This paper treats the problem of basic characteristics of quantum transitions, stimulated by a sudden disturbance V(t). The proper ''shake'' theory for quantum systems is constructed by expanding the time-displacement operator S (t, t') in a power series in ..omega..tau (h..omega.. is the typical eigenvalue of the unperturbed hamiltonian, tau is the characteristic collision time). If (V(t), V(t'))equals 0 at any Vtau/h, the ''shake'' theory gives a compact expression for the transition probabilities. Various atomic collision processes are discussed from the ''shake'' point of view. If an impulse deltap is transferred to the quantum system during the short time (..omega..tau << 11), the transition probabilities are determined by the value of universal parameter N approximately deltap. deltaR/h (uncertainty deltaR results from the slow motion of the unperturbed system).
This paper treats the problem of basic characteristics of quantum transitions, stimulated by a sudden disturbance V(t). The proper ''shake'' theory for quantum systems is constructed by expanding the time-displacement operator S (t, t') in a power series in ..omega..tau (h..omega.. is the typical eigenvalue of the unperturbed hamiltonian, tau is the characteristic collision time). If (V(t), V(t'))equals 0 at any Vtau/h, the ''shake'' theory gives a compact expression for the transition probabilities. Various atomic collision processes are discussed from the ''shake'' point of view. If an impulse deltap is transferred to the quantum system during the short time (..omega..tau << 11), the transition probabilities are determined by the value of universal parameter N approximately deltap. deltaR/h (uncertainty deltaR results from the slow motion of the unperturbed system).
A systematic theory of sudden perturbations is derived for quantum systems whose states are described both by wave functions (a pure ensemble) and by a quantum density operator (a mixed ensemble). A perturbation series is written in powers of the parameter ωτ, which is small when the perturbation is sudden; ω is the typical eigenvalue of the unperturbed system; and τ is the characteristic collision time. When the perturbation (t), taken at different times, commutes with itself, the theory yields a compact analytic expression for the probabilities for stimulated transitions for any value of Vτ /. The results of many cross-section calculations for atomic collision processes are discussed from a common standpoint: the processes are treated as jarring processes which stimulate transitions in the quantum system. If a momentum δp is rapidly transferred to the system in a collision, regardless of the physical nature of the jarring, the probabilities for the stimulated transitions are governed by the parameter N ~δpδR/ where δR is a measure of the uncertainty in the coordinates which is due to the relatively slow motions in the unperturbed system.
We now consider systems for which the Hamiltonian contains time-dependent interaction terms. In such cases we cannot reduce the time-dependent Schro¨dinger equation to an eigenvalue equation. However, if the time-dependent terms can be regarded as small perturbations, we can develop a form of perturbation theory that can be used to calculate the effects of time-dependent terms. Consider a system for which the Hamiltonian may be written as the sum of a dominant unperturbed part Hˆ0 and a small perturbation Hˆ ′ as Hˆ = Hˆ0 + λHˆ ′(t) , (10.1.1) where the unperturbed Hamiltonian Hˆ0 is independent of time, whereas the perturbation Hˆ ′ may depend on time. The parameter 0 < λ < 1 has been introduced to keep track of various orders of approximations as in the case of time-independent perturbation theory discussed in Chapter 8. The basic idea is that a small perturbation to the Hamiltonian will produce a small change in the wave function. For a time-dependent perturbation, this change means that a system, initially in a particular eigenstate of Hˆ0, will find itself in a time-dependent admixture of the eigenstates. Thus the time-dependent perturbation causes the system to undergo transition to different eigenstates of Hˆ0. So for time-dependent perturbations our interest lies in calculating the probabilities for transition to different states.
If a particle which can decay spontaneously is subjected to collisions with other particles the transition probability may be changed. The calculation of such an effect by the usual time-dependent perturbation methods is complicated by the circumstance that energy can be conserved in some intermediate states, because of the possibility of elastic scattering. The perturbation calculation is here modified in such a way that the resulting expression for the whole transition probability does not contain any divergences or spurious effects. By suitable approximations the expression is separated into two parts, one of which gives the probability of elastic scattering followed by a spontaneous transition, and the other the true stimulated transition probability which causes a change in the lifetime.
