Abstract We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a . We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a , not for vanishing a . The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.
Abstract We have often found among many of our students and colleagues the common idea that the mathematical expression for a physical quantity that is essentially of quantum nature must contain a dependence on ℏ. Conversely, a phenomenon described by classical physics should contain no explicit reference to ℏ. However, the problem of a particle encountering a discontinuous potential step, which is one of the simplest examples in quantum mechanics, contradicts this common thought: even when the particle carries enough kinetic energy to go across the step, the resulting expression for the reflection probability is non-zero—a purely quantum phenomenon—and yet it contains no reference to ℏ. We show that the absence of ℏ in this purely quantum expression is due to the idealised limit in which the potential rises sharply at a single position, thus losing any reference to a length dimension in the problem. To address the correct classical limit of the phenomenon we first regularise the discontinuity in the potential by taking a continuous linearly rising function over a distance a . We show that the correct classical limit, which is when no reflection occurs, is obtained for large values of a , not for vanishing a . The latter actually corresponds to a purely quantum case, albeit in a limiting case of a discontinuously abrupt change in the potential at a single point in space.
We have argued in previous notes that quantum formalism emerges from the one dimensional photon reflection-refraction scenario which is readily observable in the classical world. What about the situation for a particle with rest mass? In previous notes, we have considered the case of a constant E,p and V1 which changes at xo to E,p2,V2 (all again constants with E remaining unchanged). This problem is then associated with an incident and reflected particle for x<xo and a particle with p2 for x>-xo. In a classical world, however, how does one obtain reflection and refraction together? If there is a strong wall, a particle may reflect with -p and the same kinetic energyE, but it does not pass through the wall. If there is a ramp at xo leading to a horizontal floor a little higher than that for x<xo, then one has a conserved total E, but a new p2 and V2. The ramp, however, is not a sudden change at xo, but occurs over a range of x. As a result, it seems the quantum scenario, rather than leading to the classical case in a high energy limit (as in bound states), is associated with fluctuations and non-sharp boundaries. It is fine to have a sharp wall at x=xo, but in quantum mechanics, one has atoms with electrons moving in space according to probability distributions. Furthermore, there is movement due to temperature. One does not have a sharp edge or ramp scenario alone, but fluctuations which could lead to both. This suggests stochasticity at a point of discontinuity, but also stochasticity in the regions V1 and V2. In other words, V1, V2 and the sharp discontinuity points are all approximations (averages). If one accepts the fluctuations at xo (ramp and sharp wall), then classically one might impose a pressure balance equation at an idealized xo, i.e.: AAp -BBp=p2CC where AA represents a positive number etc. AA represents the incident particle, BB the reflected and CC the particle which moves up the ramp. Nonrelativistically and in the photon case, the pressure balance equation is also equivalent to a conservation of particle number or energy equation. Thus one has two unknowns (if A is known), i.e. B and C, and one equation. Here, however, we wish to stress a different, but related issue. The pressure balance equation allows for multiple solutions because it only exists at xo. For example, BB=0 is a solution as is BB=.5, CC=.5. If p2 is almost equivalent to p, one might expect that BB=0. In other words, there is a specific solution which goes beyond pressure balance, i.e. here seems to be a p (momentum) biased solution. This forces one to go beyond classical mechanical ideas, even if one introduces a fluctuating wall-ramp scenario at xo. The fluctuating scenario suggests fluctuations in V1 and V2 as well, suggesting one may have some kind of dynamical equilibrium in these regions. Would this be enough to create a momentum bias, linked say to a continuity of a function of p,x and its first derivative at xo? This extra momentum bias seems to be the extra feature which emerges in a quantum treatment of the problem, which only occurs because one introduces fluctuations at the xo point. We suggest that a dynamical equilibrium approach solves the problem if one maximizes Shannon’s entropy subject to a the constraint of a constant average classical action written as: -P(t) iEt + ipx P(x) with a Lagrange multiplier of 1. “i” is introduced to a have a periodic/dynamic probability, i.e. one that does not grow. Thus we argue that quantum mechanics for a particle with rest mass emerges because a classical pressure balance equation for a fluctuating wall-ramp picture at a V1-V2 junction allows for multiple solutions. Ifr V1 is almost equal to V2, one has a biased or selective solution which should force BB to be 0. This may be done through continuity, but there must be a physical basis for the function used and we suggest that this is maximization of Shannon’s entropy subject to an average classical action introduced to make the probability periodic due to the dynamical nature of the problem. Thus there is not just stochasticity at the discontinuity point xo, but throughout x.
In this work, the motion of a particle in one dimension under the influence of the linearly downward potential well is studied within the context of the non-relativistic quantum mechanics. The attention is paid on the paradoxical phenomenon of the reflection of a particle that is in contrast between classical and quantum physics. Classically, the reflection effect occurs only at a potential barrier. To demonstrate such counter-intuitive phenomenon, the Schrödinger equation is solved to obtain the reflection coefficient in the scattering state by considering an incident particle that is represented by a monochromatic plane wave having an energy E > 0, propagates freely from left to right, pass through the potential well. The continuity conditions at boundaries give the desired result that is expressed in terms of the Airy functions which depends on the incident energy E, the strength jV 0 j and the range L of the well. The value of the reflection coefficient R lies in the interval 0 < R < 1, and its behavior is the decreasing function with respect to the range L.
