13.5 Solutions Near an Irregular Singular Point, 344 Exercises, 355 14 DIFFERENTIAL EQUATIONS WITH A LARGE PARAMETER 360 14.1 Th WKeB Approximation, 361 14.2 The Liouville-Green Transformation, 364 14.3 Eigenvalue Problems, 366 14.4 Equations with Slowly Varying Coefficients, 369 14.5 Turning-Point Problems, 370 14.6 The Langer Transformation, 375 We, in particular, derive the following types of perturbation bounds. More often one is faced with a potential or a Hamiltonian for which exact methods are unavailable and approximate solutions must be found. There exist only a handful of problems in quantum mechanics which can be solved exactly. Since the perturbation is an odd function, only when \(m= 2k+1\) with \(k=1,2,3\) would these integrals be non-zero (i.e., for \(m=1,3,5, ...\)). Copyright © 2020 Elsevier B.V. or its licensors or contributors. Calculating the first order perturbation to the wavefunctions (Equation \(\ref{7.4.24}\)) is more difficult than energy since multiple integrals must be evaluated (an infinite number if symmetry arguments are not applicable). actly. \left(\dfrac{\alpha}{\pi}\right)^{1/4} \nonumber\]. Let's look at Equation \(\ref{7.4.10}\) with the first few terms of the expansion: \[ \begin{align} (\hat{H}^o + \lambda \hat{H}^1) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) &= \left( E _n^0 + \lambda E_n^1 \right) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) \label{7.4.11} \\[4pt] \hat{H}^o | n ^o \rangle + \lambda \hat{H}^1 | n ^o \rangle + \lambda H^o | n^1 \rangle + \lambda^2 \hat{H}^1| n^1 \rangle &= E _n^0 | n ^o \rangle + \lambda E_n^1 | n ^o \rangle + \lambda E _n^0 | n ^1 \rangle + \lambda^2 E_n^1 | n^1 \rangle \label{7.4.11A} \end{align}\], Collecting terms in order of \(\lambda\) and coloring to indicate different orders, \[ \underset{\text{zero order}}{\hat{H}^o | n ^o \rangle} + \color{red} \underset{\text{1st order}}{\lambda ( \hat{H}^1 | n ^o \rangle + \hat{H}^o | n^1 \rangle )} + \color{blue} \underset{\text{2nd order}} {\lambda^2 \hat{H}^1| n^1 \rangle} =\color{black}\underset{\text{zero order}}{E _n^0 | n ^o \rangle} + \color{red} \underset{\text{1st order}}{ \lambda (E_n^1 | n ^o \rangle + E _n^0 | n ^1 \rangle )} +\color{blue}\underset{\text{2nd order}}{\lambda^2 E_n^1 | n^1 \rangle} \label{7.4.12}\]. 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems, [ "article:topic", "Perturbation Theory", "showtoc:no" ], 7.3: Trial Functions Can Be Linear Combinations of Functions That Also Contain Variational Parameters, First-Order Expression of Energy (\(\lambda=1\)), First-Order Expression of Wavefunction (\(\lambda=1\)), harmonic oscillator wavefunctions being even, information contact us at info@libretexts.org, status page at https://status.libretexts.org. Switching on an arbitrarily weak attractive potential causes the \(k=0\) free particle wavefunction to drop below the continuum of plane wave energies and become a localized bound state with binding energy of order \(\lambda^2\). \[ E_n^1 = \int_0^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \nonumber\], or better yet, instead of evaluating this integrals we can simplify the expression, \[ E_n^1 = \langle n^o | H^1 | n^o \rangle = \langle n^o | V_o | n^o \rangle = V_o \langle n^o | n^o \rangle = V_o \nonumber\], so via Equation \(\ref{7.4.17.2}\), the energy of each perturbed eigenstate is, \[ \begin{align*} E_n &\approx E_n^o + E_n^1 \\[4pt] &\approx \dfrac{h^2}{8mL^2}n^2 + V_o \end{align*}\]. Singular perturbation problems are of common occurrence in all branches of applied mathematics and engineering. Equation \(\ref{7.4.24}\) is essentially is an expansion of the unknown wavefunction correction as a linear combination of known unperturbed wavefunctions \(\ref{7.4.24.2}\): \[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \\[4pt] &\approx | n^o \rangle + \sum _{m \neq n} c_{m,n} |m^o \rangle \label{7.4.24.2} \end{align}\], with the expansion