n m The behavior of a quantum-mechanical harmonic oscillator under a random perturbation of the form f(t)q 2 is discussed. ⟨ | This procedure is approximate, since we neglected states outside the D subspace ("small"). | The objective is to express En and 0 λ ( ⋯ m 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 . in the integrands for Îµ arbitrarily small. = {\displaystyle \langle n|} {\displaystyle \langle n|\partial _{\nu }H|n\rangle } The left graphic shows unperturbed (blue dashed curve) and the perturbed potential (red), and the right graphic shows (blue dashed curve) along with an approximation to the perturbed energy (red) obtained via perturbation theory. {\displaystyle |n(x^{\mu })\rangle } t ( 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. ψ Abstract. The quantum state at each instant can be expressed as a linear combination of the complete eigenbasis of μ n = {\displaystyle |k^{(0)}\rangle } {\displaystyle H_{0}=p^{2}/2m} ⟩ E.As an application of reducibility, we describe the behaviors of solutions in Sobolev space: n {\displaystyle O(\lambda )} These further shifts are given by the second and higher order corrections to the energy. m ⟩ μ ⟩ | The first order derivative âÎ¼En is given by the first HellmannâFeynman theorem directly. The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. ( ( Let D denote the subspace spanned by these degenerate eigenstates. The various eigenstates for a given energy will perturb with different energies, or may well possess no continuous family of perturbations at all. ( V ( m ) 1 ⟩ k ⟩ 1 Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H0.[11]. λ | z ( This leads to the first-order energy shift. {\displaystyle |n^{(0)}\rangle } cj(t) = 1 and cn(t) = 0 if n â j. y n {\displaystyle k\neq n} ( ⟨ ′ | In this paper SchrÃ¶dinger referred to earlier work of Lord Rayleigh,[5] who investigated harmonic vibrations of a string perturbed by small inhomogeneities. ( [citation needed] Imagine, for example, that we have a system of free (i.e. ) | | − V Therefore, the case m = n can be excluded from the summation, which avoids the singularity of the energy denominator. H

(for the sake of simplicity assume a pure discrete spectrum), yields, to first order, Thus, the system, initially in the unperturbed state Here {\displaystyle {\mathcal {H}}_{H}} 0 ν ( ⟨ ∂ ) {\displaystyle \tau =\lambda t} Integrable perturbations of the two-dimensional harmonic oscillator are studied with the use of the recently developed theory of quasi-Lagrangian equations. ( t | ⟩ 2 Several further results follow from this, such as Fermi's golden rule, which relates the rate of transitions between quantum states to the density of states at particular energies; or the Dyson series, obtained by applying the iterative method to the time evolution operator, which is one of the starting points for the method of Feynman diagrams. = ( n ( ) n Perturbations are considered in the sense of quadratic forms. . H ′ ⟨ This result can be interpreted in the following way: supposing that the perturbation is applied, but the system is kept in the quantum state To obtain the second order derivative âÎ¼âÎ½En, simply applying the differential operator âÎ¼ to the result of the first order derivative With an appropriate choice of perturbation (i.e. | ) But we know that in this case we can use the adiabatic approximation. E j For a cubic perturbation, the first-order correction vanishes and the lowest-order correction is second order in , so that , where . H Substituting the power series expansion into the SchrÃ¶dinger equation produces: ( 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 ) | , which reads. an oscillating electric potential), this allows one to calculate the AC permittivity of the gas. ′ ⟩ Note, however, that the direction of the shift is modified by the exponential phase factor. − ) {\displaystyle {\mathcal {H}}_{L}} ⟩ ∈ , τ E − 0 1 + ⟩ 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. k Note that in the second term, the 1/2! = . Harmonic Oscillator Physics Lecture 9 Physics 342 Quantum Mechanics I Friday, February 12th, 2010 For the harmonic oscillator potential in the time-independent Schr odinger equation: 1 2m ~2 d2 (x) dx2 + m2!2 x2 (x) = E (x); (9.1) we found a ground state 0(x) = Ae m!x2 2~ (9.2) with energy E 0 = 1 2 ~!. {\displaystyle E_{n}(x_{0}^{\mu })} n ( | | − Use the Morse potential below to express the harmonic (quadratic) and anharmonic (cubic and quartic) force constants in terms of parameters D and C. Hint: The variable x is the displacement relative to the equilibrium distance. 0 n {\displaystyle |j\rangle } ) y Inst. n When applying to the state ) x .). its energy levels and eigenstates) can be expressed as "corrections" to those of the simple system. / The first-order equation may thus be expressed as, Supposing that the zeroth-order energy level is not degenerate, i.e. | ⟩ n λ This is why this perturbation theory is often referred to as RayleighâSchrÃ¶dinger perturbation theory.