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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Matrix mechanics as a practical tool in quantum theory: The anharmonic oscillator

Ching‐Teh Li, Abraham Klein, Franz R. Krejs

Published Oct 15, 1975• 13 citations
Publisher

Abstract

For an anharmonic oscillator, it follows from the number of zeros of the exact wave functions as functions of the quantum number $n$ 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 $x$ and $p$ 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 $x$ and for $p$ and of the commutator $|x,p|$, 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.

DOI
10.1103/physrevd.12.2311

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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′〉〈n|x|{n}^{\ensuremath{'}}〉〈n∣x∣n\ensuremath′〉 and 〈n∣p∣n\ensuremath′〉〈n|p|{n}^{\ensuremath{'}}〉〈n∣p∣n\ensuremath′〉 should be rapidly decreasing functions of ∣n\ensuremath−n\ensuremath′∣|n\ensuremath{-}{n}^{\ensuremath{'}}|∣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.
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.
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.
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\%.
Matrices of dynamical variables of the one-dimensional harmonic oscillator are calculated by a new semiclassical method formulated by More and Warren. The results are semiclassical matrices Xn, m, Pn, m, etc., which i) exactly obey selection rules for zero matrix-elements, ii) are very accurate, circa 1 %, for nonzero matrix-elements, iii) which diagonalize the Hamiltonian H, and iv) which exactly satisfy the Heisenberg commutation relations and equations of motion. These statements are true for matrix-elements Fn, m with m ≽ n. The results show that most of the quantum physics is contained in the extended semiclassical theory.
The equation-of-motion method based on Heisenberg's matrix mechanics, utilized in this paper, was originally developed within the context of the nuclear many-body problem. The method was later applied successfully to anharmonic oscillators and other problems in quantum mechanics with polynomial potentials in one and two dimensions. Here we apply the method to the one-dimensional quantum system with the nonpolynomial potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ as an illustration of its applicability to a wider class of problems than has been previously considered. We examine some of the structural and numerical aspects of this one-dimensional quantum system, and show that it is essential to work with sinhx\mathrm{sinh}xsinhx instead of x as the dynamical variable for this nonpolynomial potential problem. This choice is suggested by a preliminary study of the exactly solvable Poschl-Teller potential, $\ensuremath{-}\ensuremath{\lambda}/{\mathrm{cosh}}^{2}x,$ where our methods provide an exact algebraic solution. For the potential $\ensuremath{\lambda}{\mathrm{sinh}}^{2}x,$ the same methods yield equations that can be studied numerically by a controlled sequence of nonperturbative approximations.