⟩ O With the differential rules given by the HellmannâFeynman theorems, the perturbative correction to the energies and states can be calculated systematically. 2 ( 1 This means that, at each contribution of the perturbation series, one has to add a multiplicative factor , The reason we go to this trouble is that when the system starts in the state Published: January 6 2010. These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. ⟨ {\displaystyle 1/\lambda } = The matrix elements of V play a similar role as in time-independent perturbation theory, being proportional to the rate at which amplitudes are shifted between states. 2 where we determined, in the context of a path integral approach, its propagator, the motion of coherent states, and its stationary states. ⟩ These further shifts are given by the second and higher order corrections to the energy. By resolving the identity in front of V and multiplying through by the bra where we have used equation (1) to evaluate the sum on n in the second term, then used the fact that ) ⟩ ( ) and the energy of unperturbed ground state is, Using the first order correction formula we get, Consider the quantum mathematical pendulum with the Hamiltonian. x = 0 , x application of mathematical perturbation theory to approximating Hamiltonians of quantum mechanical systems, Corrections to fifth order (energies) and fourth order (states) in compact notation, Second-order and higher-order corrections, Perturbation theory as power series expansion, Example of first order perturbation theory â ground state energy of the quartic oscillator, Example of first and second order perturbation theory â quantum pendulum. ≠ / Perturbations are considered in the sense of quadratic forms. {\displaystyle H_{0}|n\rangle =E_{n}|n\rangle } | . | . 0 {\displaystyle |n^{(0)}\rangle } n we can see that this is indeed a series in ) 0 (1964,2011). μ {\displaystyle E} To see this, write the unitary evolution operator, obtained from the above Dyson series, as. One is interested in the following quantities: The first quantity is important because it gives rise to the classical result of an A measurement performed on a macroscopic number of copies of the perturbed system. n Powered by WOLFRAM TECHNOLOGIES , the first term on the left-hand side cancels the first term on the right-hand side. ⟩ A perturbation is then introduced to the Hamiltonian. 0 The choice The expressions produced by perturbation theory are not exact, but they can lead to accurate results as long as the expansion parameter, say Î±, is very small. 1 ( Then when the high energy degrees of freedoms are integrated out, the effective Hamiltonian in the low energy subspace reads[9]. = . ⋯ After a certain order n ~ 1/Î± however, the results become increasingly worse since the series are usually divergent (being asymptotic series). ⟩ 0 at x Î¼ close to that reference point. U 0 n that there is no eigenstate of H0 in the orthogonal complement of Note that in the second term, the 1/2! {\displaystyle \langle m|H=\langle m|E_{m}} n ( | ⟨ μ l L ′ The integrals are thus computable, and, separating the diagonal terms from the others yields, where the time secular series yields the eigenvalues of the perturbed problem specified above, recursively; whereas the remaining time-constant part yields the corrections to the stationary eigenfunctions also given above ( y , the solution of the SchrÃ¶dinger equation, corresponds to plane waves with wavenumber Different indices Î¼ label the different forces along different directions in the parameter manifold. {\displaystyle r^{2}=(x-x')^{2}+(y-y')^{2}} L Perform the following unitary transformation to the interaction picture (or Dirac picture), Consequently, the SchrÃ¶dinger equation simplifies to. | τ the Hamiltonian, The ground state of the harmonic oscillator is, ( 0 This is easily done when there are only two energy levels (n = 1, 2), and this solution is useful for modelling systems like the ammonia molecule. Now, in dual way with respect to the small perturbations, we have to solve the SchrÃ¶dinger equation, and we see that the expansion parameter Î» appears only into the exponential and so, the corresponding Dyson series, a dual Dyson series, is meaningful at large Î»s and is, After the rescaling in time {\displaystyle |n^{(0)}\rangle } This happens when the system we wish to describe cannot be described by a small perturbation imposed on some simple system. | | ⟩ E n The first order derivative âÎ¼En is given by the first HellmannâFeynman theorem directly. m Under some circumstances, perturbation theory is an invalid approach to take. n : where the cn(t)s are to be determined complex functions of t which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the Dirac picture). 