Physics
Perturbation in Quantum Mechanics
In quantum mechanics, perturbation refers to a small change or disturbance applied to a system, which allows for the study of the system's behavior under these perturbations. Perturbation theory is a mathematical method used to analyze the effects of these disturbances on the system's properties and behavior, providing valuable insights into quantum systems that cannot be solved exactly.
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8 Key excerpts on "Perturbation in Quantum Mechanics"
eBook - PDFElectronic Structure Modeling
Connections Between Theory and Software
- Carl Trindle, Donald Shillady(Authors)
- 2008(Publication Date)
- CRC Press(Publisher)
8 Perturbation Theory It is often the case that we can consider properties of molecules to be defined by their response to a disturbance. For example, the dipole moment of a molecule is defined by a change in energy of the system as an electric field is applied. Much of spectroscopy is simply the study of responses of a molecu-lar system to a periodic perturbation provided by external fields. Scattering experiments similarly can be considered the study of the effects of molecu-lar collisions, momentary disturbances of the reference systems. Perturb-ation theory permits a systematic representation of the effects of small disturbances on a reference system, and is the natural language for many molecular phenomena. First-Order Correction to a Nondegenerate Reference System We may seek solution for H c ¼ E c . Assume we can solve a similar reference system H 0 c 0 n ¼ E 0 n c 0 n for which the Hamiltonian H 0 closely resembles H . H ¼ H (0) þ l V where lim l ! 0 H ¼ H 0 Here the term V is the perturbation. The key idea of perturbation theory is to represent the unknown wave function as the reference system’s wave func-tion with a series of corrections. The wave function is then c n ¼ c (0) n þ lc (1) n þ l 2 c (2) n þ l 3 c (3) n þ where again lim l ! 0 c n ¼ c (0) n Furthermore, the associated eigenvalues are each represented as the refer-ence system’s energies with a sequence of corrections. E n ¼ E (0) n þ l E (1) n þ l 2 E (2) n þ l 3 E (3) n þ 187 Note that, for instance, in l 3 E (3) n 3 is the power of l and in E (3) n 3 refers to the third-order correction to the n th energy value.
eBook - PDF- Benjamin Schumacher, Michael Westmoreland(Authors)
- 2010(Publication Date)
- Cambridge University Press(Publisher)
17 Perturbation theory 17.1 Shifting the energy levels In physics, it often happens that the problem we wish to solve is close to – but not quite the same as – a problem we actually can solve. For example, for motions close to a stable equilibrium point, a classical 1-D particle can usually be regarded as a harmonic oscillator, because the potential function U ( x ) is approximately a parabola near its minimum. This fact accounts for the very great importance of the harmonic oscillator in mechanics. The same is true for a quantum particle. 1 We can solve the classical and quantum harmonic oscillator problems very well by now. Yet most real potential functions are only approximately parabolic around an equilibrium point. How can we make our analysis more exact? The general strategy for this type of quantum problem is known as perturbation theory . We imagine that the Hamiltonian is composed of two parts, an “unperturbed” part H 0 and a perturbation H P , both of which are independent of time. (We are therefore doing time-independent perturbation theory .) That is, H = H 0 + H P . (17.1) For the harmonic oscillator example, H 0 is the exact harmonic oscillator Hamiltonian, and H P is the small difference between the real potential U and the local parabolic approxi-mation. We assume that we know the stationary states | n and the corresponding energy levels ε n for the unperturbed Hamiltonian H 0 ; for now, we shall suppose that these levels are non-degenerate. Our job in time-independent perturbation theory is to approximate the energy levels E n and stationary states | ψ n for the full Hamiltonian H . In fact, what we will do is develop a sequence of better and better approximations to the exact answer, a sort of “power series” in the perturbation. If the perturbation is small, then we may be able to make do with only the first couple of corrections to the unperturbed ε n and | n . To see how this works, introduce a parameter λ and let H λ = H 0 + λ H P .
