Tunneling: Effective Transmission and Core Transport

01What must be compared on the two sides of a barrier

When a localized matter structure approaches a barrier, ASD must address both how its accompanying field extends across the region and how the material core responds together with that field. Effective envelope theory can already answer the first question. Given an incident field and boundary conditions, reflection appears on the left and transmission can appear on the right. The relevant comparison concerns transported flux, not merely whether the amplitude at one point is nonzero.

This page uses the source-free, linear, isotropic channel of the effective wave equation. $m_0$ is the effective mass after coefficient matching, $E$ the envelope energy after the rest carrier has been removed, and $V(x)$ the corresponding effective external potential. Use a stationary form

$\psi(x)$ describes the spatial profile. The barrier equals $V_0$ for $0<x<a$ and zero outside, with width $a$. Both exterior regions have equal propagation parameters; the potential is real and nonabsorbing. The low-energy envelope approximation requires both $E$ and $V_0$ to remain well below $m_0c^2$. The following solution describes transmission within that boundary model. Transport of the nonlinear core is treated separately in the final section.

02Three solutions in the propagation and barrier regions

Substitution of the stationary form into the effective envelope equation turns the time derivative into an energy eigenvalue. For $0<E<V_0$, the exterior solutions oscillate in space, while those within the barrier vary exponentially:

Here $k$ is the exterior propagation wave number and $\kappa$ the inverse decay length inside the barrier. $\psi''$ denotes the second derivative with respect to $x$; it describes how curvature balances the difference between potential and effective energy. With incidence from the left and no additional incident wave from the right, manuscript equation (5.3.29) gives

The incident amplitude is set to 1; $r$ and $t$ are the complex reflection and transmission amplitudes. In these scattering coefficients, $t$ is an amplitude symbol, distinguished by context from time in the stationary factor; $r$ is not the radial coordinate of the gravity module. $A$ and $B$ are two complex coefficients inside the finite barrier. Keeping only a decaying branch at the outset would discard a coefficient needed to match both interfaces.

03How boundary matching determines transmitted flux

At finite potential steps without $\delta$ or $\delta'$ interface terms, $\psi$ and $\psi'$ are continuous when the mass parameter is the same. One pair of conditions connects the incident region to the barrier, and another connects the barrier to the outgoing region:

These are the four linear relations in manuscript equations (5.3.30)–(5.3.31). The right-hand pair first gives $Ae^{\kappa a}=te^{ika}(1+ik/\kappa)/2$ and $Be^{-\kappa a}=te^{ika}(1-ik/\kappa)/2$. Substituting into the left-hand pair and eliminating $r$ yields the transmission amplitude

An amplitude must be converted to flux before defining a transmission coefficient. The conserved current in this model is

The incident current is $j_{\rm in}=\hbar k/m_0$, the reflected current travels in the opposite direction, and the transmitted current is $j_{\rm tr}=(\hbar k/m_0)|t|^2$. Equal exterior parameters therefore give $T=j_{\rm tr}/j_{\rm in}=|t|^2$ and $R_{ m refl}=|r|^2$. Taking the squared modulus of the amplitude and simplifying yields

$R_{ m refl}+T=1$ expresses flux conservation in this nonabsorbing model. A single real exponential decay component has no net flux. The complex linear combination satisfying both boundaries can nevertheless carry constant nonzero transmission through $\operatorname{Im}(AB^*)$. Evanescent amplitude and transmitted flux therefore coexist without temporary negative kinetic energy or borrowed energy. If the exterior propagation parameters differ, the corresponding group-velocity or flux-normalization ratio must be included; $|t|^2$ alone is insufficient.

04Thickness, energy and continuous limits

The exact boundary solution admits two equivalent forms at every thickness. Set the dimensionless $z=e^{-2\kappa a}$ and $D=V_0^2(1-z)^2+16E(V_0-E)z$, which has units of energy squared. Using $\sinh^2(\kappa a)=(1-z)^2/(4z)$ in equation (5.3.33) gives the final manuscript's equation (5.3.34):

The two numerators sum exactly to $D$. Here $R_{\rm refl}$ denotes reflected flux, not a matter-core radius; this is an editorial symbol distinction, not a new manuscript equation. It is an exact identity for the same stationary rectangular barrier with equal exterior propagation parameters. Only in the thick-barrier limit $\kappa a\gg1$ does it reduce asymptotically to

The exponential explains strong suppression by a thick barrier, while the prefactor retains interface matching. Vanishing barrier width must recover $T\to1$. Near $E=V_0$, the thick-barrier approximation is no longer the appropriate expression.