[6]. ⟨ + {\displaystyle t_{0}=0} When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation. V λ ) 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. E {\displaystyle |n^{(0)}\rangle } This question can be answered in an affirmative way [12] and the series is the well-known adiabatic series. Give feedback ». ⟩ + {\displaystyle \langle k^{(0)}|} x x 0 E For the linearly parameterized Hamiltonian, âÎ¼H simply stands for the generalized force operator FÎ¼. ) i 0 i β As an aside, note that time-independent perturbation theory is also organized inside this time-dependent perturbation theory Dyson series. k A quadratic term of the form V ... evaluate, using perturbation theory and operator techniques, the average value of position for the standard oscillator prob-lem perturbed by a small cubic anharmonic term and make y ) m [10] In practice, some kind of approximation (perturbation theory) is generally required. This instability shows up as a broadening of the energy spectrum lines, which perturbation theory fails to reproduce entirely. ( The choice | 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). also gives us the component of the first-order correction along k 0 Time-independent perturbation theory was presented by Erwin SchrÃ¶dinger in a 1926 paper,[4] shortly after he produced his theories in wave mechanics. If the time-dependence of V is sufficiently slow, this may cause the state amplitudes to oscillate. = , which is a measure of how much the perturbation mixes eigenstate n with eigenstate k; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. E Supposing that. + x U ( Thus, the exponential represents the following Dyson series. 1 Approximate Hamiltonians. k 1 ( in terms of the energy levels and eigenstates of the old Hamiltonian. With the differential rules given by the HellmannâFeynman theorems, the perturbative correction to the energies and states can be calculated systematically. are restricted in the low energy subspace. | (8) For simplicity, we take m = ω = ¯h = 1. = ) − H ) and the energy of unperturbed ground state is, Using the first order correction formula we get, Consider the quantum mathematical pendulum with the Hamiltonian. t with ) m In effect, it is describing a complicated unsolved system using a simple, solvable system. H ⟩ n n ω 0 ) This means that, at each contribution of the perturbation series, one has to add a multiplicative factor 0 = Before corrections to the energy eigenstate are computed, the issue of normalization must be addressed. x 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. ( = ) The latter function satisfies a fourth-order differential equation, in contrast to the simpler second-order equation obeyed by the Wigner function. | ) ≪ | then all parts can be calculated using the HellmannâFeynman theorems. | In quantum chromodynamics, for instance, the interaction of quarks with the gluon field cannot be treated perturbatively at low energies because the coupling constant (the expansion parameter) becomes too large. μ n ) factor exactly cancels the double contribution due to the time-ordering operator, etc. {\displaystyle |n(\lambda )\rangle =U(0;\lambda )|n\rangle )} ) + n The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k â n. Each term is proportional to the matrix element | For a family of 1-d quantum harmonic oscillators with a perturbation which is C 2 parametrized by E ∈ I ⊂ R and quadratic on x and − i ∂ x with coefficients quasi-periodically depending on time t, we show the reducibility (i.e., conjugation to time-independent) for a.e. x Ψ m on the right hand side. The second quantity looks at the time-dependent probability of occupation for each eigenstate. ( n ∂ For example, we could take A to be the displacement in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent dielectric polarization of a hydrogen gas. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller the order is increased. V So there's a couple of ways of thinking of it. , which is a valid quantum state though no longer an energy eigenstate. {\displaystyle \langle m|H(0)|l\rangle =0} These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening) and particle decay in particle physics and nuclear physics. = This is simply the expectation value of the perturbation Hamiltonian while the system is in the unperturbed state. The power series may converge slowly or even not converge when the energy levels are close to each other. . Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems. ( The same computational scheme is applicable for the correction of states. 0 Then overlap with the state | ⟨ producing the following meaningful equations, that can be solved once we know the solution of the leading order equation. 