The temporal behavior of quantum mechanical systems is reviewed. We mainly focus our attention on the time development of the so-called “survival” probability of those systems that are initially prepared in eigenstates of the unperturbed Hamiltonian, by assuming that the latter has a continuous spectrum. The exponential decay of the survival probability, familiar, for example, in radioactive decay phenomena, is representative of a purely probabilistic character of the system under consideration and is naturally expected to lead to a master equation. This behavior, however, can be found only at intermediate times, for deviations from it exist both at short and long times and can have significant consequences. After a short introduction to the long history of the research on the temporal behavior of such quantum mechanical systems, the short-time behavior and its controversial consequences when it is combined with von Neumann’s projection postulate in quantum measurement theory are critically overviewed from a dynamical point of view. We also discuss the so-called quantum Zeno effect from this standpoint. The behavior of the survival amplitude is then scrutinized by investigating the analytic properties of its Fourier and Laplace transforms. The analytic property that there is no singularity except a branch cut running along the real energy axis in the first Riemannian sheet is an important reflection of the time-reversal invariance of the dynamics governing the whole process. It is shown that the exponential behavior is due to the presence of a simple pole in the second Riemannian sheet, while the contribution of the branch point yields a power behavior for the amplitude. The exponential decay form is cancelled at short times and dominated at very long times by the branch-point contributions, which give a Gaussian behavior for the former and a power behavior for the latter. In order to realize the exponential law in quantum theory, it is essential to take into account a certain kind of macroscopic nature of the total system, since the exponential behavior is regarded as a manifestation of a complete loss of coherence of the quantum subsystem under consideration. In this respect, a few attempts at extracting the exponential decay form on the basis of quantum theory, aiming at the master equation, are briefly reviewed, including van Hove’s pioneering work and his well-known “λ2T” limit. In the attempt to further clarify the mechanism of the appearance of a purely probabilistic behavior without resort to any approximation, a solvable dynamical model is presented and extensively studied. The model describes an ultrarelativistic particle interacting with N two-level systems (called “spins”) and is shown to exhibit an exponential behavior at all times in the weak-coupling, macroscopic limit. Furthermore, it is shown that the model can even reproduce the short-time Gaussian behavior followed by the exponential law when an appropriate initial state is chosen. The analysis is exact and no approximation is involved. An interpretation for the change of the temporal behavior in quantum systems is drawn from the results obtained. Some implications for the quantum measurement problem are also discussed, in particular in connection with dissipation.
The orders of the perturbation approximation to the Schroedinger-picture time evolution operator in powers of the presumed-small perturbation part of the Hamiltonian operator are developed by the iteration of an identity; the possibly confusing switch to the interaction picture isn't needed. Those time-evolution operator approximations are sandwiched between two orthogonal, normalized eigenstates of the unperturbed part of the Hamiltonian operator to produce the transition amplitude approximations, whose calculation is reduced to quadrature when every occurrence of the perturbation part of the Hamiltonian operator in them is expanded in the unperturbed basis. That expansion also reveals the time-dependent parts of those approximations to be multiply nested integrals which in the long-time limit approach simple products of non-singular inverses (principle value plus delta function) of differences of unperturbed energy eigenvalues. Closely related to the long-time limits of transition amplitudes are the long-time averages of transition rates. When the unperturbed basis is that of free-particle states, sums over relevant final states of transition rates from an initial state, divided by the initial state's particle flux, produce cross sections. Here the perturbation approximations to generic quantum transition rates are parlayed to the corresponding approximations to differential cross sections for nonrelativistic-particle potential scattering.
Abstract Protonium is a type of onium consisting of a proton (p) and an anti-proton ( <?CDATA $ \bar p $?> p ¯ ) which is called antiprotonic hydrogen. Since the proton and the anti-particle have the same mass, and each of them is a fermion, the Hamiltonian form of this system consists of the kinetic energy terms of each particle, the potential energy, and the spin interactions of both (hyperfine structure). To simplify the analysis, the Hamiltonian is then converted into a Hamiltonian which contains the relative motion terms of the two particles. This new Hamiltonian contains their reduced mass because the two particles orbit each other towards a certain center of mass. In the initial state, t < 0, the system is in the ground state and then at 0 < t < τ, the system is given a disturbance of the form <?CDATA $ V(t) = ({V_0} + {V_1}\,r\,\sin \theta \,{e^{i\phi }})\,\cos (\omega t)\,{\hat \sigma _p}.\hat x $?> V ( t ) = ( V 0 + V 1 r sin θ e i ϕ ) cos ( ω t ) σ ^ p . x ^ , where V 0 is a constant, V 1 is a constant and very small, <?CDATA $ {\hat \sigma _p} $?> σ ^ p is the Pauli matrix for proton, and <?CDATA $ \hat x $?> x ^ is the three-dimensional position operator in spherical coordinates. We analyze this system using a time-dependent perturbation theory that begins by defining the initial state, then calculating the transition amplitude for the new state when the perturbation is applied to the system. From the results of this calculation, we compute the system transition probability. We find that the transition probability of this system can only occur for the following quantum number terms, namely n even positive integer, l = 2,S = 1, and with the certain combinations of m and M.