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Physical Review A

Internal coordinates of quantum-mechanical systems

I. Mayer, Tibor András Rokob

Published Apr 9, 2012• 3 citations
Publisher

Abstract

It is demonstrated that the Hamiltonian for the internal motion of a nonrelativistic quantum mechanical system can be obtained just from the requirement of its overall translational momentum to be zero. Some particular cases relevant to an atomic system are also considered in some detail.
DOI
10.1103/physreva.85.044101

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Quantum mechanics is often compared with classical mechanics, but it seems that the momentum/spatial treatment of bound quantum mechanics provides for a particle-wave duality with both features having an experimental basis. In bound state quantum mechanics, one has a density pattern often similar to a two slit pattern i.e. with humps, yet at the same time one may knock a particle out of the bound state and obtain a probability distribution a(p)a(p) of momentum states. Describing a particle simply by its momentum even in the bound state seems to be classical as does using pp/2m to describe the kinetic energy of a plane wave in the decomposition of W(x), the wavefunction. Wave aspects are linked to classical particle features like momentum and do not change the particle energy pp/2m. The wave, we argue, may be related to some type of periodic cyclic forward backward motion which creates a spatial probability needed to estables resolution on the order of a wavelength proportional to 1/p. It is often not seen unless there is a potential or other probability waves which may interfere with it In this note, we consider the scenario that momentum (i.e. mov in the nonrelativistic case) is directly linked to a kind of forward-backward cyclic motion of a quantum particle regardless of its spin. The fact that spin exists suggests some kind of internal motion. In the case of a photon, it may be directly linked to angular motion of energy, although for spin ½ matters are more complicated. This motion again does not change classical energy pp/2m unless spin interacts with a magnetic field etc. We argue that a bound state is stochastic and so various free particle momentum states exist without being restricted to an x position. This is the particle scenario. The internal cycling motion, however, gives rise to a resolution of the order of 1/p suggesting a spatial distribution to account for such a resolution. Furthermore, the internal cycling (i.e. forward and backwards motion) even for a particle traveling to the right requires two dimensions to describe overall forwards and backwards motion. The spatial density of the cycling motion may be associated with internal changes in velocity, but overall there is a particle with mass mo moving with v (nonrelativistic case) and kinetic energy pp/2m. This holds for a free particle, but also for a particle in an infinite potential well with a single average |p|. An issue occurs, however, if one wishes to form an average of different p values at the same point x. In such a case the spatial probability density exp(i pj x) differs for each p at x and so the average [p1 exp(i p1 xo) + p2 exp(i p2 xo) ] / [ exp(i p1 xo) + exp(i p2 xo)] becomes relevant. In other words, one takes the average momentum, say p1, and links it to the spatial probability i.e. multiplies two averages and then divides by spatial probability to find a momentum flow at a specific point within a wavelength region. For the particle aspect, such a resolution is non-issue. The particle is the same at each x point in space.
In Part I we noted that in quantum mechanics, and even in the case of special relativity, there is an arbitrariness associated with x and t. For example, momentum in one dimension is exactly 0. One cannot arbitrarily shift it. If x=0, however, this is arbitrary, it may be shifted. We argued that this arbitrariness and the independence of x and t in a rest frame mo, x=0, t=to should carry over into the results seen in a moving frame, but in that case x’/t’=v because a Lorentz transformation links (x=0,t=0) to (x’=0, t’=0). Thus it appears that the arbitrariness of x,t and their independence have been broken, but we argue that a special Lorentz invariant -Et+px restores it by creating a time period hbar/E and wavelength hbar/p. In other words, a mathematical point is an abstraction representing a wavelength region defined by p through a complex probability exp(ipx) i.e. the wavefunction (and a similar exp(-iEt for time). In this note, we focus on these points in further detail. A point x is associated with p or -p in one dimension, but a distribution, say cos(px), combines a fixed p with probabilities and gives the impression of positive and negative momentum values when the slope is positive or negative. Thus cos(px) by itself shows no overall translational motion, but contains regions of positive or negative momentum i.e. zero point motion. One must superimpose on this zero point motion (which exists for all energies) actual “translation” to the right or left. Given that positive and negative p’s already exist in one dimension which have nothing to do with translation, one is forced to move into a second dimension to create the translational motion which is associated with a probability such that each x point has the same value. This then leads to exp(ipx). Next we consider a bound state situation. In the case of exp(ipx), cos(px) and sin(px) are linked to the positive/negative zero point type energy which has nothing to do with translation, while p has everything to do with translational motion. Thus if one is only interested in the translational motion (as Newton was) one may map exp(ipx) to p at x. An interesting feature, however, that appears in quantum mechanics is that is the zero point type motion which creates a very specific kind of averaging combines with pp/2m values to create the notion of translation(to the right and left at the same time so to speak because time is removed from the bound problem. This is very different from the quantum free particle which has p and pp/2m at each x point and is due to the two dimensional nature of exp(ipx) which leads to pexp(ipx) - pexp(-px) not equal 0. Thus the nature of “acceleration” in a bound quantum system is closely linked to zero point energy (i.e. the back and forth action-reaction momentum) which is already present in a free particle. In fact, for a bound system, this zero type motion is linked to the average -idW/dx / W (W=wavefunction) which is not the same as prms(x) obtained from -1/2m d/dx dW/dx /W.
Abstract A system of 4 point masses (without spin) is considered. The potential energy is assumed to be a function of the distances between the particles only. It is possible to express the kinetic energy relative to the center of mass as the kinetic energy of 3 "reduced" masses. The different possibilities for that procedure are discussed for systems with an arbitrary number of particles. The positions of the 3 reduced masses are described by 9 coordinates. 6 of them, the "internal" coordinates, are needed to determine the internal structure of the system, and 3, the "external" coordinates, for its orientation in space. The classical Hamiltonian in such a set of coordinates is derived. In a first step the Hamiltonian is expressed in terms of linear and angular momenta. Then these quantities are determined as functions of the coordinates and the conjugate momenta by infinitesimal trans-formations. From the classical Hamiltonian we get the quantum mechanical one by the well-known translation rule. It can be expressed in terms of the linear and angular momentum operators quite analogous to the classical case. For the solution of the time-independent Schrödinger equation the rotational symmetry of the system is used. The wave function is expanded in terms of the eigenfunctions of the total angular momentum. This leads to a system of coupled differential equations for the expansion-coefficients which depend on the 6 internal coordinates only. The further symmetries of the system (reflection and particle exchange) appear in symmetry properties of these functions. Differences and similarities in the solution theory for the 4-body-problem with respect to the 3-body-problem are discussed.