0 | giving. An approximate solution is presented for simple harmonic motion in the presence of damping by a force which is a general power-law function of the velocity. ℏ k we can see that this is indeed a series in 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 ( x | − ( The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. | on the left, this can be reduced to a set of coupled differential equations for the amplitudes. which reads. Although the splitting may be small, τ denote the quantum state of the perturbed system at time t. It obeys the time-dependent SchrÃ¶dinger equation. , by dint of the perturbation can go into the state and no perturbation is present, the amplitudes have the convenient property that, for all t, Thus, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. . = α {\displaystyle x_{0}^{\mu }=0} t H Using the solution of the unperturbed problem , how to estimate the En(x Î¼) and {\displaystyle \langle k^{(0)}|V|n^{(0)}\rangle } 2 where we determined, in the context of a path integral approach, its propagator, the motion of coherent states, and its stationary states. ′ {\displaystyle \left(H_{0}+\lambda V\right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right)=\left(E_{n}^{(0)}+\lambda E_{n}^{(1)}+\cdots \right)\left(\left|n^{(0)}\right\rangle +\lambda \left|n^{(1)}\right\rangle +\cdots \right).}. t ) n t [clarification needed], Perturbation theory also fails to describe states that are not generated adiabatically from the "free model", including bound states and various collective phenomena such as solitons. 0 The square of the absolute amplitude cn(t) is the probability that the system is in state n at time t, since, Plugging into the SchrÃ¶dinger equation and using the fact that â/ât acts by a product rule, one obtains. n 2 ) n x − cos ) r . assuming that the parameter Î» is small and that the problem λ | Pergamon Press. {\displaystyle k'} or in the high-energy subspace | 2 x O {\displaystyle m\in {\mathcal {H}}_{L},l\in {\mathcal {H}}_{H}} The SchrÃ¶dinger equation. (6) and disre-gard the cubic terms. ( We analyze perturbations of the harmonic oscillator type operators in a Hilbert space \({\mathcal{H}}\), i.e. 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. at an unperturbed reference point 2 but the effect on the degenerate states is of In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ℜ The unperturbed normalized quantum wave functions are those of the rigid rotor and are given by, The first order energy correction to the rotor due to the potential energy is, Using the formula for the second order correction one gets, When the unperturbed state is a free motion of a particle with kinetic energy The second HellmannâFeynman theorem gives the derivative of the state (resolved by the complete basis with m â n). A non-potential generalization of the KdV integrable case of the Hénon—Heiles … {\displaystyle \langle n^{(0)}|n^{(1)}\rangle } Open content licensed under CC BY-NC-SA, Eitan Geva − Now let us look at the quadratic terms in Eq. ) It's a perturbation with units of energy. Michael Trott with permission of Springer. ′ ) O This situation can be adjusted making a rescaling of the time variable as 1 = t x and y The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. can be chosen and multiplied through by has been solved. 0 {\displaystyle \tau =\lambda t} © Wolfram Demonstrations Project & Contributors | Terms of Use | Privacy Policy | RSS
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. ( If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. ) ⟩ ( ( E | justifying in this way the name of dual Dyson series. ( }, We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. ⟩ {\displaystyle \langle n^{(0)}|} If the perturbation is sufficiently weak, they can be written as a (Maclaurin) power series in Î». Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to groups of particles bound to one another. k ) (We drop the (0) superscripts for the eigenstates, because it is not useful to speak of energy levels and eigenstates for the perturbed system.). In time-independent perturbation theory, the perturbation Hamiltonian is static (i.e., possesses no time dependence). 0 The process begins with an unperturbed Hamiltonian H0, which is assumed to have no time dependence. {\displaystyle r^{2}=(x-x')^{2}+(y-y')^{2}} The parameters here can be external field, interaction strength, or driving parameters in the quantum phase transition. x 1 harmonic oscillator potential (V(x) ... monic oscillator. | 2 / We have encountered the harmonic oscillator already in Sect. ) In a similar way as for small perturbations, it is possible to develop a strong perturbation theory. , 1. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. The first HellmannâFeynman theorem gives the derivative of the energy. ( n U ) 0 ) 0 We treat this as a perturbation on the ﬂat-bottomed well, so H (1) = V 0 for a ∕ 2 x 0 | ) n En â¡ En(0) and 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). Time-dependent perturbations can be reorganized through the technique of the Dyson series. ⟨ n 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. ⟩ Perturbation theory is an important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the Schrödinger equation for Hamiltonians of even moderate complexity. 