The spin system provides a relatively simple and compact vehicle for discussion of non-degenerate perturbation theory. In fact in many such systems the terms required to write the electron's energy correct to second order may be read directly from the spin-energy matrix. In this work a hypothetical spin system and three different physical spin systems are treated in order to demonstrate the conciseness of the spin-energy-matrix approach to some perturbation problems. The combination of a perturbation problem which is calculable and which may be related to experimental data provides an approach to perturbation theory which is particularly satisfying to the student.
The spin system provides a relatively simple and compact vehicle for discussion of non-degenerate perturbation theory. In fact in many such systems the terms required to write the electron's energy correct to second order may be read directly from the spin-energy matrix. In this work a hypothetical spin system and three different physical spin systems are treated in order to demonstrate the conciseness of the spin-energy-matrix approach to some perturbation problems. The combination of a perturbation problem which is calculable and which may be related to experimental data provides an approach to perturbation theory which is particularly satisfying to the student.
Exactly solvable models in quantum mechanics are very rare.However, perturbation theory is a powerful tool to research the complex quantum systems and is widely used in theoretical physics and chemistry.For the particle weakly perturbed by the surrounding environment, its energy eigenstate and eigenvalue are separately calculated up to the first-order and second-order perturbation approximations in the undergraduate textbook of quantum mechanics.Whereas in reality, we have to calculate the high-order perturbation correction terms occasionally.Along the same lines of the low-order perturbation correction terms of energy eigenvalues and eigenstate for stationary state perturbation theory in the non-degenerate case, the third-order and fourth-order perturbation correction terms are strictly attained in this paper.Moreover, it points out the essence of perturbation theory and provides its derivation ideas.Students are no longer afraid of tedious derivation and can derive any-order perturbation correction.These results can serve as an extension and supplement for the quantum mechanics textbooks.Furthermore, it is very helpful for students to deeply understand quantum perturbation theory.
Abstract Quantum perturbation is an important tool for developing approximate solutions to quantum problems that cannot be solved exactly. This chapter provides a brief, systematic development of both time-independent and time-dependent quantum perturbation theory and the physical interpretation of its results, including both first- and second-order corrections. A quartic perturbation of the simple harmonic oscillator is used to illustrate time-independent perturbation theory. A spin-1/2 system subject to an oscillating field is used in time-dependent perturbation theory to describe the absorption and emission of energy from the quantum system.
Abstract In a previous paper the theory of perturbations was discussed for system which possess only discrete spectra. The theory is now extended to systems with both discrete and continuous spectra, and in addition a slightly more general method of perturbation is considered. The Hamiltonian H of the perturbed systems is split up into three parts H0, H1, H2 (t), two of which can be chosen arbitrarily. The solutions of the perturbed problem can then be worked out in terms of the characteristic functions corresponding to the Hamiltonian H0 + H2 (T), where T is any arbitrary fixed time. In general, however, the series of perturbations does not converge, and different modes of splitting up the Hamiltonian will lead to different formal results. The perturbation theory does usually give the correct physical results, and it is therefore necessary to give some explanation of how this happens. In (I) the asymptotic nature of certain solutions was emphasised, but this does not extend to the more general systems treated here. It is suggested that the usual boundary conditions do not give an adequate description of any but the simplest atomic problems, and that more detailed restrictions, determined by the experimental conditions, ought to be substituted. As this is impossible in practice, an alternative method is to regard the motion as a perturbation on that Hamiltonian H0 + H2 (T) which most nearly corresponds to the experimental conditions. (For example, we may regard an electron as free or bound according as the experiment determines the number of free or bound electrons respectively, whereas in the usual theory the initial Hamiltonian is taken as fundamental.) This criterion, together with the initial conditions, suffices to fix the division of H into the three parts H0, H1, H2 (t) which at first were chosen arbitrarily. The formal solution can be worked out on this basis, and it is then assumed that this solution is asymptotic to the one which would be obtained if the full boundary conditions were used. This suggestion is in accordance with the fact that a hydrogen atom in an electric field gives rise not only to the usual Stark effect, but also to an ionisation effect. In § 2 the perturbation equation are derived in a slightly more general form than has previously been given. The theories of Born,* Dirac and Oppenheimer are particular cases of that given here, and are obtained by particular choices of H0, H1, H2 (t). To obtain Born’s theory we take H1 = 0, and for Oppenheimer’s we take H2 (t) = H2 (T) and T = 0. In Dirac’s theory we have H1 = 0, H2 (0) = 0 and T = 0. Each of these theories is appropriate for the solution of special problems, but the more general method of this paper is required for the full discussion of the perturbation theory.
