In this paper, we examine transmission through one-dimensional potential barriers that are piecewise constant. The transfer matrix approach is adopted, and a new formula is derived for multiplying long matrix sequences that not only leads to an elegant representation of the wave function but also results in much faster computation than earlier methods. The proposed method covers a broad spectrum of potentials, of which multi-barrier systems are special cases. The procedure is illustrated with a finite lattice of nonuniform rectangular barriers—non-uniformity being a novel feature, as the uniform case has been solved exactly by Griffiths and Steinke. For the nonuniform multi-barrier problem, the intervening wells strongly influence the transmission probability. Surprisingly, we find that the wells act “individually,” i.e., their influence is a function only of their width and is independent of their exact locations in a multi-barrier system. This finding leads to an observation that we have termed the “alias effect.” The exact solutions are supplemented with asymptotic formulas.

## References

*κ–τ*plane.

There is no loss of generality. One might as well choose right incidence. Since Eq. (7) is a forward difference equation, it is amicable to left incidence calculations. Because the transfer matrix is nonsingular, one can readily invert it on to the other side, followed by a re-labeling of the index in Eq. (7) (i.e., *j* → *j* – 1). This can then be used in an analogous fashion to handle the case of right incidence and *A*_{1} must be set to zero (instead of *B _{N}*

_{+1}). Later, the reflection and transmission coefficients must be redefined as $T=|B1/BN+1|2$ and $\mathbb{R}=|AN+1/BN+1|2$.

This result can be derived explicitly from the continuity equation. A particularly neat (and equivalent) way of deriving the same is from the conservation of the average momentum $\u27e8p\u27e9$ associated with the wave function ψ (Ref. 9). Note: The momentum associated with the wave function(s) $A\xb1e\xb1i\kappa x$ is $\xb1\u210f\kappa $, with A_{±} being the probability amplitude of finding the particle with $p=\xb1\u210f\kappa $. Hence the average momentum associated with ψ_{j} [Eq. (2)] is $\u27e8p\u27e9=(+\u210f\kappa j)AjAj\u2217+(\u2212\u210f\kappa j)BjBj\u2217=\u210f\kappa j(|Aj|2\u2212|Bj|2)=const\xb7\kappa 1=\kappa N+1$. Thus $|A1|2\u2212|B1|2=|AN+1|2\u2212|BN+1|2$. Since $BN+1=0,\u2009|AN+1|2/|A1|2+|B1|2/|A1|2=1$ or $T+\mathbb{R}=1$.

All the plots in this paper have been generated from programs written in MATLAB version 7.12.0.635 (R2011a).

Only the uniform MBP admits well-defined bands (a band being a local group of *m* – 1 resonant peaks). R. Gilmore in his book (Ref. 9) terms these bands as *N tuplets*, *N* = *m* – 1. The *β* of Eq. (28) essentially gives the total *number of bands* of a uniform MBP (below *V*_{0}). For an asymmetric MBP, the band structure is completely lost and is replaced by a stray collection of resonant peaks. The parameter *α* (defined in Eq. (29)) gives the *total number of resonant peaks* for an asymmetric MBP. Thus for a uniform MBP, *α* = (*m* – 1)*β*.

By classical scattering we specifically mean $T=0$ for $\kappa <V0$ and $T=1$ for $\kappa >V0$.

We choose *δ* = 1 (instead of 0.5) for the colored plots, to bring out the correspondence better. If one takes *δ* = 0.5 the curves would overlap. A higher value of delta lowers the value of $T$ and has no effect on the positions of the peaks. Hence the correspondence between the resonant peaks of the asymmetric MBP and the uniform MBPs is still preserved.

Although we have (*m* – 1)! distinct permutation-equivalent MBPs, some of them are not aliased solutions. For instance, in a 4BP with three distinct well widths *τ*_{1}, *τ*_{2}, and *τ*_{3}, consider the permutations *τ*_{3}-*τ*_{1}-*τ*_{2} and *τ*_{2}-*τ*_{1}-*τ*_{3}. They would result in the same transmission coefficient, since a particle incident from left on the former barrier configuration is equivalent to a particle incident from right on the latter. And the transmission coefficient of the barrier must not depend on the direction from which the particle approaches. In fact, this feature is inherently embedded in the structure of the transfer matrices and can be rigorously proven.

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