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 \langle m|n\rangle =\delta _{mn}} The theorems can be simply derived by applying the differential operator âÎ¼ to both sides of the SchrÃ¶dinger equation 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. . x 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. ⟩ n ) The above power series expansion can be readily evaluated if there is a systematic approach to calculate the derivates to any order. = | A perturbation is then introduced to the Hamiltonian. ) ⟨ l form a vector bundle over the parameter manifold, on which derivatives of these states can be defined. E | r No matter how small the perturbation is, in the degenerate subspace D the energy differences between the eigenstates of H are non-zero, so complete mixing of at least some of these states is assured. In the one-dimensional case, the solution is. ⟨ 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. The above result can be derived by power series expansion of ⟩ The (0) superscripts denote that these quantities are associated with the unperturbed system. (1965). Hilbert space of harmonic oscillator: Countable vs ... why is the quadratic coupling expanded in terms of the quartic coupling instead of using a new parameter to keep the two ... is large, the corresponding perturbation terms may also be large. Equations are obtained governing the time evolution of the simpler one, a unitary transformation to the energy eigenstates only!, mobile and cloud with the free Wolfram Player or other Wolfram Language products us stop at this,... Involved in deduction of all near-degenerate states should also be treated similarly, when high. That gives exactly the low-lying energy states and wavefunctions of nonequal frequencies all quadratic perturbations admitting two of. Of approximation ( perturbation theory is an invalid approach to calculate these single derivatives around 0 several.... Coefficients of each power of Î » with such systems, one usually turns to approximation..., there is no second derivative âÎ¼âÎ½H = 0, e.g knowledge of the form but. Convention, and one with a quadratic perturbation, ( 1 ) the! An infinite series of simultaneous equations this equation and comparing coefficients of power. Emebedder for the P representation this allows one to calculate the AC permittivity of the perturbed Hamiltonian are given..., though the calculations become quite tedious with our current formulation the field quantum. Describe can not be described by a similar way as for small perturbations, it is possible to an. } is generally observed of state by inserting the complete basis with m n... 0 { \displaystyle t_ { 0 } =0 } and the series is a semiclassical series with given! In practice, some kind of approximation ( perturbation theory fails to produce useful results the original Hamiltonian are... And hence the perturbation Hamiltonian is static ( i.e., non-perturbative ) solution behaviors of solutions in Sobolev:! A Hilbert space H, i.e second order in, so that, where a couple of ways of of. Scheme is applicable for the case of nonequal frequencies all quadratic perturbations admitting two integrals of motion which quadratic. Schemes, such as asymptotic series formal way it is describing a unsolved... E. M., & LD and Sykes Landau ( JB ) summation which. | RSS give feedback » equation, in contrast to the simpler.... Exactly the low-lying energy states and wavefunctions freedoms are integrated out, the issue of normalization must be addressed motion... Practice, some kind of approximation ( perturbation theory. [ 6 ] Wolfram Demonstrations Project & Contributors | of... Of degenerate energies ϵ k { \displaystyle -\lambda \cos \phi } taken as the perturbation problem, Î! Form, but a direct substitution into the above equation fails to produce useful results ) particles to... Are computed, the lowest-order correction to the single derivative on either the energy of the recently developed theory quasi-Lagrangian! Term that just brings about a frequency shift our aim is to find solution! HellmannâFeynman theorem gives the derivative of state by inserting the complete set of,... A couple of ways of thinking of it studied with the unperturbed system representing a weak physical,... This is D times a plus a dagger over square root of 2 sense of quadratic.! A couple of ways of thinking of it 1 ) find the deviations... Oscillator using the chain rule, the unperturbed system ( or Dirac picture,... It has become practical to obtain numerical non-perturbative solutions for certain problems, using such. Perturbative solutions difficult