We give a brief and basic introduction to perturbation theory. The main idea is to attempt to consider the situation of interest as a small perturbation of a simpler situation (which can be understood completely), and in particular to consider a system with a weak interaction as a perturbation of a non-interacting system. We develop the interaction picture, which allows approximating the time-evolution of an interacting system by the partial sums of the Dyson series, a fundamental tool for the sequel. We illustrate these ideas on a rudimentary model of the interaction of electrons and “photons”.
The physics of strongly correlated materials poses one of the most challenging problems in condensed-matter sciences. Standard approximations applicable to wide classes of materials such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons. Instead, we resort to non-perturbative many-body methods. The calculations are, however, only feasible for rather small model systems. The full Hamiltonian of a real material is approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons is included in an average way by renormalizing the parameters. Realistic calculations of strongly correlated materials need to include sufficiently many of these electronic degrees of freedom. The new generation of massively parallel supercomputers allows for these realistic calculations. However, exploiting their computational power requires newly devised algorithms. As a solver we use the Lanczos method, which needs the full many-body state of the correlated system. It is thus limited by the available main memory. The foremost problem for a distributed-memory implementation is that the multiplication of the Hamiltonian to the many-body state leads to highly non-local memory access patterns. A simple yet important observation leads to an efficient solution: in the kinetic term of the Hamiltonian the electron-spin is conserved. Thus, writing the many-body vector as a matrix v(iup,idn), where the indices label spin-configurations, we find that the hopping term only connects vector elements that differ in one index. Hence, storing entire slices v(iup,:) on one node, the kinetic term for the spin-down electrons is local to that process. After transposing v, the same is true for the hopping of the spin-up electrons. With an efficient matrix transpose, implemented via MPI collectives, we thus obtain a version of the Lanczos method that efficiently well on JUBL, Jülich's Blue Gene/L. Using the code we treated two-orbital Hubbard Models which show interesting Mott-band insulator transitions. Kohn's criterium which determines the phase of a system by ground-state properties alone, namely the response of the ground-state energies to a vector potential, was employed. This vector potential can be equivalently regarded as a change in boundary conditions, requiring complex wave vectors. We found that half-filled systems become metallic in the region between a band and Mott insulating phase. To extrapolate to infinite system sizes and therefore getting access to an arbitrarily high resolution in reciprocal space, we use cluster perturbation theory (CPT). It works by diagonalizing a finite cluster exactly and then treating the hopping between identical clusters perturbatively in strong coupling perturbation theory. This leads to an effectively infinite chain. Due to our efficient Lanczos implementation it is possible to calculate relatively large clusters, minimizing finite-size effects. The combination of our Lanczos code and CPT allowed for a treatment of the the organic metal TTF-TCNQ. We were able to solve a long-standing problem in the interpretation of its experimental spectra. Being an interesting one-dimensional metal TTF-TCNQ has been studied thoroughly for more than 30 years. Theoretical estimates as well as experimental measurements suggest values for the parameters of the Hubbard model t and U. With these values Hubbard model calculations show qualitatively the same features as APRES experiments, namely the signatures of spin-charge separation at the Gamma-point. But in order to fit the experimental data to the numerical calculations t has to be doubled, which translates into a broadening of the spectra. This ad-hoc adjustment is not only unsatisfactory but also the temperature dependence of the spectral function is incorrectly reproduced. We resolved this problem by properly including nearest-neighbor Coulomb interactions. Recent DFT calculations show that indeed the commonly used values of t and U are correct. But Coulomb interactions between electrons on neighboring sites are also significant and must not be neglected. Thus, we studied the effect of the next-neighbor interaction parameter V being at about U/2. Including V in our Lanczos calculations for obtaining the angular-integrated spectral function shows that its effect is to broaden the spectra, comparable to increasing the value of t. Moreover, we found that doubling t indeed mimics the effect of Vapprox U/2. Surprisingly, the broadening of the spectrum as a result of V can be understood in first-order Rayleigh-Schrödinger perturbation theory.