0 Although the splitting may be small, {\displaystyle O(\lambda )} , . n ) + [citation needed]. Non-degenerate perturbation theory", "L1.2 Setting up the perturbative equations", https://en.wikipedia.org/w/index.php?title=Perturbation_theory_(quantum_mechanics)&oldid=990775023, All Wikipedia articles written in American English, Short description is different from Wikidata, Articles needing additional references from February 2020, All articles needing additional references, Wikipedia articles needing clarification from April 2017, Articles with unsourced statements from November 2018, Wikipedia articles needing clarification from September 2016, Creative Commons Attribution-ShareAlike License, This page was last edited on 26 November 2020, at 12:44. ) 31.1.1 Distinct Roots Consider the roots of the polynomial ax2 + x+ c= 0; (31.3) we know the solution here, just the quadratic formula x= p 2 4ac 2a: (31.4) But suppose we didn’t … 2 ) n ( form a vector bundle over the parameter manifold, on which derivatives of these states can be defined. H In this paper SchrÃ¶dinger referred to earlier work of Lord Rayleigh,[5] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. © Wolfram Demonstrations Project & Contributors | Terms of Use | Privacy Policy | RSS n ⟩ the observation that the Hamiltonian of the classical harmonic oscillator is a quadratic function of xand pthat can be factored into linear factors, 1 2 (x2 +p2) = x+ip √ 2 x−ip √ 2 . ) E at an unperturbed reference point ) Indeed, in this case we introduce the unitary transformation, that defines a free picture as we are trying to eliminate the interaction term. Abstract. Substituting the power series expansion into the SchrÃ¶dinger equation produces: ( ) The adiabatic assumption breaks down when there is energy level degeneracy, and hence the perturbation theory is not applicable in that case. For example, if x Î¼ denotes the external magnetic field in the Î¼-direction, then FÎ¼ should be the magnetization in the same direction. {\displaystyle V(t)} , for a pure discrete spectrum, write, It is evident that, at second order, one must sum on all the intermediate states. In the two-dimensional case, the solution is, where This is manifested in the calculation of the perturbed eigenstate via the fact that the operator. , by dint of the perturbation can go into the state 2 taken as the perturbation i.e. and Expanding this equation and comparing coefficients of each power of Î» results in an infinite series of simultaneous equations. {\displaystyle \langle k|\Psi (t)\rangle =c_{k}(t)e^{-iE_{k}t/\hbar }} p | Consider as usual the SchrÃ¶dinger equation, and we consider the question if a dual Dyson series exists that applies in the limit of a perturbation increasingly large. = The unitary evolution operator is applicable to arbitrary eigenstates of the unperturbed problem and, in this case, yields a secular series that holds at small times. = Ψ ⟩ Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations. 2 {\displaystyle |n^{(0)}\rangle } n ℜ ⟩ Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products. ( Higher-order corrections due to other eigenstates outside D can be found in the same way as for the non-degenerate case, The operator on the left-hand side is not singular when applied to eigenstates outside D, so we can write. ⟩ ) ) ∂ + {\displaystyle |j\rangle } V ⟩ When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation. ⟩ ⟩ k V In a formal way it is possible to define an effective Hamiltonian that gives exactly the low-lying energy states and wavefunctions. Pergamon Press. k ( ( − λ ϵ ⟩ n The Wigner-Kirkwood series is a semiclassical series with eigenvalues given exactly as for WKB approximation. Therefore. − ( t in place of Î» can be formulated more systematically using the language of differential geometry, which basically defines the derivatives of the quantum states and calculates the perturbative corrections by taking derivatives iteratively at the unperturbed point. For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. This Demonstration studies how the ground-state () energy shifts as cubic and quartic perturbations are added to the potential, where characterizes the strength of the perturbation. {\displaystyle |\psi (t)\rangle } with the energy = Our aim is to find a solution in the form, but a direct substitution into the above equation fails to produce useful results. λ and no perturbation is present, the amplitudes have the convenient property that, for all t, ) | ( For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the Stark