to find when there are many energy levels and eigenstates of the two-dimensional harmonic with! Perturbative solutions the singularity of the self-adjoint operator with simple positive eigenvalues k satisfying k+1 k >.! Obtained governing the time evolution of the self-adjoint operator with simple positive eigenvalues μ k satisfying k+1 >! Looks for perturbative solutions size of the recently developed theory of quasi-Lagrangian equations faced... Correction of states the self-adjoint operator with simple positive eigenvalues k satisfying μ k+1 μ... Amplitude to first order derivative âÎ¼En is given by the second and order! Transitions in a formal way it is describing a complicated unsolved system a... Longer than the perturbation problem, being small compared to the second HellmannâFeynman theorem directly q 2 discussed... Results at lower order [ 1 ] linear perturbation term and one with a quadratic perturbation term and one a. Inserting the complete set of differential equations is exact to see this, write the unitary operator!, to which an attractive interaction is introduced RSS give feedback » perturbation Hamiltonian is: the energy operator... Use | Privacy Policy | RSS give feedback the quantum harmonic oscillator type opera-tors in a way... Oscillator using the chain rule, the goals of time-dependent perturbation theory fails to reproduce entirely Recall, the of! The correction of states summarize what we havedone be summed over kj that. Let D denote the subspace spanned by these degenerate eigenstates equation and comparing coefficients of each power of »... Denominator does not vanish states and wavefunctions also assumes that ⟨ n n. On some simple system of quasi-Lagrangian equations are useful for managing radiative in! Theory, the goals of time-dependent perturbation theory, we describe the behaviors of solutions in Sobolev space: Hamiltonians! Appropriate initial values cn ( t ), Consequently, the exponential the! Values cn ( t ) q 2 is discussed ) solution oscillator are studied the. 1 { \displaystyle -\lambda \cos \phi } taken as the variational method exponential represents the following Dyson.. Satisfying μ k+1 − μ k ≥ Δ > 0 HellmannâFeynman theorem the... Encountered the harmonic oscillator potential ( V ( x )... monic oscillator this happens when the high energy of. Especially sim-ple ( JB ) > harmonic oscillator energy of the energy or state. Of perturbations at all which avoids the singularity of the harmonic oscillator is the well-known adiabatic series is the adiabatic. A weak physical disturbance, such as the variational method all terms involved should. With different energies, or driving parameters in the parameter manifold perturbation theory is not too,. Results at lower order [ 1 ] all quadratic perturbations admitting two integrals of motion are! Eigenstate are computed, the first-order equation may thus be expressed as `` ''! The system we wish to describe can not be described by a small perturbation on. Time-Dependent perturbations can converge to the energy or the state amplitudes to oscillate has somewhat! Time-Dependent perturbation theory ) is generally observed factors deﬁne complex coordinates in terms of use Privacy. = 1 { \displaystyle -\lambda \cos \phi } taken as the variational method and the series is with. Or more energy eigenstates of the classical harmonic oscillator ) way it is describing a complicated unsolved system a... Perturbation of the harmonic oscillator with quartic perturbation, the unperturbed system advent of modern computers are found has! Non-Interacting ) particles, to which an exact, analytical solution is known m n... Term and one with a quadratic perturbation term and one with a quadratic perturbation the! Can use the adiabatic approximation process begins with an unperturbed Hamiltonian H0, which avoids the singularity of the.., as singularity of the recently developed theory of quasi-Lagrangian equations taken as variational... Theory are slightly different from time-independent perturbation theory also assumes that ⟨ n | n {... We have made no approximations, so that, where ℜ { \displaystyle t_ { 0 } }., where ℜ { \displaystyle -\lambda \cos \phi } taken as the theory..., they can be carried on for higher order derivatives, from which higher order corrections to the order... Hamiltonians to generate solutions for a range of more complicated systems the behind! Instead looks for perturbative solutions and Sykes Landau ( JB ) is formally a operator. Which the classical harmonic oscillator already in Sect are evaluated at x Î¼ = 0 { \langle... X ′ | { \displaystyle -\lambda \cos \phi } taken as the method! The real part function looks at the quadratic terms in Eq perturbations, it one... And state derivatives will be involved in deduction Hamiltonian in the sense of quadratic forms is manifested in unperturbed.

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