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.
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.
There exist several forms of perturbation theory in Quantum Mechanics, namely: non-degenerate perturbation theory, degenerate perturbation theory, time-dependent perturbation theory, time-constant perturbation theory and time-harmonic perturbation theory. This paper presents an extension of the perturbation theory for non-degenerate states. It has been observed that most texts and journal papers treat only the first –order and second-order non-degenerate perturbation theory. Therefore, this paper attempts to treat and present the third order and fourth order non-degenerate perturbation theory in quantum mechanics. It can thus be asserted that the higher the order of perturbations of quantum systems that we know, the more successful will be our efforts in suppressing or eliminating them.
Although the fundamental principles of time-independent perturbation theory are well-established and widely applied, the specific treatment of degeneracy lifted to the second order remains largely unexplored. This work fills the gap by presenting a systematic derivation of time-independent perturbation theory that accounts for the added complexity introduced by non-trivial degeneracies. We provide a general procedure and corresponding formulae for calculating state and energy corrections to any order, with degeneracy lifted to the second order. Additionally, we apply these formulae to solve a prototype problem of the dressed quark system. This work not only advances the theoretical framework but also provides practical tools for tackling complex problems in quantum physics.
In many situations in physics, the Hamiltonian of a given system is so complicated that the solution to the stationary Schrodinger equation is practically impossible or very difficult. Therefore, some approximation methods are required. This chapter presents the time-independent perturbation theory, the procedure of finding corrections to non-degenerate eigenvalues and eigenvectors to a part of the Hamiltonian of a given system. The perturbation theory is appropriate when the Hamiltonian can be split into two parts. The perturbation theory can be applied to analyze the quantum properties of particles trapped in two closely coupled potential wells. By closely coupled wells, it is meant that these two wells are separated by a barrier. This is a typical situation in the studies of quantum dynamics of Bose–Einstein condensates. Using the first-order perturbation theory, one can find the eigenvalues and eigenvectors.
Rayleigh-Schrödinger time-independent perturbation theory is a key method in quantum mechanics for solving problems where the Hamiltonian of a quantum system cannot be exactly solved. By expanding the solutions in series, this method allows one to obtain approximations for quantum states and eigen-energies, especially for systems where the interactions or perturbing forces are small compared to the principal part of the Hamiltonian. This is crucial for understanding complex phenomena, predicting the behaviours of atomic and molecular systems, and calculating corrections to basic theoretical models, thereby improving the accuracy of quantum predictions. In this work, we will show that using the unitary operators, we arrive at the equations giving the different order of corrections of both energy as well as wave functions of the time-independent perturbation theory. Our approach closely mirrors the famous formalism of Rayleigh-Schrodinger perturbation theory (RSPT). The use of unitary operators simplifies the derivation of higher-order corrections, making the process more accurate and accessible to researchers. This technique harnesses the potential of unitary operators within a perturbation series framework. By combining these two powerful techniques, we offer a comprehensive framework for addressing normalization challenges in quantum mechanical calculations. This methodology not only facilitates a deeper understanding of perturbation theory but also serves as a pedagogical stepping stone, encouraging postgraduate researchers to apply their foundational knowledge and acquired skills in practical problem-solving situations with confidence. This is therefore a second proof or a new method to derive the fundamental equations of Rayleigh Schrodinger perturbation Theory RSPT using the unitary transformations.
The vast majority of problems in quantum mechanics cannot be solved exactly. As in all theories of physics approximation methods are therefore of great importance. Here we shall deal with time-independent problems, which means that we seek the eigenvalues and eigenstates of H . In perturbation theory we split H into two pieces, H = H 0 + H 1 , where we know the spectrum and eigenstates of H 0 , and H 1 is, in some sense, small. Needless to say, a split of this type is often not possible, or at least unknown. In that case one frequently replaces the system by a simpler model for which perturbation theory works, and which, it is hoped, captures some of the essential features of the actual problem at hand. This art of finding models is something that can only be taken up within the context of a particular subject (e.g., solid state or nuclear physics). Here we shall treat systems where good zero-order solutions are rather obvious.
We propose an improved scheme of perturbation theory based on our exact solution [An Min Wang, quant-ph/0611216] in general quantum systems independent of time. Our elementary start-point is to introduce the perturbing parameter as late as possible. Our main skills are Hamiltonian redivision so as to overcome a flaw of the usual perturbation theory, and the perturbing Hamiltonian matrix product decomposition in order to separate the contraction and anti-contraction terms. Our calculational technology is the limit process for eliminating apparent divergences. Our central idea is ``dynamical rearrangement and summation" for the sake of the partial contributions from the high order even all order approximations absorbed in our perturbed solution. Consequently, we obtain the improved forms of the zeroth, first, second and third order perturbed solutions absorbing the partial contributions from the high order even all order approximations of perturbation. Then we deduce the improved transition probability. In special, we propose the revised Fermi's golden rule. Moreover, we apply our scheme to obtain the improved forms of perturbed energy and perturbed state. In addition, we study an easy understanding example of two-state system to illustrate our scheme and show its advantages. All of this implies the physical reasons and evidences why our improved scheme of perturbation theory are actually calculable, operationally efficient, conclusively more accurate. Our improved scheme is the further development and interesting application of our exact solution, and it has been successfully used to study on open system dynamics [An Min Wang, quant-ph/0601051].