Abstract Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.
Abstract Having examined free particles and particles that are confined in space by a potential energy term, we now consider the impact of a disturbance in the flat energy landscape for a free particle. By disturbance we means some kind of fixed “obstacle” which is either a positive (repelling) or negative (attractive) potential. We are interested in determining the impact on the free particle. Continuing to work mostly in one dimension, the particle described by a plane wave corresponding to momentum moving in the positive direction (a positive k−vector in the x−direction), we study elastic scattering. In one dimension, this means that we determine the probability that the particle is transmitted (continuing in the forward direction) or reflected (now moving in the backward direction.) We will also determine the nature of the solution inside the potential and in the case that the potential energy maximum is greater than the kinetic energy of the particle, we will show that the particle tunnels through the barrier. Interestingly, when we have two barriers, we can find conditions where the probability that the particle is transmitted is unity. This is the result of resonance, a feature of the wave-like nature of the particle’s wave function.
In this note, we try to understand why tunneling should be part of a quantum mechanical treatment of a one-dimensional bound state problem. We suggest that the ideas behind tunneling begin with the directional probability exp(ipx) used to describe conservation of momentum in Newtonian 2-body scattering. Given the notion of a wavelength in space, there must be a probability scheme in space which “physically” creates the wavelength, as we have mentioned before. This means one cannot have uniform probability in x and so some directional probability (linked to exp(ipx)) must be removed from one x region and moved to another. Thus, one cannot point to a particular x1 point, for which cos(px1)=0 and say that the particle has disappeared. Probability is simply shifted from one point to another and this leads to the idea of positive and negative values linked to this probability as one sees from the negative and positive values of sin(px) and cos(px). As mentioned in Part 4, one may create various averages of functions of p including the function 1 which is then linked to particle presence (i.e. spatial density although it does not directly equal this). Given the periodicity of exp(ipx), W(x) =Sum over a(p)exp(ipx) usually tends to be periodic as well, even if one considers a(p)=a(-p) and a real W(x). It is possible, however, to have smooth valued functions for {Sum over p f(p)exp(ipx) a(p)} / W(x). This might lead one to think that one has a classical scenario with no wavelength effects present, but we argue here that this is not the case. Furthermore, we argue that this type of argument seems to explain why tunneling is linked with a one-dimensional bound state solution. We noted in Part 4, that one may solve an equation like the time-independent Schrodinger one to obtain a linear combination solution of directional probabilities exp(ipx)s, i.e. W(x). In Part 4, we were concerned with what to do with W(x) and argued for imposing boundary conditions- constraints -continuity on W(x). Here we consider the case of classical turning point, for which KEave(x)=0. For a classical particle, delta x being tiny is huge compared with the wavelength of exp(ipx). Thus, p→0 leads to wavelength = hbar/p, but for a tiny p, a large wavelength is not a problem as it is still not greater than delta x. Thus, one may have a p (momentum) become very small at a classical turning point (and even argue that it is 0.) In the quantum region, one cannot do this because a delta x near the classical turning point x1 is very small compared to hbar/p for p close to zero. One cannot have a single or a few small value p’s, but must rather have sizable p values. This means that the particle is in one of these p states or another and has sizable impulse and kinetic energy pp/2m. As a result, the potential barrier is not a problem- it does not hold the particle back. What is zero is the average value of KE, i..e KEave(x) = { Sum over p a(p)pp/2m exp(ipx) } / W(x). As we have argued above, for a(p)=a(-p), exp(ipx)) becomes cos(px) and even with positive a(p), one has negative and positive directional probability values which allows sizable p, pp/2m’s to have an average of zero. As noted above, however, an average kinetic energy of 0 at a single x1 does not mean that it is zero at x1+delta. Just as with cos(px), removing values from some x region means that they appear elsewhere. KEave(x)=0 is not a constraint for x outside the classical turning point region. The quantum average KEave(x)=0 describes various probabilities for sizable momenta and pp/2m and so must deal with a particle moving in and out of the so-called bound region. The question becomes: How does one make this appear like a bound state problem, i.e. one localized in space? This occurs by imposing the constraint W(x)=0 as x→ +/- infinite as noted in Part 4. It is this constraint which shows that even though one has sizable p values at the classical turning point (despite KEave(x)=0 but W(x) not=0), one must have the average W(x) drop to 0 as x→ +/- infinite. As a result, given the nature of the quantum calculation, it is physically expected that tunneling should occur for a one-dimensional bound problem because there is no constraint on a single p or a few p values becoming very small (or 0) at the classical turning point. Thus, they should keep moving past the classical turning point. This is the physical event which should occur (i.e. tunneling).
