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Bound states in one dimension

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

Abstract In this and in the following chapter, it reviews an important part of undergraduate material, namely solving for bound states and for scattering in one-dimension for the case where the potential is constructed from different constant values in various regions pieced together.
DOI
10.1093/acprof:oso/9780198749349.003.0008

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In (1) a particle with rest mass and momentum magnitude p1 moves from a region with a constant potential V1 into a region with a constant potential V2 with the interface being along the x axis. Overall energy is conserved and it is noted that conservation of the x-component is analogous to Snell’s law. ((1) goes on to use a variational approach to argue conservation.) In this note we begin with Newton’s idea that momentum changes in the direction of a force and that this force is -dV/dx where V(x) is a conservative potential. For constant V then, forces only arise from discontinuities in potential. We note that for a two dimensional problem (x,y) choosing a starting point of (x=0, y=Y) and changing the incident angle (measured from they axis, leads to different x values. Thus x is a piece of information in the two dimensional problem. In one dimension (along the x axis), however, momentum is conserved because there is no discontinuity in potential. In this one dimension there is x invariance and x should not be a piece of information. The question becomes: Is there a function in two dimensions that when applied to the two regions of this problem leads to x information disappearing? We consider an ending point (L-x, -Y) where L and Y are constant and note immediately that: x+ L-x=L i.e. a constant so that x disappears. Thus if x is multiplied by a constant (namely the x component of p) then the function A1=(px) x + (py)y meets the requirement i.e. A1+A2 loses all information about x because of the conservation of (px). As a result one sees the association of p and x which in Newton’s theory would normally not be combined. Furthermore if one uses the constraint that L=1/(px) (and drops the y=Y,-Y requirement), then (px)L = 1 which holds for all px values. This is linked to the idea of having one constant (namely 1 here) representing all constant motion as discussed in (2). In this way one may see a characteristic length proportional to 1/px emerge so that (px)(1/px) is the same for all constant motion. This characteristic length is the wavelength which appears in experiments. A second issue we wish to consider is the quadratic form of pp in both nonrelativistic E= pp/2m + V(x) and relativistic (E-V)(E-V) = pp + momo (c=1) energy conservation equations. We use the refraction type problem (two constant potentials) together with conservation of (px) = p sin(theta). We use Newton’s idea that force changes momentum in the direction of -dV/dx, (py) and hence p magnitude must change, but (px) must be a constant. Thus one may write: p1p1 - p2p2 = (p1yp1y-p2yp2y) = C1 ((A)). We have used p1x=p2x, but ((A)) leads again to this result so the general result p1p1=C2 should hold. A similar argument holds relativistically. Thus without integrating one is led to the idea of a quadratic p (i.e. p magnitude p magnitude) as part of a constant equation (becuase the potential is constant). This we argue is conservation of energy.
Abstract In preceding chapters, while deriving the scaling behaviour of correlation functions, we have always kept only the leading term in the critical region. We examine now the different corrections to the leading behaviour. For instance, when we have solved the renormalizaton group (RG) equations, so far, we have neglected the small deviation of the effective coupling constant from its fixed-point value. Moreover, to establish RG equations, we have neglected corrections subleading by powers of the cut-off, and effects of other couplings of higher canonical dimensions. Subleading terms related to the value of the effective coupling constant which give the leading corrections, at least near four dimensions, can easily be derived from the solutions of the renormalization group (RG) equations and are discussed first. The situations below and at four dimensions (the upper-critical dimension) have to be examined separately. The second type of corrections involves additional considerations and is examined in the second part of the chapter. The last section is devoted to one physics application, provided by systems with strong dipolar forces, which have 3 as upper-critical dimension.