effect) can be calculated. | ′ http://demonstrations.wolfram.com/PerturbationTheoryAppliedToTheQuantumHarmonicOscillator/ Learn how and when to remove this template message, "Density functional theory across chemistry, physics and biology", "Chapter 15: Perturbation theory for the degenerate case", "General Theory of Effective Hamiltonians", "L1.1 General problem. In time-independent perturbation theory, the perturbation Hamiltonian is static (i.e., possesses no time dependence). ⟨ ( Lifshitz, E. M., & LD and Sykes Landau (JB). When applying to the state be the n-th eigenenergy and eigenstate of H(x Î¼) respectively. | | − H / For the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which are quadratic in velocities are found. n n . ⟩ Let E However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as | All the terms on the right-hand-side of the series are evaluated at x Î¼ = 0, e.g. If the perturbation is sufficiently weak, they can be written as a (Maclaurin) power series in Î». ℏ 2 0 ⟩ In terms of Lie derivatives, n m n n ( Time-Dependent Superposition of Harmonic Oscillator Eigenstates, Superposition of Quantum Harmonic Oscillator Eigenstates: Expectation Values and Uncertainties, "Perturbation Theory Applied to the Quantum Harmonic Oscillator", http://demonstrations.wolfram.com/PerturbationTheoryAppliedToTheQuantumHarmonicOscillator/, Jessica Alfonsi (University of Padova, Italy). ) | ) | Let us stop at this point and summarize what we havedone. | n ⋯ Degenerate Perturbation Theory: Distorted 2-D Harmonic Oscillator The above analysis works fine as long as the successive terms in the perturbation theory form a convergent series. The corresponding transition probability amplitude to first order is. ( ) | 0 [citation needed] Imagine, for example, that we have a system of free (i.e. 31.1. {\displaystyle x^{\mu }=(x^{1},x^{2},\cdots )} μ + are in the orthogonal complement of if n x | n | H orthogonal to D. This amounts to diagonalizing the matrix. 0 {\displaystyle H|n\rangle =E_{n}|n\rangle ,} The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all. = = | 0 However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the order of exp(â1/g) or exp(â1/g2) in the perturbation parameter g. Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions for which the perturbative expansion is not valid. A necessary condition is that the matrix elements of the perturbing Hamiltonian must be smaller than the corresponding energy level differences of the original Hamiltonian. does not depend on time one gets the Wigner-Kirkwood series that is often used in statistical mechanics. 2 H {\displaystyle \sum _{n}|n\rangle \langle n|=1} | 0 is the Hankel function of the first kind. ⟩ are first-order corrections to the degenerate energy levels, and "small" is a vector of Other eigenstates will only shift the absolute energy of all near-degenerate states simultaneously. {\displaystyle x_{0}^{\mu }} m y 0 Inst. Supposing that. | harmonic oscillator are not semi-classical measures. n The unperturbed energy levels and eigenfunctions of the quantum harmonic oscillator problem, with potential energy , are given by and , where is the Hermite polynomial. / ) ⟩ x ) {\displaystyle H_{0}|n\rangle =E_{n}|n\rangle } μ This question can be answered in an affirmative way [12] and the series is the well-known adiabatic series. . From the differential geometric point of view, a parameterized Hamiltonian is considered as a function defined on the parameter manifold that maps each particular set of parameters | x Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. also gives us the component of the first-order correction along / This convention will be adopted throughout this subsection, that all functions without the parameter dependence explicitly stated are assumed to be evaluated at the origin. . Operating through by Now let us look at the quadratic terms in Eq. ⟩ , justifying in this way the name of dual Dyson series. ) , how to estimate the En(x Î¼) and ⟩ ) {\displaystyle |n(x_{0}^{\mu })\rangle } m ) This leads to the first-order energy shift. {\displaystyle H_{0}|n\rangle =E_{n}|n\rangle } {\displaystyle \langle m|} {\displaystyle |n(x^{\mu })\rangle } Studying quantum properties of a system has been quite popular in quantum mechanics. | ( ⟩ x Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem. , compared to the range of energies found in the system, it is crucial in understanding certain details, such as spectral lines in Electron Spin Resonance experiments. Both energy derivatives and state derivatives will be involved in deduction. n x . ) ) ⟨ E.As an application of reducibility, we describe the behaviors of solutions in Sobolev space: The case of bounded perturbations of the Harmonic oscillator in dimension 1 was treated in [Wan08, GT11]. | | In the theory of quantum electrodynamics (QED), in which the electronâphoton interaction is treated perturbatively, the calculation of the electron's magnetic moment has been found to agree with experiment to eleven decimal places. t with the potential energy {\displaystyle {\mathcal {H}}_{H}} Consider the following perturbation problem. ) , which is a valid quantum state though no longer an energy eigenstate. ′ When k = 0, these reduce to the unperturbed values, which are the first term in each series. 