Newtonian mechanics often employs a potential V(x) such that .5m v(x)v(x) + V(x) =E ((1)) where v(x) is velocity. Mathematically V(x) is defined at each x, but in reality there should be a tiny cutoff i.e. finite dx. Newtonian mechanics makes use of the idea of small dx lengths which tend to 0 mathematically. To first order v(x) is constant within a dx range, but the overall mathematics is continuous instead of discrete. Thus there is a continuous equation ((1)) for conservation of energy. A photon may be described by an eclectic field varying as cos(-Et+px). In terms of wavelengths exp(ipx) is often used as a mathematical convenience and noncontinous behaviour is not an issue. For example in refraction, momentum = hbar/wavelength abruptly changes and one straight line path is joined to another with a different slope. exp(ipx) is associated with repeatable wavelength patterns with a single wavelength representing a spatial distribution/ (kind of probability) region. A particle with rest mass may also be described by exp(ipx) which suggests a characteristic length hbar/p. Within this length (i.e at each x point which exist within hbar/p) the particle has p and pp/2m. Thus for a square well potential there is conservation of energy at each x point even though there is a wavelength in the problem. This is fine for a constant momentum, but what about a potential V(x) which implies p must change at each dx region? exp(ipx) is an eigenfunction of -id/dx which is linked to spatial invariance. It manifests itself physically by interference (e.g. two slits) which again involves non-smooth motion (i.e. one compares rays with completely different slopes). The characteristic length hbar/p is fixed and associated with a single p, yet momentum is supposed to change from one x point to the next. hbar/p does not fit within a single dx and this is a problem. Quantum mechanics remedies this issue by allowing for a superposition of exp(ipx)s i.e. W(x) = wavefunction = Sum over p a(p) exp(ipx) such that energy conservation is forced at each x. Thus one combines the idea of wavelength (which is physical) with energy conservation which is also physical. A consequence of this, however, is that -id/W/dx / W leads to a strange average momentum which is not the square root of -1/2m d/dx dW/dx/ W as well as discrete energies. A mathematical benefit is that one has a continuous differential equation, namely the time-independent Schrodinger equation. The question we ask is: What happens in the so-called classical limit? In the nonrelativstic case this leads to Newtonian mechanics in which there is no wavelength. How does the wavelength disappear so to speak? We suggest (as has been suggested often in the literature and textbooks) that the wavelengths in W(x) may become extremely tiny so as to not be detected easily in classical length scales. Mathematically the time-independent Schrodinger equation holds and there is a smooth W(x) at all points, but we suggest that this smoothness may not be necessary because it is not found in refracting light. If the wavelength is tiny compared to small dx regions in the classical limit, then many wavelengths of a given p fit into dx. Thus the idea of spatial invariance approximately holds within dx and there is no issue of the wavelength spilling out of the dx region as it does when it is comparable to the size of the system as in quantum bound problems. Thus within dx a single p with many (or at least several) wavelengths holds and there is a single p. The kinetic energy is pp/2m and the potential V(x) where x is in the center of dx. One may think of p changing abruptly (nonsmoothly) at the boundary of one dx with another such that conservation of energy holds. Thus one has p wavelengths which do not spill out of dx and give an approximate sense of spatial invariance together with energy conservation. The only loss is that of a smooth wavefunction, but it is not clear that such smoothness is a requirement. V(x) is physically associated with a limiting dx length (at which the concept of V(x) does not hold, but rather there are stochastic hits) and if one expands this length i.e. chooses a much bigger dx in keeping with classical scales, it seems that there should not be a problem. It as if V(x) delivers an impulse hit at each dx junction (like a series of different media in refraction).. This approach leads to the decoherence of wavelengths in the classical limit.
Abstract We consider the probabilities of transmission p T and reflection p R of a quantum particle incident upon potential drops with significant qualitative differences. We look at a smooth potential drop for which the potential and the derivative of the potential are continuous. We also consider a potential drop which is continuous but has discontinuities in its derivative. The two cases give markedly different results for the limiting values of p T and p R with increasing values of the total potential drop V 0 . We explore the difference in the results using potentials that can be adjusted to go from continuous to discontinuous first derivatives. We also investigate potentials with more severe discontinuities in the derivative of the potential.
This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schrödinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau. PACS: 03.65.-w, 03.65.Nk, 01.30.Rr. Key words: Schrödinger equation; potential step; confining potential; reflection and transmission coefficients. 1
We discuss a phenomenon of elementary quantum mechanics that is counterintuitive, non-classical, and apparently not widely known: the reflection of a particle at a downward potential step. In contrast, classically, particles are reflected only at upward steps. The conditions for this effect are that the wavelength is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. The phenomenon is suggested by non-normalizable solutions to the time-independent Schrödinger equation. We present numerical and mathematical evidence that it is also predicted by the time-dependent Schrödinger equation. The paradoxical reflection effect suggests and we confirm mathematically that a particle can be trapped for a long time (though not indefinitely) in a region surrounded by downward potential steps, that is, on a plateau.