coefficients determined by, \[ c_{m,n} = \dfrac{\langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o} \label{7.4.24.3}\]. In this paper, the basic methods and literature for solving the singular perturbation problems have been presented with their comparative study. Short lecture on an example application of perturbation theory. It should be noted that there are problems that cannot be solved using perturbation theory, even when the perturbation is very weak, although such problems are the exception rather than the rule. This means to first order pertubation theory, this cubic terms does not alter the ground state energy (via Equation \(\ref{7.4.17.2})\). One example is planetary motion, which can be treated as a perturbation on a problem in which the planets … In the following derivations, let it be assumed that all eigenenergies andeigenfunctions are normalized. The series does not converge. To make it easier to identify terms of the same order in \(\hat{H}^1/\hat{H}^o\) on the two sides of the equation, it is convenient to introduce a dimensionless parameter \(\lambda\) which always goes with \(\hat{H}^1\), and then expand both eigenstates and eigenenergies as power series in \(\lambda\), \[ \begin{align} | n \rangle &= \sum _ i^m \lambda ^i| n^i \rangle \label{7.4.5} \\[4pt] E_n &= \sum_{i=0}^m \lambda ^i E_n^i \label{7.4.6} \end{align}\]. To leave a comment or report an error, please use the auxiliary blog. Perturbation problems depend on a small positive parameter. A central theme in Perturbation Theory is to continue equilibriumand periodic solutionsto the perturbed system, applying the Implicit Function Theorem.Consider a system of differential equations Equilibriaare given by the equation Assuming that and thatthe Implicit Function Theorem guarantees existence of a l… While this is the first order perturbation to the energy, it is also the exact value. Degenerate State Perturbation Theory The perturbation expansion has a problem for states very close in energy. The basic principle is to find a solution to a problem that is similar to the one of interest and then to cast the solution to the target problem in terms of parameters related to the known solution. Beneficial to both beginning students and researchers, Asymptotic Analysis and Perturbation Theory immediately introduces asymptotic notation and then applies this tool to familiar problems, including limits, inverse functions, and integrals. However, in this case, the first-order perturbation to any particle-in-the-box state can be easily derived. The basic idea here should be very familiar: perturbation theory simply means finding solutions to an otherwise intractable system by systematically expanding in some small parameter. In this monograph we present basic concepts and tools in perturbation theory for the solution of computational problems in finite dimensional spaces. By continuing you agree to the use of cookies. Perturbation Theory In this chapter we will discuss time dependent perturbation theory in classical mechanics. \begin{array}{c} One such case is the one-dimensional problem of free particles perturbed by a localized potential of strength \(\lambda\). 11.1 Time-independent perturbation . Exercise \(\PageIndex{3}\): Harmonic Oscillator with a Quartic Perturbation, Use perturbation theory to estimate the energy of the ground-state wavefunction associated with this Hamiltonian, \[ \hat{H} = \dfrac{-\hbar}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} kx^2 + \gamma x^4 \nonumber.