The Schrödinger equation is reformulated as a universal continuity equation, which connects between changes in the particles probability density distribution to probability current densities (fluxes). The formulation of particle conservation in terms of stationary fluxes enables one to associate stationary wave functions also to open quantum systems characterized by stationary particle currents. These functions are (improper) solutions of the stationary Schrödinger equation, obtained under scattering boundary conditions. These boundary conditions can be fulfilled for any positive asymptotic kinetic energy, hence, the energy spectrum of the scattering states is continuous. We demonstrate flux calculations in scattering through a one-dimensional potential energy well/barrier, focusing on transmission and reflection probabilities. Nonclassical phenomena such as transmission at energies below a potential energy barrier (quantum tunneling), or reflections at energies above a potential energy well are analyzed. The phenomenon of full transmission through a double barrier structure (resonant tunneling) is introduced in the context of nanoscale transport.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
We have argued in previous notes that it is special relativity which gives rise to the free particle probability exp(-iEt+ipx). This probability is directly associated with physical behaviour because the somewhat random variables are x and t and while p and E are exact. One has probability in t and x for the interaction variables E,p. This suggests a kind of resonance feature, we argue.This idea of resonance seems to carry over into a single bound state and helps to explain why such a quantum system is associated with more than simply a statistical calculation. Using exp(-ipx) as a probability for a time-independent problem (i.e considering impulse hits in x only), one may mathematically express the average kinetic energy as: {Sum over p a(p)pp/2m exp(ipx) }/W(x), where W(x)=Sum over p a(p)exp(ipx). One may then use: KEave(x) + V(x) = En. As an average, one has different p values creating KEave(x), with only the average KE plus V(x) equalling En. The notion of conservation of energy, however, must hold and this begs the question: How is the energy for each p obtained such that energy is conserved. We suggest that the only source is V(x) and so if one moves from a p1 to a p2, at some x, then if p2>p1, the extra kinetic energy must be borrowed from the potential energy. This is not simply an average calculation, energy conservation must be imposed. This suggests that a physical resonance is established, which allows for high p values using borrowed V(x) energy. As a result, a quantum particle is not stopped at a classical turning point, but it receives its high p and passes this point using borrowed energy. The energy belongs to the potential V(x) and the particle is not free to move off with the borrowed energy. We suggest that tunneling occurs with borrowed potential energy and that the simplest condition W(x)->infinite shows that a particle which escapes the classical turning point cannot go too far with its borrowed energy. The catch is that if another potential is placed near the classical turning point of the first, it can lend energy to the particle so that the borrowed energy from the first potential is returned and the particle is pulled into the second system, as seen in experiments. Thus, borrowed potential which is part of a resonance mechanism allows a quantum particle to bypass the classical turning point restriction and one can swap borrowed energy from one system to another to actually have a particle tunnel out of one bound system into another system.
Abstract This paper examines tunnelling through potential well protected by double barriers as well as the motion of a wave packet encountering this type of potential. Various options for the left and right barriers’ penetration factors are taken into consideration. When one of the penetration factors has limiting value of 0.5, the transmission coefficients turns out to be independent of incoming energy of the particle and the width of potential well. When both penetration factors have a limiting value of 0.5, a scenario similar to resonance tunneling is achieved. Furthermore, by doing the wave packet analysis, we found that the width and height of potential barriers have a negligible impact on the probability of finding the particle inside the potential well in the limit of large penetration factors.