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arXiv:1002.3421 · Physical Review A

Gauge-invariant hydrogen-atom Hamiltonian

Weimin Sun, Xiang-Song Chen, Xiaofu Lü, Fan Wang

Published Jul 22, 2010• 13 citations
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Abstract

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.

DOI
10.1103/physreva.82.012107

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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.
The Dirac equation for a charged particle in a time-dependent electromagnetic field is reduced to the Pauli equation in the non-relativistic limit by making a time-dependent Foldy-Wouthuysen transformation in a completely gauge-invariant way. In a time-dependent field, the expectation value of the Hamiltonian is dependent on the gauge, and so it cannot be the energy. The energy is obtained by taking the expectation value of the energy operator, the time rate of change of which is the average of the quantum mechanical power operator. The Hamiltonian formulation of a charged particle in a time-dependent external electromagnetic field is satisfactory when this distinction between the Hamiltonian, which describes the time evolution of the system, and the energy operator, which is gauge invariant, is made.
Hamiltonian operators are gauge dependent. For overcome this difficulty we reexamined the effect of a gauge transformation on Schrödinger and Dirac equations. We show that the gauge invariance of the operator H−iℏ∂∂tH-i\hbar\frac{\partial}{\partial t}H−iℏ∂t∂​ provides a way to find the energy operator from first principles. In particular, when the system has stationary states the energy operator can be identified without ambiguities for non-relativistic and relativistic quantum mechanics. Finally, we examine other approaches finding that in the case in which the electromagnetic field is time independent, the energy operator obtained here is the same as one recently proposed by Chen et al. [1].
A manifestly gauge-invariant formulation of quantum mechanics is applied to a charged isotropic harmonic oscillator in a time-varying magnetic field in the magnetic dipole approximation. The energy operator for the problem is the sum of the kinetic and potential energies. The kinetic energy operator is the square of the gauge-invariant kinetic momentum operator divided by twice the mass. The energy eigenvalues and state probabilities are calculated and are shown to be the same in all gauges. In this problem there is no gauge in which the energy operator reduces to the unperturbed Hamiltonian, as there is in the electric dipole approximation. Consequently, eigenvalues of the unperturbed Hamiltonian and corresponding (gauge-dependent) state 'probabilities' are different from the gauge-invariant quantities in all gauges.
The external electromagnetic interactions of hydrogenic atoms, which exhibit global Galilean invariance but preserve the relativistic description of the electron, are investigated. The Hamiltonian for the free atom consists of the Breit Hamiltonian for a relativistic electron interacting with a nonrelativistic spinless proton. Approximate generators of the Galilean group are constructed in terms of individual-particle position, momentum, and spin operators in several unitarily equivalent representations. In one of these the Hamiltonian is separable in conventional relative and center-of-mass coordinates. This separability facilitates the treatment of a number of external-field problems. An effective Hamiltonian for the interaction of the atom with a constant external electric field is given and compared to that obtained by other authors. The low-energy theorem for Compton scattering from the composite system is explicitly verified including corrections due to atomic binding. This also gives an independent verification of recently obtained bound-state corrections to the atomic magnetic moment. Finally, an effective interaction of a charged Dirac particle with a hydrogenic atom is given in the representation in which the atomic Hamiltonian is separable.
A relativistically invariant representation of the generalized momentum of a particle in an external field is proposed. In this representation, the dependence of the potentials of the interaction of the particle with the field on the particle velocity is taken into account. The exact correspondence of the expressions of energy and potential energy for the classical Hamiltonian is established, which makes identical the solutions to the problems of mechanics with relativistic and nonrelativistic approaches. The invariance of the proposed representation of the generalized momentum makes it possible to equivalently describe a physical system in geometrically conjugate spaces of kinematic and dynamic variables. Relativistic invariant equations are proposed for the action function and the wave function based on the invariance of the representation of the generalized momentum. The equations have solutions for any values of the constant interaction of the particle with the field, for example, in the problem of a hydrogen-like atom, when the atomic number of the nucleus is Z > 137. Based on the parametric representation of the action, the expression for the canonical Lagrangian, the equations of motion, and the expression for the force acting on the charge are derived when moving in an external electromagnetic field. The Dirac equation with the correct inclusion of the interaction for a particle in an external field is presented. In this form, the solutions of the equations are not limited by the value of the interaction constant. The solutions of the problem of charge motion in a constant electric field, the problems for a particle in a potential well and the passage of a particle through a potential barrier, the problems of motion in an exponential field (Morse), and also the problems of a hydrogen atom are given.
A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.