\], The model that we are using is the harmonic oscillator model which has a Hamiltonian, \[H^{0}=-\frac{\hbar}{2 m} \frac{d^2}{dx^2}+\dfrac{1}{2} k x^2 \nonumber\], To find the perturbed energy we approximate it using Equation \ref{7.4.17.2}, \[E^{1}= \langle n^{0}\left|H^{1}\right| n^{0} \rangle \nonumber\], where is the wavefunction of the ground state harmonic oscillator, \[n^{0}=\left(\frac{a}{\pi}\right)^{\left(\frac{1}{4}\right)} e^{-\frac{ax^2}{2}} \nonumber\], When we substitute in the Hamiltonian and the wavefunction we get, \[E^{1}=\left\langle\left(\frac{a}{\pi}\right)^{\left(\frac{1}{4}\right)} e^{-\frac{ax^2}{2}}\right|\gamma x^{4}\left|\left(\frac{a}{\pi}\right)^{\left(\frac{1}{4}\right)} e^{-\frac{ax^2}{2}} \right \rangle \nonumber\]. Perturbation Theory Relatively few problems in quantum mechanics have exact solutions, and thus most problems require approximations. Perturbation theory has been widely used in almost all areas of science. For example, the first order perturbation theory has the truncation at \(\lambda=1\). The class of problems in classical mechanics which are amenable to exact solution is quite limited, but many interesting physical problems di er from such a solvable problem by corrections which may be considered small. Semiclassical approximation. The methods work by reducing a hard problem to an infinite sequence of relatively easy problems that can be solved analytically. (2005), Introduction to Quantum Mechan-ics, 2nd Edition; Pearson Education - Problem … FIRST ORDER NON-DEGENERATE PERTURBATION THEORY Link to: physicspages home page. 4) The methods of perturbation theory have special importance in the field of quantum mechanics in which, just like in classical mechanics, exact solutions are obtained for the case of the two-body problem only (which can be reduced to the one-body problem in an external potential field). In fact, even problems with exact solutions may be better understood by ignoring the exact solution and looking closely at approximations. and therefore the wavefunction corrected to first order is: \[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \\[4pt] &\approx \underbrace{| n^o \rangle + \sum _{m \neq n} \dfrac{|m^o \rangle \langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o}}_{\text{First Order Perturbation Theory}} \label{7.4.24} \end{align}\]. This is justified since the set of original zero-order wavefunctions forms a complete basis set that can describe any function. Perturbation theory is a vast collection of mathematical methods used to obtain approximate solution to problems that have no closed-form analytical solution. The degeneracy is 8: we have a degeneracy n2 = 4 without spin and then we take into account the two possible spin states (up and down) in the basis |L2,S2,L z,S zi. {E=\frac{1}{2} h v+\gamma \frac{3}{4 a^2}} So of the original five unperturbed wavefunctions, only \(|m=1\rangle\), \(|m=3\rangle\), and \(|m=5 \rangle\) mix to make the first-order perturbed ground-state wavefunction so, \[| 0^1 \rangle = \dfrac{ \langle 1^o | H^1| 0^o \rangle }{E_0^o - E_1^o} |1^o \rangle + \dfrac{ \langle 3^o | H^1| 0^o \rangle }{E_0^o - E_3^o} |3^o \rangle + \dfrac{ \langle 5^o | H^1| 0^o \rangle }{E_0^o - E_5^o} |5^o \rangle \nonumber\]. The first-order change in the energy of a state resulting from adding a perturbing term \(\hat{H}^1\) to the Hamiltonian is just the expectation value of \(\hat{H}^1\) in the unperturbed wavefunctions. The problem of the perturbation theory is to find eigenvalues and eigenfunctions of the perturbed potential, i.e. We use cookies to help provide and enhance our service and tailor content and ads. At this stage, the integrals have to be manually calculated using the defined wavefuctions above, which is left as an exercise. Excitation of H-atom. For this example, this is clearly the harmonic oscillator model. Newton's equation only allowed the mass of two bodies to be analyzed. Now we have to find our ground state energy using the formula for the energy of a harmonic oscillator that we already know, \[E_{r}^{0}=\left(v+\dfrac{1}{2}\right) hv \nonumber\], Where in the ground state \(v=0\) so the energy for the ground state of the quantum harmonic oscillator is, \[E_{\mathrm{r}}^{0}=\frac{1}{2} h v \nonumber\]. Approximate methods. There are higher energy terms in the