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Progress of Theoretical Physics

Quantum Theory of Anharmonic Oscillators: Some Exact Relations between Matrix Elements and Their Use for Various Approximation Methods

K. Yamazaki

Published Sep 1, 1983• 1 citation
Publisher

Abstract

This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.

DOI
10.1143/ptp.70.629

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This is an investigation of an anharmonic oscillator characterized by the potential (ω02/ 2) q2 + λ′ q4. By using the exact relations obtained by evaluating ≪i|[H, O]∓|j > where O is an arbitrary operator, H is our total Hamiltonian and |i > and |j > are exact eigenstates of H, we derive an exact recurrence formula. This formula allows us to express τ-functions with a higher power of the variables through τ-functions with a lower power of the variables and energy eigenvalues. In this way we derive several exact relations, which are, in a sense, generalizations of the virial theorem and sum rules (Eqs. (4·5), (4·6) and (5·4)). These exact relations are the central equaions of this paper. On the basis of these exact relations we propose our “nearest neighbour level” (N. N. L) approximation, which seems to provide a good approximation scheme. We can also use our exact relations to test the validity of various approximation methods, and as an example, we discuss the “New-Tamm-Dancoff” (N. T. D)-type of approximation in detail.
This is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1/2 (omega 2 q 2 + lambda q 4 ). As a lowest-order approximation an extremely simple formula for energy levels, E i (0) = (i + 1/2)1/4(3/alpha i + alpha i ), is derived (i being the quantum number of the energy level), which covers any (lambda, i). alpha i is the real positive root of a cubic equation: y i alpha i 3 + alpha i 2 - 1 = 0, with y i = 6lambda(2i 2 + 2i + 1)/(2i+1). This formula reproduces the exact energy levels within an error of about 1% for any ( lambda ,i) (the worst case is 2% for i = 0, lambda to infinity). Systematically higher orders of this perturbation theory are developed, which contains the 'usual' perturbation theory for the limiting case of small lambda , but this perturbation theory is valid for any (i, lambda). The second-order perturbation theory reduces the errors of our lowest-order results by a factor of about 1/5 in general. Various ranges (large, intermediate, small) of (i, lambda ) are investigated and compared with the exact values obtained by the Montroll group (1975, 1978). For i = 0, 1, even the fourth-order perturbation calculation can be elaborated explicitly, which reduces the error to about 0.01% for any lambda . For small lambda it gives correct numerical coefficients up to lambda 4 terms, as it should.
As in the usual quantum field theory, the states, and therefore also the eigenvalue spectrum of an anharmonic oscillator can be characterized by means of the so-called τ-functions, that is the matrix element of the type 〈0 | qn | j〉. For the calculation of these matrix elements, the equation of motion of the anharmonic oscillator can be used to obtain an infinite set of equations, which define an eigenvalue problem. To solve it a new set of functions, the so-called φ-functions, are introduced by means of a transformation, whose matrix corresponds formally to the WICK rule. An analysis of this infinite system of φ-equations shows that a convergent secular polynomial can be obtained, which exists as a limiting value of the polynomials for the truncated N φ -equationsystems in the limit N → ∞ . It is therefore permissible to calculate the eigenvalues of the infinite system in an approximate way from the truncated systems. Such an approximation procedure is the essential content of the so-called TAMM-DANCOFF method. The above mentioned convergence of the determinants therefore provides its justification. The convergence of the eigenvalues of the truncated systems to the exact oscillator values is numerically examined up to N= 20. The results are satisfactory.
For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number nnn that the matrix elements $〈n|x|{n}^{\ensuremath{'}}〉$ and $〈n|p|{n}^{\ensuremath{'}}〉$ should be rapidly decreasing functions of $|n\ensuremath{-}{n}^{\ensuremath{'}}|$. Matrix elements of polynomials in xxx and ppp should therefore be well approximated by a finite number of terms in their sum-rule decomposition. From the matrix elements of the equations of motion for xxx and for ppp and of the commutator ∣x,p∣|x,p|∣x,p∣, one thereby obtains closed sets of nonlinear algebraic equations to characterize subspaces of the Hilbert space of exact eigenfunctions. The approximations are also derived from a novel variational principle, and numerous variant approximation schemes are suggested. Essentially exact numerical results are obtained and compared with previous work. The broad applicability of the techniques is emphasized.