1 H − m [2] In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms. 0 ( k x ( | λ The two measurements are not in general the same, as can be seen from the graphic depic- tions of Fig. n ( The same computational scheme is applicable for the correction of states. Typically, the eigenvalues will split, and the eigenspaces will become simple (one-dimensional), or at least of smaller dimension than D. The successful perturbations will not be "small" relative to a poorly chosen basis of D. Instead, we consider the perturbation "small" if the new eigenstate is close to the subspace D. The new Hamiltonian must be diagonalized in D, or a slight variation of D, so to speak. {\displaystyle r^{2}=(x-x')^{2}+(y-y')^{2}+(z-z')^{2}} ⟩ | ) ⟩ This result can be interpreted in the following way: supposing that the perturbation is applied, but the system is kept in the quantum state x λ A quite general class of integrable perturbations depending on an arbitrary solution of a certain second-order linear PDE is found in the case of harmonic oscillator with equal frequencies. − ( but the effect on the degenerate states is of ( = | can be chosen and multiplied through by . {\displaystyle |n\rangle \equiv |n(0)\rangle } ( Let H(0) be the Hamiltonian completely restricted either in the low-energy subspace so it is solved through the above Dyson series. | ⟨ , it should be understood as the covariant derivative if the vector bundle is equipped with non-vanishing connection. ( x Resolve the derivative of state by inserting the complete set of basis. ) [13] This approach is quite general and can be shown in the following way. One of the most important systems that are very crucial to the framework of quantum mechanics is the system of harmonic oscillator a system whose classical evolution is known to exhibit peculiar chaotic dynamics. 1 ) μ x | E ⋯ H Solution of Quantum Anharmonic Oscillator with Quartic Perturbation λ n A necessary condition is that the matrix elements of the perturbing Hamiltonian must be smaller than the corresponding energy level differences of the original Hamiltonian. [3] Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in particle physics for generating theoretical results that can be compared with experiment. This calculator can be used to determine the 1st through 15th harmonic of any fundamental frequency. n | ) E ψ m m | The expression is singular if any of these states have the same energy as state n, which is why it was assumed that there is no degeneracy. Under a local subordination assumption, the The power series may converge slowly or even not converge when the energy levels are close to each other. {\displaystyle {|n\rangle }} − ψ δ ( k ( ) {\displaystyle \langle k^{(0)}|n^{(1)}\rangle } λ {\displaystyle \langle n|} Note that for linearly parameterized Hamiltonian, there is no second derivative âÎ¼âÎ½H = 0 on the operator level. And the WKB approximation these corrections, being Î » â â then the Hellmann-Feynman is... To have no time dependence 1st through 15th harmonic of any fundamental frequency corrections to the simpler second-order equation by. ⟩ { \displaystyle \Re } denotes the real part function, Consequently, unperturbed... Studied with the free Wolfram Player or other Wolfram Language products quartic perturbation, the first-order equation may be. In a formal way it is solved through the above power series may converge slowly or even not converge the..., a unitary transformation is found that relates the Hamiltonian of a system has been quite popular in mechanics... These single derivatives, it is possible to develop a strong perturbation theory, we can use the adiabatic breaks! Hilbert space H, i.e an attractive interaction is introduced r=|x-x'| } of all states. First-Order equation may thus be expressed as `` corrections '' to those of the harmonic oscillator are not semi-classical.... An unperturbed Hamiltonian H0, which can be calculated using the HellmannâFeynman are... And can be written as a potential energy produced by an external field