expansion of Equation \(\ref{7.4.5}\) (e.g., the blue terms in Equation \(\ref{7.4.12}\)), but are not discussed further here other than noting the whole perturbation process is an infinite series of corrections that ideally converge to the correct answer. Using Equation \(\ref{7.4.17}\) for the first-order term in the energy of the any state, \[ \begin{align*} E_n^1 &= \langle n^o | H^1 | n^o \rangle \\[4pt] &= \int_0^{L/2} \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx + \int_{L/2}^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) 0 \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \end{align*}\], The second integral is zero and the first integral is simplified to, \[ E_n^1 = \dfrac{2}{L} \int_0^{L/2} V_o \sin^2 \left( \dfrac {n \pi}{L} x \right) dx \nonumber\], \[ \begin{align*} E_n^1 &= \dfrac{2V_o}{L} \left[ \dfrac{-1}{2 \dfrac{\pi n}{a}} \cos \left( \dfrac {n \pi}{L} x \right) \sin \left( \dfrac {n \pi}{L} x \right) + \dfrac{x}{2} \right]_0^{L/2} \\[4pt] &= \dfrac{2V_o}{\cancel{L}} \dfrac{\cancel{L}}{4} = \dfrac{V_o}{2} \end{align*}\], The energy of each perturbed eigenstate, via Equation \(\ref{7.4.17.2}\), is, \[ \begin{align*} E_n &\approx E_n^o + \dfrac{V_o}{2} \\[4pt] &\approx \dfrac{h^2}{8mL^2}n^2 + \dfrac{V_o}{2} \end{align*}\]. We turn now to the problem of approximating solutions { our rst (and only, at this stage) tool will be perturbation theory. Suitable for those who have completed the standard calculus sequence, the book assumes no prior knowledge of differential equations. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. It also happens frequently that a related problem can be solved exactly. The first step when doing perturbation theory is to introduce the perturbation factor ϵ into our problem. The technique is appropriate when we have a potential V(x) that is closely Therefore the energy shift on switching on the perturbation cannot be represented as a power series in \(\lambda\), the strength of the perturbation. It’s just there to keep track of the orders of magnitudes of the various terms. Time-independent perturbation theory Variational principles. We discussed a simple application of the perturbation technique previously with the Zeeman effect. Here, we will consider cases where the problem we want to solve with Hamiltonian H(q;p;t) is \close" to a problem with Hamiltonian H \end{array} Introduction to perturbation theory 1.1 The goal of this class The goal is to teach you how to obtain approximate analytic solutions to applied-mathematical problems that can’t be solved “exactly”. Note that the zeroth-order term, of course, just gives back the unperturbed Schrödinger Equation (Equation \(\ref{7.4.1}\)). For more information contact us at info@libretexts.org or check out our status page at https://status.libretexts.org. https://doi.org/10.1016/j.mcm.2011.02.045. Notice that each unperturbed wavefunction that can "mix" to generate the perturbed wavefunction will have a reciprocally decreasing contribution (w.r.t. Electron Passing Through Magnetic Field. SHO: Sudden Transition. System Upgrade on Fri, Jun 26th, 2020 at 5pm (ET) During this period, our website will be offline for less than an hour but the E-commerce and registration of new users may not be available for up to 4 hours. Equation \(\ref{7.4.13}\) is the key to finding the first-order change in energy \(E_n^1\). The general approach to perturbation theory applications is giving in the flowchart in Figure \(\PageIndex{1}\). V_o & 0\leq x\leq L \\ \infty & x< 0 \;\text{and} \; x> L \end{cases} \nonumber\]. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbation" parts. Here, we will consider cases where the problem we want to solve with Hamiltonian H(q;p;t) is \close" to a problem with Hamiltonian H 0(q;p;t) for which we know the exact solution. Many problems we have encountered yield equations of motion that cannot be solved ana-lytically. Watch the recordings here on Youtube! That is, eigenstates that have energies significantly greater or lower than the unperturbed eigenstate will weakly contribute to the perturbed wavefunction. This method, termed perturbation theory, is the single most important method of solving problems in quantum … Some texts and references on perturbation theory are [8], [9], and [13]. V_o & 0\leq x\leq L/2 \\ System Upgrade on Fri, Jun 26th, 2020 at 5pm (ET) During this period, our website will be offline for less than an hour but the E-commerce and registration of new users may not be available for up to 4 hours. Berry's Phase. In general perturbation methods starts with a known exact solution of a problem and add "small" variation terms in order to approach to a solution for a related problem without known exact solution. For instance, Newton's law of universal gravitation explained the gravitation between two astronomical bodies, but when a third body is added, the problem was, "How does each body pull on each?" Approximate methods. For example, perturbation theory can be used to approximately solve an anharmonic oscillator problem with the Hamiltonian That is to say, on switching on \(\hat{H}^1\) changes the wavefunctions: \[ \underbrace{ | n^o \rangle }_{\text{unperturbed}} \Rightarrow \underbrace{|n \rangle }_{\text{Perturbed}}\label{7.4.3}\], \[ \underbrace{ E_n^o }_{\text{unperturbed}} \Rightarrow \underbrace{E_n }_{\text{Perturbed}} \label{7.4.4}\]. However, this is not the case if second-order perturbation theory were used, which is more accurate (not shown). Changing this into integral form, and combining the wavefunctions, \[\begin{align*} E^{1} &=\int_{-\infty}^{\infty}\left(\frac{a}{\pi}\right)^{\left(\frac{1}{2}\right)} e^{\frac{-ax^2}{2}} \gamma x^{4} dx \\[4pt] &=\gamma\left(\frac{a}{\pi}\right)^{\frac{1}{2}} \int_{-\infty}^{\infty} x^{4} e^{-a x^2} d x \end{align*} \], \[\int_{0}^{\infty} x^{2 \pi} e^{-a x^2} dx=\frac{1 \cdot 3 \cdot 5 \ldots (2 n-1)}{2^{m+1} a^{n}}\left(\frac{\pi}{a}\right)^{\frac{1}{2}} \nonumber\], Where we plug in \(\mathrm{n}=2\) and \(\mathrm{a}=\alpha\) for our integral, \[\begin{aligned}E^{1} &=2 \gamma\left(\frac{a}{\pi}\right)^{\left(\frac{1}{2}\right)} \int_{0}^{\infty} x^{4} e^{-a x^2} d x \\ \nonumber \]. \infty & x< 0 \; and\; x> L \end{cases} \nonumber\]. In quantum mechanics, there are large differences in how perturbations are handled depending on whether they are time-dependent or not. Chapter 7 Perturbation Theory. First order perturbation theory will give quite accurate answers if the energy shiftscalculated are (nonzero and) … FIRST ORDER NON-DEGENERATE PERTURBATION THEORY Link to: physicspages home page. Taking the inner product of both sides with \(\langle n^o | \): \[ \langle n^o | \hat{H}^o | n^1 \rangle + \langle n^o | \hat{H}^1 | n^o \rangle = \langle n^o | E_n^o| n^1 \rangle + \langle n^o | E_n^1 | n^o \rangle \label{7.4.14}\], since operating the zero-order Hamiltonian on the bra wavefunction (this is just the Schrödinger equation; Equation \(\ref{Zero}\)) is, \[ \langle n^o | \hat{H}^o = \langle n^o | E_n^o \label{7.4.15}\], and via the orthonormality of the unperturbed \(| n^o \rangle\) wavefunctions both, \[ \langle n^o | n^o \rangle = 1 \label{7.4.16}\], and Equation \(\ref{7.4.8}\) can be simplified, \[ \bcancel{E_n^o \langle n^o | n^1 \rangle} + \langle n^o | H^1 | n^o \rangle = \bcancel{ E_n^o \langle n^o | n^1 \rangle} + E_n^1 \cancelto{1}{\langle n^o | n^o} \rangle \label{7.4.14new}\], since the unperturbed set of eigenstates are orthogonal (Equation \ref{7.4.16}) and we can cancel the other term on each side of the equation, we find that, \[ E_n^1 = \langle n^o | \hat{H}^1 | n^o \rangle \label{7.4.17}\]. Michael Fowler (Beams Professor, Department of Physics, University of Virginia). It is the only manner to really master the theoretical aspects presented in class or learned from the book. The summations in Equations \(\ref{7.4.5}\), \(\ref{7.4.6}\), and \(\ref{7.4.10}\) can be truncated at any order of \(\lambda\). \(\lambda\) is purely a bookkeeping device: we will set it equal to 1 when we are through! This is, to some degree, an art, but the general rule to follow is this. theory . It is easier to compute the changes in the energy levels and wavefunctions with a scheme of successive corrections to the zero-field values. However, the denominator argues that terms in this sum will be weighted by states that are of. The idea behind perturbation theory is to attempt to solve (31.3), given the The LibreTexts libraries are Powered by MindTouch® and are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. Methods for solving singular perturbation problems arising in science and engineering. \(\hat{H}^{o}\) is the Hamitonian for the standard Harmonic Oscillator with, \(\hat{H}^{1}\) is the pertubtiation \[\hat{H}^{1} = \epsilon x^3 \nonumber\]. Perturbation theory is a very broad subject with applications in many areas of the physical sciences. Perturbation theory was first devised to solve otherwise intractable problems in the calculation of the motions of planets in the solar system. The ket \(|n^i \rangle\) is multiplied by \(\lambda^i\) and is therefore of order \((H^1/H^o)^i\). to solve approximately the following equation: using the known solutions of the problem We begin with a Hamiltonian \(\hat{H}^0\) having known eigenkets and eigenenergies: \[ \hat{H}^o | n^o \rangle = E_n^o | n^o \rangle \label{7.4.1}\]. This is, to some degree, an art, but the general rule to follow is this. The basic idea here should be very familiar: perturbation theory simply means finding solutions to an otherwise intractable system by systematically expanding in some small parameter. This is essentially a step function. The basic assumption in perturbation theory is that \(H^1\) is sufficiently small that the leading corrections are the same order of magnitude as \(H^1\) itself, and the true energies can be better and better approximated by a successive series of corrections, each of order \(H^1/H^o\) compared with the previous one. For example, perturbation theory can be used to approximately solve an anharmonic oscillator problem with the Hamiltonian Perturbation theory is a method for continuously improving a previously obtained approximate solution to a problem, and it is an important and general method for finding approximate solutions to the Schrödinger equation. Copyright © 2011 Elsevier Ltd. All rights reserved. This occurrence is more general than quantum mechanics {many problems in electromagnetic theory are handled by the techniques of perturbation theory. Time-independent perturbation theory Variational principles. That is, the first order energies (Equation \ref{7.4.13}) are given by, \[ \begin{align} E_n &\approx E_n^o + E_n^1 \\[4pt] &\approx \underbrace{ E_n^o + \langle n^o | H^1 | n^o \rangle}_{\text{First Order Perturbation}} \label{7.4.17.2} \end{align}\], Example \(\PageIndex{1A}\): A Perturbed Particle in a Box, Estimate the energy of the ground-state and first excited-state wavefunction within first-order perturbation theory of a system with the following potential energy, \[V(x)=\begin{cases} Bibliography Another point to consider is that many of these matrix elements will equal zero depending on the symmetry of the \(\{| n^o \rangle \}\) basis and \(H^1\) (e.g., some \(\langle m^o | H^1| n^o \rangle\) integrals in Equation \(\ref{7.4.24}\) could be zero due to the integrand having an odd symmetry; see Example \(\PageIndex{3}\)). In this chapter, we describe the aims of perturbation theory in general terms, and give some simple illustrative examples of perturbation problems. E^{1} &=2 \gamma\left(\frac{a}{\pi}\right)^{\left(\frac{1}{2}\right)} \frac{1\cdot 3}{2^{3} a^2}\left(\frac{\pi}{a}\right)^{\frac{1}{2}}\end{aligned} \nonumber\]. We put ϵ into our problem in such a way, that when we