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Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation

Richard M. Friedberg, Lee Tsung Dao, Zhao Wei Qin

Published Oct 13, 1999• 13 citations
OpenAlex

Abstract

We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is $\\geq 0$, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter $g^{-1}$ and a much smaller quantum tunneling parameter $\\epsilon$ between these different minima. Expanding either the wave function or its energy as a formal double power series in $g^{-1}$ and $\\epsilon$, we show how the coefficients of $g^{-m}\\epsilon^n$ in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential $V={1/2}g^2(x^2-a^2)^2$.

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Relations between low-lying quantum wave functions and solutions of the Hamilton-Jacobi equation

R. Friedberg, T. D. Lee, W. Q. Zhao

arXiv:quant-ph/9910047•Nuovo cimento della Società italiana di fisica. A, Nuclei, particles and fields•Oct 1, 1999•14 citations
We discuss a new relation between the low lying Schroedinger wave function of a particle in a one-dimentional potential V and the solution of the corresponding Hamilton-Jacobi equation with -V as its potential. The function V is geq0\\geq 0geq0, and can have several minina (V=0). We assume the problem to be characterized by a small anhamornicity parameter g−1g^{-1}g−1 and a much smaller quantum tunneling parameter epsilon\\epsilonepsilon between these different minima. Expanding either the wave function or its energy as a formal double power series in g−1g^{-1}g−1 and epsilon\\epsilonepsilon, we show how the coefficients of g−mepsilonng^{-m}\\epsilon^ng−mepsilonn in such an expansion can be expressed in terms of definite integrals, with leading order term determined by the classical solution of the Hamilton-Jacobi equation. A detailed analysis is given for the particular example of quartic potential V=1/2g2(x2−a2)2V={1/2}g^2(x^2-a^2)^2V=1/2g2(x2−a2)2.
Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be shown that there is one-to-one physical correspondence between basic solutions (represented always by one Hamiltonian eigenfunction only) and classical ones, as the non-zero quantum potential has not any physical sense, representing only the "numerical" difference between Hamilton principal function and the phase of corresponding wave function in the case of non-inertial motion. Possible interpretation of superposition solutions will be then discussed in the light of this fact. And also different interpretation alternatives of the quantum-mechanical model will be newly analyzed and new attitude to them will be reasoned.
The numerical version of the Hamilton-Jacobi quantization method, recently proposed, is applied to the one dimensional quartic oscillator. A suitable quantization condition is formulated and various energy levels and wave functions are computed. The results very well agree with those obtained by means of the Schroedinger equation, and confirm that the Quantum Hamilton Jacobi approach, which is the exact version of the semiclassical WKB scheme, is a self-contained quantization procedure, equivalent and independent from the Schoedinger's one but more general. Indeed, with respect to this latter, the Quantum Hamilton-Jacobi equation can be used to investigate the limit h->0, where the Schroedinger equation loses its significance, and explains how the fundamental quantities of the classical mechanics, the Hamilton's characteristic function and the classical momentum, are generated from the corresponding quantum ones.
General analytical solutions of the Quantum Hamilton Jacobi Equation for conservative one-dimensional or reducible motion are presented and discussed. The quantum Hamilton's characteristic function and its derivative, i.e. the quantum momentum function, are obtained in general, and it is shown that any one-dimensional wave function can be exactly represented in a WKB-like form. The formalism is applied to the harmonic oscillator and to the electron's motion in the hydrogen atom, and the above mentioned functions are computed and discussed for different quantum numbers. It is analyzed how the quantum quantities investigated tend to the corresponding classical ones, in the limit ℏ→0\hbar\to 0ℏ→0. These results demonstrate that the Quantum Hamilton Jacobi Equation is not only completely equivalent to the Schrödinger Equation, but allows also to fully investigate the transition from quantum to classical mechanics.
Recent work shows that the Schrödinger equation can be solved exactly based only on classical least action. The computation is based on first computing the multi-valued action solutions of the Hamilton-Jacobi equation, computing the classical propagated density along all stationary action multipaths, and finally constructing the exact kernel and wave function based on these classical multi-valued quantities alone. The method requires that the propagated classical square root density along each stationary action path is harmonic, as is the case for basic examples such as e.g., the double-slit, quantum tunneling, or the Einstein-Podolsky-Rosen experiment. For more complex cases, residual Bohm-like quantum potentials along each stationary action path can be eliminated through a time-rescaling. This paper proposes an extended version of the above construction which avoids the need for time-rescaling even in the most general cases. This is achieved by introducing in the Hamilton-Jacobi equation a real viscosity term, which classically simply corresponds to replacing a lumped parameter model with a continuum model including a volumetric stress tensor. This keeps the earlier construction unchanged, except that the real stationary action multipaths are now real stationary viscous action multipaths defined with respect to an associated modified Lagrangian. As in the earlier version, the construction extends straightforwardly to the special and general relativistic contexts, thus describing in a common language the quantum, classical, and relativistic cases. Note that the global Bohm potential term in the lumped Hamilton-Jacobi equation associated with a Madelung transform does not have a classical equivalent, while the volumetric stress tensor term in our viscous Hamilton-Jacobi equation directly represents the real paths of a classical deformed particle field.
By means of numerical solutions of the quantum Hamilton Jacobi equation, a general WKB-like representation for one-dimensional wave functions is obtained. This representation is unique in the classically forbidden regions, while in the allowed one, each wave function corresponds to a one parameter family of solutions of the QHJE. The method has been applied to various systems, with different energies and initial conditions. In all investigated cases, the wave functions so obtained accurately reproduce the solutions of the Schrödinger equation, analytically or numerically computed by other ways. Some results for harmonic oscillator and radial Coulomb motion are presented.