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Multiple-Scale Analysis of the Quantum Anharmonic Oscillator

Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which is solved exactly. The solution provides an operator mass renormalization of the theory.

Multiple-scale perturbation theory (MSPT) is a powerful and sophisticated perturbative method for solving physical problems having a small parameter ǫ [1,2]. MSPT is applicable to both linear and nonlinear problems. Indeed, it is so general that other perturbative methods such as WKB theory and boundary-layer theory, which are useful in more limited contexts, may be viewed as special cases of MSPT [1].
MSPT recognizes that dynamical systems exhibit characteristic physical behaviors at various length or time scales. The problem with conventional perturbation theory is that there is often a resonant coupling between successive orders. This coupling gives rise to secular terms (terms that grow rapidly with the length or time variable) in the perturbation series. Secular terms conflict with physical requirements that the solution be finite. MSPT reorganizes the conventional perturbation series to eliminate secular terms, and in doing so it describes quantitatively the behaviors that occur at many scales.
Ordinarily, MSPT is applied to classical differential equations such as Duffing's equation (the nonlinear equation of motion for the classical anharmonic oscillator): y ′′ + y + 4ǫy 3 = 0 (ǫ ≥ 0). (1) The positivity of ǫ ensures that y(t) is bounded [1]. The classical harmonic oscillator (ǫ = 0) has only one time scale, the period of oscillation. However, when ǫ = 0, the nonlinear term in Eq. (1) introduces many time scales. Using MSPT one can show that on a long-time scale there is a frequency shift of order ǫ.
In this letter we generalize MSPT and apply it to the Heisenberg operator equations of motion of the quantum anharmonic oscillator (the quantum version of Duffing's equation).
This generalization of MSPT techniques is nontrivial because it gives rise to a nonlinear system of operator differential equations [3]. We find the exact closed-form solution to this system and thereby obtain the quantum operator analog of the classical frequency shiftan operator mass renormalization that expresses the first-order shift of all energy levels.
We begin our presentation by reviewing the difficulties one encounters when one tries to solve Duffing's equation using a conventional perturbation series, y(t) = ∞ n=0 ǫ n y n (t). We choose as initial conditions y(0) = 1 and y ′ (0) = 0, which translate into y n (0) = δ n,0 and y ′ n (0) = 0, and substitute y(t) into Eq. (1). To zerothand first-order in powers of ǫ, we have The solution to Eq. (3) satisfying the initial conditions is y 0 (t) = cos t. Introducing this solution into Eq. (4), we obtain y ′′ 1 + y 1 = − cos(3t) − 3 cos t, which represents a forced harmonic oscillator whose driving term has frequencies 3 and 1. A harmonic oscillator when driven at its natural frequency, which in this case is 1, exhibits resonance. As a result, the solution y 1 (t) = 1 8 cos(3t) − 1 8 cos t − 3 2 t sin t contains a secular term that grows linearly with time t. The function y 1 (t) cannot be valid for long times because the exact solution to Duffing's equation remains bounded for all t [1]. Hence, the conventional perturbation expansion is sensible only for short times t << ǫ −1 .
If one is clever, one can use the conventional perturbation series to determine y(t) for long times, say of order ǫ −1 . To do so we note [1] that the structure of the most secular (highest power in t) term in y n (t) has the form 1 2 (3it/2) n e it /n! + c.c.; we then approximate y(t) by summing the most secular term in every order and the result is a cosine function that remains bounded for all times t: Hence, on the long-time scale τ = ǫt we see a frequency shift in the oscillator of order 3 2 ǫ. This result is not exact because there are less secular terms to all orders in the perturbation expansion; such terms give rise to frequency shifts of order ǫ 2 , ǫ 3 , ... .
The advantage of MSPT is that it reproduces Eq. (5) directly and bypasses the elaborate procedure of summing the conventional perturbation series to all orders by excluding ab initio secular terms from the perturbation expansion. MSPT assumes the existence of many time scales (t, τ = ǫt, σ = ǫ 2 t, ...), which can be temporarily treated as independent variables.
Here, we illustrate by performing just a first-order calculation. We use only the two variables t and τ = ǫt and seek a perturbative solution to Eq. (1) of the form The chain rule and the identity dτ dt = ǫ convert Eq. (1) to a sequence of partial differential equations for the dependent variables Y 0 , Y 1 , ... . The first two are The general solution to Eq.
into the right side of Eq. (8) and use triple-angle formulas such as cos 3 t = 1 4 cos(3t) + 3 4 cos t to simplify the result. To determine the functions A(τ ) and B(τ ) we demand that there be no resonant coupling between zeroth and first order in perturbation theory so that no secular terms appear in Y 1 . That is, we require that the coefficients of sin t and cos t vanish: To solve this system we multiply the first equation by B(τ ), the second by A(τ ), and add the resulting equations. Letting Thus, C(τ ) is the constant C(0) and the differential equation system (9) becomes linear: The solution to this system that satisfies the initial conditions is C(0) = 1 2 and Y 0 (t, τ ) = cos (1 + 3 2 ǫ)t , which reproduces the approximate solution in Eq. (5). While conventional perturbation theory is valid for t << ǫ −1 , the MSPT approximation is valid for t << ǫ −2 .
We now consider the quantum anharmonic oscillator, whose Hamiltonian is H(p, q) = 1 2 p 2 + 1 2 q 2 + ǫq 4 . Here, ǫ ≥ 0 so that the spectrum of H(p, q) is bounded below and p and q are operators satisfying the canonical equal-time commutation relation [q(t), p(t)] = ih.
