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.
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.
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 consider a quantum particle with energy E incident upon a one-dimensional potential. We show that the probabilities of transmission and reflection are the same for incidence upon a general potential from either side (from ‘the left’ or ‘the right’). This equality holds true for any potential which goes to constant values as x → ± ∞ and is finite for all x . We present a remarkably simple proof that the probabilities are equal. The simplicity of our proof is the most important pedagogical result of our paper, and will be easily understood by undergraduate students in second to fourth year. We discuss several cases, including the step potential and the potential barrier.
We investigate the transmission and reflection of a quantum particle incident upon a step potential increase, a step potential decrease, a square well, and a square barrier, all well studied in undergraduate quantum mechanics. We are especially interested in the extreme where the change in the potential is arbitrarily large, but with the difference in the energy of the particle and the potential held fixed, if possible. We obtain the probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation. In the time-dependent case, we have the particle initially in a Gaussian wave packet or a similar localized state. We obtain results that fall into three categories: results that are not surprising, results where time-dependent and time-independent agree surprisingly well, and results that are very different. We discuss the unexpected results. Our work may be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.
We argue that continuity and discontinuity in special relativity play an important role in the probabilistic wave properties of quantum mechanics. Special relativity is involved with transformations from a rest frame to a frame moving with a constant -v, for example . The discontinuity appears in the form of a change from v=0 to v, while the continuity arises from Lorentz invariants which indicate one has the same physical state whether it is viewed from a frame with one velocity or another. This invariance applied to E, p (momentum) as a modulus forces a minus sign between the two, in order to allow for linkage with E=mo (c=1) and p=0, i.e. -EE + pp = -momo (c=1). Another invariant, however, is A= -Et’ + px’, and in this case the minus sign and the discontinuity lead to a new x’,t’ (in comparison with the rest frame for a particle with rest mass mo), but also to t’ -> t’+ b/E and x->x’ +b/p leaving A unchanged (b=constant). The discontinuity in v, which leads to a jump in x’,t’ (as seen in the moving frame) also manifests itself in a time-length interval which ultimately leads to exp(-iEt+ px), we argue. In other words, classically one considers constant velocity as smooth continuous motion, but the introduction of two frames, v=0 and -v, leads to a discontinuity which leads to “chunkiness” in the form of a frequency and wavelength. If these cannot be perceived classically (as is often the case), then one considers only smoothness. Relative differences in time or length or in frequency or wavelength, however, can appear in nature for length scales of the order of the frequency/length scale. In the case of n-slit interference-diffraction, a relative difference leads to a “jump” (so to speak) which manifests the wavelength. In particular, a photon moves along the y direction with a phase px (considering a steady state picture). It reaches a two-slit system along the x-axis and there is probability for it to interact with both slits. If one considers two parallel rays from each slit moving towards a point on a screen very far away (as is done traditionally), there is again smooth motion with a rotated p vector (same magnitude), but for a point on the screen to the left of the center, there is a jump in the length associated with the right slit. In other words, there is a jump similar to the v=0, -v jump. This leads to a relative difference in length associated with both rays (linked to an OR probability situation using exp(ip dot r). This relative difference in length displays the phase difference initiated by special relativity. In the case of a bound state, different impulse hits by a potential create different momenta which have different phases exp(ipx) because of special relativity and one has interference. The same phase A= -Et + p dot r, which is linked with a chunkiness due to a discontinuity, is also associated with continuity because A is a Lorentz invariant. Thus A=constant may be used to solve for refraction/reflection problems with or without moving mirrors/interfaces with different indices of refraction. Continuity appears further in a 1-dimensional reflection/refraction problem when applied to the exp(ipx), i.e. Aexp(ipx) + Bexp(-ipx) = C exp(ip2x) where E=pc = p2 c2, p2=n2p and c2=c/n2 with n1=1, together with a continuity equation for the first derivative. Thus, both continuity and discontinuity (or jumps), which follow from special relativity, seem to play an important role in quantum mechanics.
Quantum tunneling is a phenomenon in quantum mechanics. In this phenomenon, Quantum particle can go through potential barrier that potential energy is higher than total energy of quantum particle. Physicists are interested in transmitted and reflected value of Quantum particle that is transmission and reflection probabilities. Then, Schrödinger equation is used to find transmission and reflection probabilities by directly solving. However, some potential barriers are too complex to solve directly. Then, we use another method to solve more complex problem. That method is called "quasilinearization''. Quasilinearization method (QLM) is a numerical method to solve nonlinear differential equations. In this thesis, we compare the quasilinearization method with the well-known WKB approximation method. The results show that the transmission and reflection probabilities calculated by the QLM and WKB approximation have the same trend as the transmission and reflection probabilities calculated from the exact solution.
We investigate the transmission and reflection of a quantum particle incident upon a step potential decrease and a square well. The probabilities of transmission and reflection using the time-independent Schrödinger equation and also the time-dependent Schrödinger equation are in excellent agreement. We explain why the probabilities agree so well. In doing so, we make use of an exact analytical expression for the square well for time-dependent transmission and reflection, which reveals additional interesting and unexpected results. One such result is that transmission of a wave packet can occur with the probability of transmission depending weakly on the initial spread of the packet. The explanations and the additional results will be of interest to instructors of and students in upper year undergraduate quantum mechanics courses.