A self-consistent, nonperturbative approximation scheme is proposed which is potentially applicable to arbitrary interacting quantum systems. For the case of self-interaction, the scheme consists in approximating the original interaction HI(ϕ) by a suitable "potential" V(ϕ) which satisfies the following two basic requirements, (i) exact solvability (ES): the "effective" Hamiltonian H0 generated by V(ϕ) is exactly solvable i.e., the spectrum of states |n〉 and the eigenvalues En are known and (ii) equality of quantum averages (EQA): 〈n|HI(ϕ)|n〉 = 〈n|V(ϕ)|n〉 for arbitrary n. The leading order (LO) results for |n〉 and En are thus readily obtained and are found to be accurate to within a few percent of the "exact" results. These LO-results are systematically improvable by the construction of an improved perturbation theory (IPT) with the choice of H0 as the unperturbed Hamiltonian and the modified interaction, λH′(ϕ)≡λ(HI(ϕ) - V(ϕ)), as the perturbation where λ is the coupling strength. The condition of convergence of the IPT for arbitrary λ is satisfied due to the EQA requirement which ensures that 〈n|λH′(ϕ)|n〉 = 0for arbitrary λ and n. This is in contrast to the divergence (which occurs even for infinitesimal λ!) in the naive perturbation theory where the original interaction λHI(ϕ) is chosen as the perturbation. We apply the method to the different cases of the anharmonic and the double well potentials, e.g. quartic-, sextic- and octic-anharmonic oscillators and quartic-, sextic-double well oscillators. Uniformly accurate results for the energy levels over the full allowed range of λ and n are obtained. The results compare well with the exact results predicted by supersymmetry for the case of the sextic anharmonic potential and the double well partner potential. Further improvement in the accuracy of the results by the use of IPT, is demonstrated. We also discuss the vacuum structure and stability of the resulting theory in the above approximation scheme.
The finite-element approach to lattice field theory is both highly accurate (relative errors ∼1/N2\sim1/N^2∼1/N2, where NNN is the number of lattice points) and exactly unitary (in the sense that canonical commutation relations are exactly preserved at the lattice sites). In this talk I construct matrix elements for dynamical variables and for the time evolution operator for the anharmonic oscillator, for which the continuum Hamiltonian is H=p2/2+λq4/4.H=p^2/2+λq^4/4.H=p2/2+λq4/4. Construction of such matrix elements does not require solving the implicit equations of motion. Low order approximations turn out to be extremely accurate. For example, the matrix element of the time evolution operator in the harmonic oscillator ground state gives a result for the anharmonic oscillator ground state energy accurate to better than 1\%, while a two-state approximation reduces the error to less than 0.1\%.
Matrix mechanic methods are used to find approximate equations and solutions for quantum anharmonic oscillator problems. A series of hypotheses are introduced that truncate and partially decouple the infinite set of coupled equations that specify the problem in the matrix mechanics formulation. The dependent variables or unknowns in these equations are the matrix elements of the coordinate and momentum operators. The independent variables are the matrix indices and coupling strengths. The equations themselves specify that the off diagonal matrix elements of the Hamiltonian and the commutators expressed in terms of the unknowns vanish and that the diagonal commutator matrix elements vanish except for canonical pairs in which case they are equal to −ih/. The truncation and decoupling hypotheses offer an orderly procedure for dealing approximately with the vast array of equations of the exact problems. Only the leading behavior of the coordinate and momentum operator matrix elements is found in terms of the matrix indices and coupling parameters. Although general techniques are presented to find the equations, the solutions discussed and the applications are brief extensions of problems that have already been treated.