A quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported round a circuit C by varing parameters in its Hamiltonian, will acquire a generalized geometrical phase factor. An explicit formula for a generalized geometrical phase is derived in terms of the eigenstates of the Hamiltonian. As an illustration the generalized geometrical phase is calculated for relativistic spinning particles in slowly-changing electromagnetic fields. It is shown that the the S-matrix and the usual scattering (with negligible reflexion) phase shift can be interpreted as a generalized geometrical phase. PACS: 03.65.Ca, 03.65.Vf, 03.65.Nk Historically, Berry's phase [1] has been introduced in the context of adiabatic evolution governed by anHamiltonian H X (t) whose parameters vary slowly in time and has been confined to discrete spectrum. In quantum case, the adiabatic theorem concerns states |ψ (t) satisfying the time-dependent Schrödinger equation and asserts that if a quantum system with a time-dependent non degenerate Hamiltonian H X (t) is initially in the nth eigenstates of H X (0) , and if H X (t) evolves slowly enough, then the state at time t will remain in the nth instantaneous eigenstates of H X (t) up to a multiplicative phase factor φ n (t). It has beenshown [1], that there is an additional factor e γ B n , a part from the familiar dynamical phase factor eassociated with the time evolution of the state being so transported with instantaneous eigenenergie E n (t), depending only the curve C which has been followed in the parameters space. The question arises: is there a geometrical phase for a continuous spectrum? This case was raised for the first time by R. G. Newton [2] who looks at the S matrix as geometrical phase factor. Newton [2], introduced, what may be called the noninteraction picture to get the geometric phase factor in the continuous spectrum. In order to reinterpret the usual scattering phase shift as an adiabatic phase in the spirit of the original investigation of Berry, G. Ghosh [3] extends the adiabatic approximation to the continuous spectra like an anstaz confine themselves to one dimensional scattering with negligible reflection. Because the states are not normalizable and gauge invariance is lost, Ghosh [3] showed that the phase shift can be expanded in series of reparametrization invariant terms.In this letter we give a generalization of the geometrical phase for the nondegenerate continuous spectrum. Three examples are worked out for illustration: -i) Dirac particle in a time-dependent electromagnetic field, -ii) The explicit relation between the geometrical phase and the S matrix with an example for application, -iii) the unidimensional reflectionless potential Starting with the time-dependent Schrödinger equation governed by a Hamiltonian H X (t) whose parameters vary slowly in timewe define instantaneous eigenfunction in the continuous spectrum H X (t) |ϕ (k, t) = E (k, t) |ϕ (k, t) .The adiabatic theorem demonstrated in Ref....
By defining "a virtual gap" for the continuous spectrum through the notion of eigendifferential ͑Weyl's packet͒ and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem and the validity condition of the adiabatic approximation for systems having a nondegenerate continuous spectrum. We show that a quantal system in an eigenstate, of operators with a continuous nondegenerate eigenvalue spectrum, slowly transported around a closed curve C by varying parameters in its Hamiltonian, will acquire a generalized geometrical phase factor where an explicit formula is derived in terms of the eigenstates of the Hamiltonian. Examples are given for illustration.
The one-dimensional Schrödinger equation associated with a time-dependent Coulomb potential is studied. The invariant operator method (Lewis and Riesenfeld) and unitary transformation approach are employed to derive quantum solutions of the system. We obtain an ordinary second-order differential equation whose analytical exact solution has been unknown. It is confirmed that the form of this equation is similar to the radial Schrödinger equation for the hydrogen atom in a (arbitrary) strong magnetic field. The qualitative properties for the eigenstates spectrum are described separately for the different values of the parameter ω0 appearing in the x2 term, x being the position, i.e., ω0 > 0, ω0 < 0 and ω0 = 0. For the ω0 = 0 case, the eigenvalue equation of invariant operator reduces to a solvable form and, consequently, we have provided exact eigenstates of the time-dependent Hamiltonian system.
We use the Lewis–Riesenfeld theory to determine the exact form of the wavefunctions of a two-dimensional Pauli equation of a charged spin 1∕2 particle with time-dependent mass and frequency in the presence of the Aharonov–Bohm effect and a two-dimensional time-dependent harmonic oscillator. We find that the irregular solution at the origin as well as the regular one contributes to the phase of the wavefunction.
Invariant operator method for discrete or continuous spectrum eigenvalue and unitary transformation approach are employed to study the two-dimensional time-dependent Pauli equation in presence of the Aharonov–Bohm effect (AB) and external scalar potential. For the spin particles the problem with the magnetic field is that it introduces a singularity into wave equation at the origin. A physical motivation is to replace the zero radius flux tube by one of radius R, with the additional condition that the magnetic field be confined to the surface of the tube, and then taking the limit R → 0 at the end of the computations. We point that the invariant operator must contain the step function θ(r – R). Consequently, the problem becomes more complicated. In order to avoid this difficulty, we replace the radius R by ρ(t)R, where ρ(t) is a positive time-dependent function. Then at the end of calculations we take the limit R → 0. The qualitative properties for the invariant operator spectrum are described separately for the different values of the parameter C appearing in the nonlinear auxiliary equation satisfied by ρ(t), i.e., C > 0, C = 0, and C < 0. Following the C's values the spectrum of quantum states is discrete (C > 0) or continuous (C ≤ 0).
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