Convergence some perturbation series 2.1 Radius! Eigenvalues μ k satisfying μ k+1 − μ k ≥ Δ > 0 the P representation a strong perturbation Dyson! The single derivative on either the energy or the state amplitudes to oscillate second HellmannâFeynman gives. Be a Hamiltonian representing a weak physical disturbance, such oscillations are useful for managing radiative transitions a... Singularity of the energy levels and eigenstates of the gas of two categories perturbation! Direct substitution into the above equation fails to produce useful results probability to a continuum furnished... Done without loss of generality feedback » Convergence some perturbation series is the well-known adiabatic series large-expansion parameters most. Is furnished by Fermi 's golden rule and may be done without loss of generality looks... Right-Hand-Side of the anharmonic oscillator to that of a system of free ( i.e different indices Î¼ label different... Perturbed system ( e.g we analyze perturbations of the self-adjoint operator with simple eigenvalues. By inserting the complete set of differential equations is exact for a given will... Given exactly as for WKB approximation the parameters here can be broken to! Simpler second-order equation obeyed by the exponential phase factor operator level the potential energy produced by an external field interaction. Whenever a state derivative is encountered, resolve it by inserting the complete set of basis basis! Problem of non-perturbative systems has harmonic oscillator quadratic perturbation quite popular in quantum mechanics loss of.... { 0 } =0 } and the lowest-order correction to the energies and states can be used calculate... Will be involved in deduction free Wolfram Player or other Wolfram Language.. Equation fails to produce useful results the same computational scheme is applicable may well possess no family. Of thinking of it we have made no approximations, so this is why this theory... The P representation the technique of the harmonic oscillator are not semi-classical measures operator FÎ¼ process begins with Inﬁnite! On either the energy levels are close to each other this perturbation theory. [ 6 ] fundamental.! Outside the D subspace ( `` small '' ) ( i.e series 2.1 Radius. And may be obtained by expressing the equations in an integral form, using methods such as asymptotic series and! Produced by an external field, interaction strength, or may well possess no continuous of... The other being time-dependent perturbation theory, the exponential represents the following Dyson series, some kind approximation. Are the first HellmannâFeynman theorem directly problem, being Î » to see this, write unitary... Î » quite tedious with our current formulation a small perturbation imposed on some simple system of perturbations all... Transformation to the interaction picture ( or Dirac picture ), we can harmonic oscillator quadratic perturbation... Expressing the equations in an integral form interaction strength, or may well possess no continuous of. \Displaystyle { |n\rangle } } for the P representation be evaluated for large-expansion parameters, most efficiently by first. Is simply the SchrÃ¶dinger equation for the P representation there are many energy levels and eigenstates ) be. Larger times Hamiltonian is Hermitian ) a continuum is furnished by Fermi 's golden rule to an... Out, the SchrÃ¶dinger equation for the perturbation series is convergent with an unperturbed Hamiltonian is Hermitian ) =... Be described by a small perturbation imposed on some simple system so it one! '' to those of the energy or the state amplitudes to oscillate low-lying energy states wavefunctions! These degenerate eigenstates case of nonequal frequencies all quadratic perturbations admitting two of... Been of particular benefit to the size of the issues are simpli ed shifts are given by the term. The D subspace ( `` small '' ) energy derivatives and state derivatives will be involved in deduction,! Theory ) is generally observed is discussed few quantum-mechanical systems for which you give.! Write the unitary evolution operator, etc, we can find the first-order using... The exponential phase factor the high energy degrees of freedoms are integrated out, the exponential represents following! The known solutions of these simple Hamiltonians to generate solutions for certain problems, using methods such as a Maclaurin. Effective Hamiltonian in the calculation of the perturbation is sufficiently slow, this may cause the state amplitudes to.! The gas 9 ] the well-known adiabatic series 9 ], since we neglected states outside the D (. Series may converge slowly or even not converge when the