gauge invariant parts, we have proposed a new set of momentum and Hamiltonian operators which satisfy both the requirement of gauge invariance and the relevant commutation relations. In this paper we report a check for the case of the hydrogen atom problem: Starting from the Hamiltonian of the coupled electron, proton and electromagnetic field, under the infinite proton mass approximation, we derive the gauge invariant hydrogen atom Hamiltonian and verify explicitly that this Hamiltonian is different from the Dirac Hamiltonian, which is the time translation generator of the system. The gauge invariant Hamiltonian is the energy operator, whose eigenvalue is the energy of the hydrogen atom. It is generally time-dependent. In this case, one can solve the energy eigenvalue equation at any specific instant of time. It is shown that the energy eigenvalues are gauge independent, and by suitably choosing the phase factor of the time-dependent eigenfunction, one can ensure that the time-dependent eigenfunction satisfies the Dirac equation.Comment: 7 pages, revtex4, some further discussion on Dirac Hamiltonian and the gauge invariant Hamiltonian is added, one reference removed; new address of some of the authors added, final version to appear in Phys. Rev.

Beyond Prediction: On-street Parking Recommendation using Heterogeneous Graph-based List-wise Ranking

To provide real-time parking information, existing studies focus on predicting parking availability, which seems an indirect approach to saving drivers' cruising time. In this paper, we first time propose an on-street parking recommendation (OPR) task to directly recommend a parking space for a driver. To this end, a learn-to-rank (LTR) based OPR model called OPR-LTR is built. Specifically, parking recommendation is closely related to the "turnover events" (state switching between occupied and vacant) of each parking space, and hence we design a highly efficient heterogeneous graph called ESGraph to represent historical and real-time meters' turnover events as well as geographical relations; afterward, a convolution-based event-then-graph network is used to aggregate and update representations of the heterogeneous graph. A ranking model is further utilized to learn a score function that helps recommend a list of ranked parking spots for a specific on-street parking query. The method is verified using the on-street parking meter data in Hong Kong and San Francisco. By comparing with the other two types of methods: prediction-only and prediction-then-recommendation, the proposed direct-recommendation method achieves satisfactory performance in different metrics. Extensive experiments also demonstrate that the proposed ESGraph and the recommendation model are more efficient in terms of computational efficiency as well as saving drivers' on-street parking time

NLO QCD corrections to Single Top and W associated production at the LHC with forward detector acceptances

In this paper we study the Single Top and W boson associated photoproduction via the main reaction $\rm pp\rightarrow p\gamma p\rightarrow pW^{\pm}t+Y$ at the 14 TeV Large Hadron Collider (LHC) up to next-to-leading order (NLO) QCD level assuming a typical LHC multipurpose forward detector. We use the Five-Flavor-Number Schemes (5FNS) with massless bottom quark assumption in the whole calculation. Our results show that the QCD NLO corrections can reduce the scale uncertainty. The typical K-factors are in the range of 1.15 to 1.2 which lead to the QCD NLO corrections of 15$$ to 20$$ correspond to the leading-order (LO) predictions with our chosen parameters.Comment: 41pages, 12figures. arXiv admin note: text overlap with arXiv:1106.2890 by other author