eBook - PDF- Mohsen Razavy(Author)
- 2011(Publication Date)
- World Scientific(Publisher)
Chapter 11 Perturbation Theory In their pioneering work on matrix mechanics, Born, Heisenberg and Jordan among other original contributions also developed a systematic method of ap-proximate calculation of the eigenvalues based on perturbation theory [1]. In “three men’s paper” this perturbation theory was formulated in the following way: Let us write the Hamiltonian of the system which is not explicitly time-dependent as H = H 0 ( p, q ) + λH 1 ( p, q ) + λ 2 H 2 ( p, q ) + · · · , (11.1) where λ is a small dimensionless parameter. We assume that the solution for the unperturbed Hamiltonian H 0 is known and we use a representation in which H 0 ( p, q ) is diagonal, i.e. h n | H 0 ( p 0 , q 0 ) | j i = E (0) j δ nj (11.2) where p 0 and q 0 are matrices which make H 0 diagonal. Here p 0 = lim p and q 0 = lim p as λ → 0. In order to diagonalize H ( p, q ) we choose a unitary transformation U such that p = Up 0 U -1 , and q = Uq 0 U -1 , (11.3) and then the Hamiltonian H ( p, q ) = UH ( p 0 , q 0 ) U -1 , (11.4) becomes a diagonal matrix h n | H | j i = E j δ nj . (11.5) 309 310 Heisenberg’s Quantum Mechanics To determine the form of the transformation, U , we write it as a power series in λ U = 1 + λU 1 + λ 2 U 2 + · · · , (11.6) with its inverse given by U -1 = 1 -λU 1 + λ 2 ( U 2 1 -U 2 ) + · · · . (11.7) Next we substitute (11.1), (11.6) and (11.7) in (11.4) and equate different powers of λ and we obtain the following set of equations: h n | H 0 ( p 0 , q 0 ) | j i = E (0) j δ nj , (11.8) h n | U 1 H 0 -H 0 U 1 + H 1 | j i = E (1) j δ nj , (11.9) h n | U 2 H 0 -H 0 U 2 + H 0 U 2 1 -U 1 H 0 U 1 + U 1 H 1 -H 1 U 1 + H 2 | j i = E (2) j δ nj , (11.10) · · · · · · · · · · · · · · · · · · · · · · · · · · · h n | U r H 0 -H 0 U r + F r ( H 0 · · · H r , U 0 · · · U r -1 ) | j i = E ( n ) j δ nj . (11.11) In these relations all of the operators are functions of p 0 and q 0 .
eBook - PDF- F. Constantinescu, E. Magyari, J.A. Spiers(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
CHAPTER VIII Perturbation Theory. The Variational Method 1 1. Stationary State Perturbation Theory Let H = Ho+H' be the Hamiltonian operator of a physical system, where H 0 is a time-independent operator (whose eigenfunctions are known), and H' is also a time-independent operator, called the perturbation. Through perturbation theory the energy spectrum and the eigenfunctions of the Hamiltonian H can be determined to various orders of useful approximation, provided that certain conditions are satisfied. 1. The spectrum of Ho is discrete and non-degenerate. Let φ η and E^ be the wavefunction and the energy respectively of a stationary state of the unperturbed Hamiltonian H 0 , and ψ η , Ε η the corresponding quantities of the Hamiltonian H. To the first order of approximation of perturbation theory we have then that E n = Eg»+H' m9 Ψη = φ η + Σ -Ewhw^ ( VIIL1 ) where H' mn = (m H' | n) = j φ^Η'φη dV are the matrix elements of the perturbation. To the second order of approximation E n = Ep+H' nn + Σ lf m L ■ (VIII.2) A necessary condition for these results to be valid is that H' mn «Ef^-E^ (VIII.3) for any m and n. 2. The spectrum of the Hamiltonian Ho is discrete and degenerate. Let us suppose that the energy level £* 0) is/-fold degenerate. In general, the introduction of a perturbation removes or partly removes the degeneracy of degenerate energy levels. t For other problems in perturbation theory and other methods of approximation, see Chapter XI. 203 Problems in Quantum Mechanics The distinct energies which result from the initial level E^ through the introduction of the perturbation are the solutions of the secular equation: Hft-E 2 1 3 1 1 2 Hg-E H® H{f... H&... H®-E.. = 0. (VIII.4) where H& = (nl H'nk) = J φ* η1 Η'φ η1ζ dV.
