Gauge-invariant hydrogen-atom Hamiltonian

Journal Article · Thu Jul 15 00:00:00 EDT 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.82.012107· OSTI ID:21442885

Weimin, Sun; Fan, Wang ^[1]; Xiangsong, Chen ^[2]; Xiaofu, Lue ^[3]

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1. Department of Physics, Nanjing University, CPNPC, Nanjing 210093 (China)
2. Department of Physics, Huazhong University of Science and Technology, Wuhan 430074 (China)
3. Department of Physics, Sichuan University, Chengdu 610064 (China)

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 was recently provided by us [X.-S. Chen et al., Phys. Rev. Lett. 100, 232002 (2008)]. 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.

• OSTI ID: 21442885
• Journal Information: Physical Review. A, Vol. 82, Issue 1; Other Information: DOI: 10.1103/PhysRevA.82.012107; (c) 2010 The American Physical Society; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
APPROXIMATIONS
ATOMS
CHARGED PARTICLES
COMMUTATION RELATIONS
DIRAC EQUATION
EIGENFUNCTIONS
EIGENVALUES
ELECTROMAGNETIC FIELDS
ELECTRONS
GAUGE INVARIANCE
HAMILTONIANS
HYDROGEN
MATRIX ELEMENTS
POTENTIALS
PROTONS
QUANTUM MECHANICS
TIME DEPENDENCE
BARYONS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
ELEMENTARY PARTICLES
ELEMENTS
EQUATIONS
FERMIONS
FIELD EQUATIONS
FUNCTIONS
HADRONS
INVARIANCE PRINCIPLES
LEPTONS
MATHEMATICAL OPERATORS
MECHANICS
NONMETALS
NUCLEONS
PARTIAL DIFFERENTIAL EQUATIONS
QUANTUM OPERATORS
WAVE EQUATIONS