The Heisenberg operator equations of motion, , combine to give the quantum Duffing's equation (1): Since p(t) and q(t) are operators, we cannot impose numerical initial conditions like those in Eq. (2); rather, we enforce a general operator initial condition at t = 0: where p 0 and q 0 are time-independent operators obeying the Heisenberg algebra [q 0 , p 0 ] = ih.
We now apply MSPT to Eq. (12). Assuming that q(t) exhibits characteristic behavior on the short-time scale t and on the long-time scale τ = ǫt, we write This equation is analogous to Eq. (6) but here Q 0 and Q 1 are operator-valued functions.
We substitute q(t) in Eq. (14) into Eq. (12), collect the coefficients of ǫ 0 and ǫ 1 , and obtain operator differential equations analogous to Eqs. (7) and (8): Because Eq. (15) is linear, it is easy to find its general solution, and from p = dq dt we obtain the momentum operator p(t) = B(τ ) cos t − A(τ ) sin t + O(ǫ). It is now necessary to find the coefficient functions A(τ ) and B(τ ), which are operators.
The canonical commutation relation in [q(t), p(t)] = ih implies that these operators satisfy We determine A(τ ) and B(τ ) by evaluating the right side of Eq. (16) and expanding the cubic term, taking care to preserve the order of operator multiplication. As in the classical case, we simplify the result using triple-angle formulas. To prevent secularity in Q 1 (t, τ ) we set the coefficients of cos t and sin t to zero and obtain This system of operator-valued differential equations is the quantum analog of Eq. (9).
The operator ordering W[q 0 f (Hτ )] is defined as follows: (1) Expand f (Hτ ) as a Taylor series in powers of the operator Hτ ; (2) Weyl-order the Taylor series term-by-term: Using this definition it is easy to verify that Eq. (22) is indeed the exact operator solution to Eq. (21) satisfying the initial conditions (18). We simplify the formal solution in Eq. (22) and re-express it in closed form by observing that if we reorder W(q 0 H n ) by commuting q 0 symmetrically to the left and to the right to maintain the Hermitian form we generate a set of polynomials [4] of degree n: We identify E n as the nth Euler polynomial [5] in which the argument is shifted by 1 2 : and so on. The generating function for these nonorthogonal polynomials is This generating function allows us to express the following Weyl-ordered product compactly: W q 0 e Hτ = q 0 e Hτ + e Hτ q 0 2 cosh(τh/2) .
The time dependence of a matrix element reveals the energy-level differences of the quantum system. It is easy to construct a set of states because the operators q 0 and p 0 satisfy the Heisenberg algebra [q 0 , p 0 ] = ih. Hence, appropriate linear combinations of q 0 and p 0 may be used as raising and lowering operators to generate a Fock space consisting of the states |n . By construction, these states are eigenstates of the operator H: H|n = n + 1 2 h|n . Evaluating Eq. (24) between the states n − 1| and |n and allowing the operator H to act to the left and the right, we obtain which predicts that the energy-level differences of the quantum oscillator are 1 + 3nhǫ. We may verify this result by recalling that the first-order correction to the energy eigenvalues [6] is E n = n + 1 2 + 3 4 ǫh(2n 2 + 2n + 1) + O(ǫ 2 ). Thus, E n − E n−1 = 1 + 3nhǫ + O(ǫ 2 ). We conclude by noting that in addition to the operators q(t) and p(t), the wave function ψ(x) for the quantum anharmonic oscillator also exhibits multiscale behavior. Specifically, to all orders in conventional weak-coupling Rayleigh-Schrödinger perturbation theory, ψ(x) behaves like the Gaussian e −x 2 /4 for large x; however, a geometrical-optics approximation to ψ(x) from WKB theory predicts that for large x, ψ(x) decays like the exponential of a cubic e − √ ǫ|x| 3 /6 . We can resolve this discrepancy by using a MSPT approach; we reorder the perturbation series by resumming secular terms [7].
The recursion relation for the polynomials P n (x) is typical of all perturbative calculations; the homogeneous part of this recursion relation is independent of n while the inhomogeneous part contains all previous polynomials. This recursive structure is responsible for successive orders of perturbation theory being resonantly coupled and causes the degree of the polynomials to grow with n.
To study the behavior of the wave function ψ(x) for large x we approximate ψ(x) by resumming the perturbation series and keeping just the highest power of x in every order.
[This is an exact analog of finding the coefficient of the highest power of t (most secular and a new set of polynomials. It seems impossible for this approach to give cubic exponential behavior because at each stage in the reorganization of the perturbation series the variable x appears only in even powers. However, we recognize that the exponent in Eq. (27) is the beginning of a binomial series whose sum is 1 6ǫ 1 − (1 + ǫx 2 ) 3/2 . If we now let x be large (ǫx 2 >> 1), we recover the correct asymptotic behavior of ψ(x) [8].
The approach used above for the anharmonic oscillator wave function has been used in perturbative quantum field theory to sum leading-logarithm divergences [9] and leading infrared divergences [10]. It is our hope that in the future the direct nonperturbative multivariate approach of MSPT will provide a framework to simplify such schemes.
CMB thanks the Department of Theoretical Physics at Imperial College, London, for its hospitality and the Fulbright Foundation, the PPARC, and the U.S. Department of Energy for financial support. LMAB thanks JNICT-Programa Praxis XXI for financial support under contract BD 2243/92.