eBook - PDF- Alexander Altland, Ben Simons(Authors)
- 2006(Publication Date)
- Cambridge University Press(Publisher)
11 In contrast, low-order expansions in the external perturbation (e.g. experimentally applied electric or magnetic fields or whatever) are usually secure; see Chapter 7. 212 Perturbation theory system. However, before plunging into the technicalities of this analysis, it is worthwhile discussing some qualitative aspects of the problem. 5.2.1 Qualitative aspects A principal question that we will need to address is under r s what physical conditions are interactions “weak” (in com- parison to the kinetic energy), i.e. when does a perturbative approach with the interacting electron system make sense at all? To estimate the relative magnitude of the two contribu- tions to the energy, let us assume that each electron occupies an average volume r 3 0 . According to the uncertainty relation, the minimum kinetic energy per particle will be of order 2 /mr 2 0 . On the other hand, assuming that each parti- cle interacts predominantly with its nearest neighbors, the Coulomb energy is of order e 2 /r 0 . The ratio of the two energy scales defines the dimensionless density parameter e 2 r 0 mr 2 0 2 = r 0 a 0 ≡ r s where a 0 = 2 /e 2 m denotes the Bohr radius. 12 Physically, r s is the radius of the spheri- cal volume containing one electron on average; for the Coulomb interaction, the denser the electron gas, the smaller r s . We have thus identified the electron density as the relevant parameter controlling the relative strength of electron–electron interactions. 13 Eugene P. Wigner 1902–95 Nobel Laureate in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles." (Image © The Nobel Foundation.) Below, we will be concerned with the regime of high density, r s 1, or weak Coulomb interaction. In the opposite limit, r s 1, properties become increasingly dom- inated by electronic correlations.
eBook - PDF- Roberto Casalbuoni(Author)
- 2011(Publication Date)
- WSPC(Publisher)
Chapter 8 Perturbation theory Relativistic quantum field theory does not offer simple calculable examples and therefore one has to develop perturbative methods in order to dis-cuss interacting fields. In the following we will be interested in describing scattering processes, since they are the typical processes appearing in the experiments in particle physics. The methods we will use for introducing the perturbation theory are quite general but, for sake of simplicity and for the particular physical interest, we will discuss in detail only the case of the electromagnetic interactions of charged spin 1/2 particles described by a Dirac field. The results we will obtain can be easily generalized to other interacting theories. 8.1 The electromagnetic interaction As we have already discussed, the electromagnetic interaction of an ar-bitrary charged particle is obtained through the minimal substitution or, equivalently, by invoking the gauge principle ∂ μ → ∂ μ + ieA μ . (8.1) For a charged Klein-Gordon particle, following this prescription, we get the following Lagrangian density L free = ∂ μ φ † ∂ μ φ -m 2 φ † φ -1 4 F μν F μν → [( ∂ μ + ieA μ ) φ )] † [( ∂ μ + ieA μ ) φ ] -m 2 φ † φ -1 4 F μν F μν . (8.2) The interacting part is given by the following two terms L int . = -ie φ † ∂ μ φ -( ∂ μ φ † ) φ A μ + e 2 A 2 φ † φ. (8.3) 177 178 Introduction to Quantum Field Theory In the first term the gauge field is coupled to the current j μ = ie φ † ∂ μ φ -( ∂ μ φ † ) φ , (8.4) but another interacting term appears. This term is a straight consequence of the gauge invariance. In fact, the current j μ , which is conserved in the absence of the interaction, is neither conserved, nor gauge invariant, when the electromagnetic field is turned on.
