Equivalence of a Harmonic Oscillator to a Free Particle

1019 Prog. Theor. Phys. Vol. 84, No.6, December 1990, Progress Letters Equivalence of a Harmonic Oscillator to a Free Particle Shin TAKAGI Department of Physics, Tohoku University, Sendai 980 (Received August 20, 1990) It is shown that a time· dependent scaling of space and time coordinates transforms the time· dependent Schrodinger equation for a (time-dependent) harmonic oscillator to that for a free particle. This equivalence gives a particularly straightforward way to compute the Feynman kernel. Other applications are also suggested. This paper proves the equivalence of a time-dependent harmonic oscillator (TDHO) to a free particle. It is not only of conceptual interest but also of practical use in solving initial-value problems. By the word equivalence we mean that the time-dependent Schrodinger equation for a TDHO is converted into that for a free particle under a certain transformation. (Identity of the spectrum of Hamiltonian is not implied of course; Hamiltonian changes under the transformation.) Although this equivalence is the only explicit result obtained in this paper, there are various interesting features associated with the transformation used here which seem to have potential applications. Therefore we bigin with a discussion of a somewhat broader scope than a TDHO. A one-body SchrOdinger equation with a potential of fixed shape whose width and/ or height depen~ on time could serve as a reference for various many-body situations, and could have some relevance to such problems as tunneling and coherence in an environment. I) Also it might be of interest in connection with the issue of the so-called tunneling time. 2 ) Even a direct experiment with such a potential might not be inconceivable with modern technology; a ultra-cold neutron under a magnetic field where the Zeeman energy provides a (spin-dependent) potential for its orbital motion, an electron injected into a so-called quantum well structure, and so· on. Regardless of these expectations being sound or not, such a Schrodinger equation is interesting in its own right. Now, it is an elementary fact that the change of width of a potential by a constant factor can be converted into the change of its height through a constant rescaling of space and time coordinates. It is then natural to ask how this statement is generalized to the case of a time-dependent· width. This question motivates us to go over to a comoving frame of reference in which the width appears constant. (The phrase comoving is not meant to imply that a particle is at rest in such a frame. See below.) The original problem is then shown to be transformed to that of a new potential, to be called a comoving potential, which consists of two terms; the original potential with the time-dependence transferred from the width to the height, and a harmonic inertial force. Quantum mechanics of a particle in comoving frame Specifically we consider the following SchrOdinger equation: 1020 Progress Letters Vol. 84, No.6 (1) where V is an arbitrary potential whose width a and height h depend on time. For convenience another arbitrary potential W has been added. The dimension D of the space is arbitrary, and r is the Cartesian coordinate with D components, while {7 r denotes the gradient with respect to it. We use units such that m=ti=l, where m is the mass of the particle. We first perform the time-dependent scale transformation x=.I.. a' (t at' r= )to [a(t'))2 , (2) where to is an arbitrarily chosen initial time. The re-normalization factor a D / 2 is to preserve the normalization. The new coordinates are analogous to the comoving spatial coordinates and the conformal (or arc-parameter) time frequently used in cosmology.3) (The exponent 2 on a in the definition of r is dictated by the fact that the Schrodinger equation is diffusion-like, which in turn reflects the Galilean covariance for a free particle; in a relativistic case it will be 1.) For this reason we shall call them the comoving coordinates; they comove with the potential as it were, but not necessarily with the particle. This transformation, which is well-defined so long as the scale factor a (i.e., width) does not vanish, converts Eq. (1) into i 1r W=[ -tc{7x-iaax)2+a2hV(x)- ~ (aax)2+a 2W(ax, OJ w, (3) where the overdot denotes differentiation with respect to t (not r). Next we eliminate the longitudinal vector potential aax by the gauge transformation ¢lex, r)=exp [ - ~ aax 2JW(x, r), (4) where the phase is nothing but Jpodr with p=(a!a)r being the momentum with respect to the inertial frame of a particle at rest at x in the co moving frame (Hubble's law), to find (5) where (6) The remarkable feature as already announced is the appearance of a harmonic inertial force in the comoving potential v. It is interesting to note in passing that v takes relatively simple forms in some special classes of problems with W=O, although it is not the purpose of the present work to analyze them in detail. Class (0. If aa= - kh, where k is a constant, then v=a 2h[ V(x)- k1;2 /2]. The December 1990 }Jro~ress ]Setters 1021 time dependence appears only in the height of the original potential augmented by a constant harmonic term. A special subclass occurs when a is linear in t (hence k vanishes); the time dependence has been simply transferred from the width to the height. !his result also holds if a is piecewise linear in t. (It would be of interest to analyze the case of a piecewise linear periodic function and compare it with the case of a potential of periodically varying height. 2)) Class (ii). If the height synchronizes with the width so that a 2 h is a constant, say unity, then v= V(x)+a 3 ax 2 /2. Class (iii). If the conditions of the above two classes are met simultaneOusly, that is, if a2 is a polynomial of t of degree 2 or 1, then v does not depend on time. This class may be further classified according to k=l, 0 or -1 (in suitable units) just like the Friedmann-Robertson-Walker universe. 3 ) In the special case of vanishing k, it is just the original potential with all the time dependence removed. The effect of the original time variation would then be encapsulated in the transformations (2) and (4). Equivalence of a harmonic oscillator to a free particle When V is itself harmonic, there is no distinction between height and width. A suitable choice of width makes V cancel with the harmonic inertial force. Let us consider a time-dependent isotropic oscillator with the potential (7) with n greater than 2. Introducing a scale factor aU), whose time-dependence is to be specified later, we rewrite it as (8) This is in accordance with Eq. (1), and the transformation to the comoving frame leads to ·Eq. (5) with ~9) Now we choose a so that (10) then the harmonic term disappears. Thus in this comoving frame TDHO becomes free. One need not worry that the transformation would become invalid within a time of order of (0-1 when a chosen solution of Eq. (10) vanished (...truncated)