For $E>V_0$, the interior solution becomes oscillatory. Defining $q=\sqrt{2m_0(E-V_0)}/\hbar$ and continuing through $\kappa\to iq$ gives the above-barrier result. The threshold follows from the continuous limit on either side:

Perfect transmission at $qa=n\pi$ comes from phase matching between the interfaces of one finite barrier. It is a different boundary mechanism from resonant tunneling through a quasibound state inside a double barrier. The threshold expression checks that the decaying and oscillatory regimes join continuously.

The interactive curve uses this ideal rectangular-barrier model to show how its parameters affect envelope transmission. It is neither a core trajectory nor a fit to a particular material or scanning-tunneling-microscope dataset.

05Slowly varying barriers and the WKB approximation

A real barrier need not be rectangular, and the channel's directional response may vary with position. Let the positive dimensionless coefficient $b(x)$ describe the effective response along the propagation direction. The equation must retain the divergence form of manuscript equation (5.3.37):

This form makes $\psi$ and $b\psi'$ continuous at finite interfaces, instead of importing the constant-coefficient derivative condition. In the forbidden region take $\psi=A(x)\exp[\pm\int^x\kappa(x')\,dx']$. The WKB approximation separates the local decay scale in the exponent from a slowly varying amplitude: the highest order determines $\kappa$, and the next order determines amplitude transport.

The label $T_{\rm WKB}$ explicitly denotes the first-order WKB estimate, and $C_0$ is the integration constant of the amplitude transport equation. The equals sign defines this selected approximation; it does not turn $T_{\rm WKB}$ into the exact transmission of an arbitrary variable barrier. The exponent alone does not determine the interface prefactor.

Specifically, the next-order equation is $2b\kappa A'+(b'\kappa+b\kappa')A=0$, which integrates to the amplitude factor shown. $x_1$ and $x_2$ are turning points bounding the forbidden region, and $\mathcal P$ is the prefactor determined by boundary and turning-point matching. Here $A(x)$ is a varying amplitude, not the constant coefficient used inside the rectangular barrier.

The approximation requires $|\kappa'|/\kappa^2\ll1$ and $|b'|/(b\kappa)\ll1$. These conditions fail near turning points, where local connection solutions are needed. A single real exponential branch still has no net flux. The transmission exponent is sensitive to $V$ and $b$, but additional directional response must first be calculated from the same ASD constitutive model. Ordinary crystal anisotropy, surface potentials and thickness errors also affect transmission, so direction dependence alone is not a distinct ASD prediction.

06Connecting envelope transmission to the matter core

In the ASD picture, an evanescent accompanying field establishes a phase connection across the barrier region. Whether a localized topological core can form a transmitted state through continuous deformation, energy exchange and transfer of topological density is a nonlinear dynamics problem. The effective transmission coefficient is a result that the total scattering statistics should recover. It does not by itself establish a core trajectory across the barrier.

Core reflection, transmission and detection must enter an energy account that includes field, barrier and source back-action. Conserved topological density must also obey its corresponding continuity relation. OP-QM-005: core–field coupling in tunneling addresses that connection, while individual-event statistics connect to the detector response.

Scanning tunneling microscopy provides a practical application of transmission across a vacuum gap, but its current also depends on electrode state densities, occupation and surface barriers. Agreement in mathematical form with a known transmission relation is not independent experimental confirmation of ASD.

Phase delay, dwell time, packet-peak position and detector click time have different definitions. Spectral selection can shift a transmitted peak without advancing a new information front. Sudden preparation or rapid boundary changes require the original second-order propagation equation and actual detector response; a stationary low-energy envelope cannot describe those transient processes on its own.

The same effective equation also gives bound spectra created by external boundaries. Take zero potential for $0<x<L$, ideal infinite barriers outside, and $\psi(0)=\psi(L)=0$. The oscillating solution then requires $kL=n\pi$. Chapter 5, equation (5.3.40), gives

Here $L>0$ is the well width, distinct from the earlier barrier width $a$. The normalization $\int_0^L|\psi_n|^2dx=1$ gives $\psi_n$ units of $\mathrm{m^{-1/2}}$, while $E_n$ has energy units. These results still use this page's low-energy envelope approximation, so the retained levels require $E_n\ll m_0c^2$. Their discreteness comes from the external boundary. It is not the closure classification of a matter core and does not determine electron or proton species or internal spectra.