eBook - PDF- John Mcgervey(Author)
- 2012(Publication Date)
- Academic Press(Publisher)
Atomic Structure The triumphs of the new quantum theory, or 4 'quantum mechanics as it is now called, described in the previous two chapters, leave no doubt that this theory correctly de-scribes atomic phenomena. Unfortunately, there are very few systems for which the Schrodinger equation can be solved exactly; even the hydrogen atom requires an approx-imation treatment if one is to account for the effects of electron spin (Section 7.3). Therefore we begin our study of atomic structure by in-troducing approximation methods that permit us to obtain, to a high degree of accuracy, the energy levels of a three-body system (the helium atom), and we shall then see, qualitatively, how the structure of heavier atoms is deter-mined. In the course of this study, we shall see that there is a fundamental difference between the classical and the quan-tum view of systems containing two or more identical parti-cles. The quantum-mechanical view of identical particles is required not only for atomic physics, but also for the expla-nation of a wide variety of phenomena in solid-state and molecular physics. 279 280 ATOMIC STRUCTURE 8.1 TIME-INDEPENDENT PERTURBATION THEORY Suppose that the potential energy function of a system is such that we cannot solve the Schrodinger equation exactly, but that we can solve it exactly for a potential V(x) which differs slightly 1 from the actual potential. We can write the potential energy of the system as V(x) + v(x) where v(x) is a small 1 perturbation added to the unperturbed potential V{x). The Hamiltonian operator of the unperturbed system is 2 2m dx 2 so that the Schrodinger equation of that system is Η 0 ψ ι = Ε ι ψ ι with a known set of eigenvalues E x and eigenfunctions φ χ . The Schrodinger equation of the perturbed system contains the addi-tional term v(x) in the Hamiltonian: ΙΗ 0 + ν(χ)-]ψ' η = Ε' η ψ' η (1) and as a result we cannot find the perturbed eigenfunctions ψ' η and eigen-values E' n directly.
- Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
- 1997(Publication Date)
- WSPC(Publisher)
Chapter 6 TIME-DEPENDENT PERTURBATION THEORY, TRANSITION PROBABILITIES, AND SCATTERING The central tool for calculating transition rates in quantum mechanics is known as Fermi's golden rule] it is a result that comes directly from time-dependent perturbation theory. To develop the relevant formalism, it is convenient to introduce the interaction picture of quantum mechanics, an alternative to the Schrodinger and Heisenberg pictures first presented in Chapter 3. A major application of the golden rule is to the calculation of scattering cross-sections. As an example, in this chapter we will derive the so-called double-differential cross-section for thermal neutron scattering. It will be seen that the probability that a neutron transfers a particular momentum and energy to the scattering medium depends on a knowledge of dynamic structure factors] these are functions that appear in light and x-ray scattering calculations as well. 6.1 The Interaction Picture in Quantum Mechanics In the Schrodinger picture of quantum mechanics, the time-evolution of a system's state vector ip s (t)) is dictated by the Schrodinger equation, i.e., Eq. 3.120. In Section 3.3 it was shown that given the state vector at t = 0, i.e., |'0 S (O)), the solution for the state vector at time t can be expressed as IV'SW) = U(t) |t/> 5 (0)), where U(t) is a unitary time-evolution operator satisfying the differential equation ih^-Uit) = HU(t). (6.1) at In cases when the Hamiltonian H is independent of time, Eq. 6.1 can be integrated and the solution is simply U(t) = exp 1—iHt/h). However, certain situations, e.g., many problems encountered in quantum electrodynamics, share the common feature that the Hamiltonian can be split into two pieces, namely, a time-independent Hamil-tonian H 0 and a time-dependent potential V(t), i.e., H = H 0 + V(t). (6.2) Think of H 0 as the Hamiltonian for the system when no disturbance, or perturbation, is present; we assume that the eigenstates of H 0 are known.
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