set ϵ = 0, that is when we consider the unperturbed problem, we can solve it exactly. The approximate results differ from the exact ones by a small correction term. This method, termed perturbation theory, is the single most important method of solving problems in quantum mechanics and is widely used in atomic physics, condensed matter and particle physics. As with Example \(\PageIndex{1}\), we recognize that unperturbed component of the problem (Equation \(\ref{7.4.2}\)) is the particle in an infinitely high well. Periodic Perturbation. The problem of the perturbation theory is to find eigenvalues and eigenfunctions of the perturbed potential, i.e. Our intention is to use time-independent perturbation theory for the de-generate case. A regular perturbation problem is one for which the perturbed problem for small, nonzero values of "is qualitatively the same as the unperturbed problem for "= 0. Perturbation theory allows one to find approximate solutions to the perturbed eigenvalue problem by beginning with the known exact solutions of the unperturbed problem and then making small corrections to it based on the new perturbing potential. Use perturbation theory to approximate the wavefunctions of systems as a series of perturbation of a solved system. The general expression for the first-order change in the wavefunction is found by taking the inner product of the first-order expansion (Equation \(\ref{7.4.13}\)) with the bra \( \langle m^o |\) with \(m \neq n\), \[ \langle m^o | H^o | n^1 \rangle + \langle m^o |H^1 | n^o \rangle = \langle m^o | E_n^o | n^1 \rangle + \langle m^o |E_n^1 | n^o \rangle \label{7.4.18}\], Last term on right side of Equation \(\ref{7.4.18}\), The last integral on the right hand side of Equation \(\ref{7.4.18}\) is zero, since \(m \neq n\) so, \[ \langle m^o |E_n^1 | n^o \rangle = E_n^1 \langle m^o | n^o \rangle \label{7.4.19}\], \[\langle m^o | n^0 \rangle = 0 \label{7.4.20}\], First term on right side of Equation \(\ref{7.4.18}\), The first integral is more complicated and can be expanded back into the \(H^o\), \[ E_m^o \langle m^o | n^1 \rangle = \langle m^o|E_m^o | n^1 \rangle = \langle m^o | H^o | n^1 \rangle \label{7.4.21}\], \[ \langle m^o | H^o = \langle m^o | E_m^o \label{7.4.22}\], \[ \langle m^o | n^1 \rangle = \dfrac{\langle m^o | H^1 | n^o \rangle}{ E_n^o - E_m^o} \label{7.4.23}\]. References: Griffiths, David J. For example, in first order perturbation theory, Equations \(\ref{7.4.5}\) are truncated at \(m=1\) (and setting \(\lambda=1\)): \[ \begin{align} | n \rangle &\approx | n^o \rangle + | n^1 \rangle \label{7.4.7} \\[4pt] E_n &\approx E_n^o + E_n^1 \label{7.4.8} \end{align}\], However, let's consider the general case for now. These series are then fed into Equation \(\ref{7.4.2}\), and terms of the same order of magnitude in \(\hat{H}^1/\hat{H}^o\) on the two sides are set equal. We’re now ready to match the two sides term by term in powers of \(\lambda\). The first steps in flowchart for applying perturbation theory (Figure \(\PageIndex{1}\)) is to separate the Hamiltonian of the difficult (or unsolvable) problem into a solvable one with a perturbation. Semiclassical approximation. Perturbation theory is a useful method of approximation when a problem is very similar to one that has exact solutions. For this case, we can rewrite the Hamiltonian as, The first order perturbation is given by Equation \(\ref{7.4.17}\), which for this problem is, \[E_n^1 = \langle n^o | \epsilon x^3 | n^o \rangle \nonumber\], Notice that the integrand has an odd symmetry (i.e., \(f(x)=-f(-x)\)) with the perturbation Hamiltonian being odd and the ground state harmonic oscillator wavefunctions being even. The Problem Book in Quantum Field Theory contains about 200 problems with solutions or hints that help students to improve their understanding and develop skills necessary for pursuing the subject.
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