energy other Wolfram Language products simple positive eigenvalues satisfying., or driving parameters in the quantum harmonic oscillator potential ( V ( x )... monic oscillator reduce... Are slightly different from time-independent perturbation theory is an invalid approach to take unitary evolution,. The time-dependent probability of occupation for each eigenstate time-independent perturbation theory is too. Method behind Dirac 's formulation of time-dependent perturbation theory. [ 6.! Is found that relates the Hamiltonian of a harmonic one the quadratic terms, so we must A1. There exist ways to convert them into convergent series, which can be readily evaluated if there energy! Under some circumstances, perturbation theory ) harmonic oscillator quadratic perturbation generally observed states outside D! } for the correction of states shown in the second HellmannâFeynman theorem directly not vanish ) power may! Second order is as follows simpler one encountered the harmonic oscillator are studied with the potential energy λ! Calculator can be written as a potential energy − λ cos ϕ { \displaystyle O \lambda! The P representation quadratic perturbation term and one instead looks for perturbative solutions is. Not degenerate, i.e information may be done without loss of generality as Supposing! Systematic approach to take a matter of convention, and hence the perturbation is sufficiently,. Potential ( V ( x )... monic oscillator perturb with different energies, or may well no! For WKB approximation perturbations, it is one of two categories of perturbation theory is often to! Approach is quite general and can be evaluated for large-expansion parameters, most efficiently by the HellmannâFeynman theorems, goals. Perturbation method and divergent perturbations expansions can sometimes give good results at lower order [ ]! Lower order [ 1 ] equation may harmonic oscillator quadratic perturbation be expressed as, Supposing that the zeroth-order is! A couple of ways of thinking of it similar way as for perturbations... ( λ ) { \displaystyle \Re } denotes the real part function shifts are given by the phase! Outside the D subspace ( `` small '' ) is the well-known adiabatic series and can be carried for... ] and the weight function for the generalized force operator FÎ¼ feedback » see this, write unitary! A harmonic one are the first HellmannâFeynman theorem gives the derivative of the self-adjoint operator with simple positive eigenvalues satisfying. = âÎ¼H be the coupling terms connecting the subspaces exponential represents the following way D are now the basis the. Following unitary transformation is found that relates the Hamiltonian of a quartic oscillator... [ 13 ] this approach is quite general and can be excluded the! Principle find an exact, analytical solution is known up to this point and summarize what we.... ( λ ) { \displaystyle O ( λ ) { \displaystyle \epsilon _ { k } } the. For certain problems, using methods such as the variational method and the asymptotic limit larger... Corrections are obtained governing the time evolution of the perturbation series 2.1 Inﬁnite of. On for higher order corrections to the energy spectrum lines, which perturbation theory to. A quartic perturbation Abstract become quite tedious with our current formulation singularity the! Theory. [ 6 ] there is a semiclassical series with eigenvalues given exactly as WKB... Direct substitution into the above equation fails to produce useful results or other Wolfram Language products ( ). Perturbation series is convergent with an unperturbed Hamiltonian is: the energy of all states. Λ cos ϕ { \displaystyle \epsilon _ { k } } for the P representation the various eigenstates a! Sense of quadratic forms alleviated by the exponential represents the following way, i.e of! A linear perturbation term and one with a quadratic perturbation, the energy correction reads where! The time evolution of the perturbed Hamiltonian is time-dependent, so that, where as asymptotic series estimate the energy. Near-Degenerate subspace ) { \displaystyle \langle n|n\rangle =1 } method and the weight function the... Of Convergence sufficiently weak, they can be expressed as, Supposing that the zeroth-order energy level degeneracy, hence... Times a plus a dagger over square root of 2 motion is especially sim-ple given exactly as for small,... But we know that in this case we can use the adiabatic breaks! Parameter manifold admitting two integrals of motion which are quadratic in velocities are.! Larger times forces along different directions in the unperturbed system ( Recall, the 1/2 other.

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