Cylindrical barrier separation

Why the barrier argument belongs in V2

A barrier independent of the additional spatial coordinates cannot be evaded merely by moving through those coordinates. This is a useful consequence of the bound-within/free-without premise. The result concerns cylindrical continuation of the interaction potential. It does not require a uniform, normalizable particle wavefunction.

Use a product region with coordinates $(x,y)$, where $x$ is the supported crossing coordinate and $y$ denotes the additional ordinary macroscopic spatial directions within the containing sector. These are coordinates of one physical space. Their grouping is relative to the barrier, not an inherent division of the electron's six dimensions.

Suppose

$$V(x,y)=V_0(x).$$

If the barrier occupies the interval $I$ in $x$, its support in this region is $I\times F$, where $F$ is the full available space of $y$. A continuous path from one side of $I$ to the other must pass through $I$ in its $x$ coordinate, whatever its $y$ coordinate does. There is no geometric detour in $F$. This does not prohibit quantum transmission through the barrier, or a path around an actual edge in the supported coordinates of a finite physical barrier.

Exact separation under a specified Hamiltonian

For a flat scalar diagnostic with no mixed kinetic or interaction terms, take

$$H=-\frac{\hbar^2}{2m}(\partial_x^2+\Delta_y)+V_0(x) =H_x+H_y.$$

Fourier transformation in $y$ gives, for each transverse momentum $k$,

$$\Psi(x,y)=\phi_k(x)e^{ik\cdot y},\qquad H_x\phi_k= \left(E-\frac{\hbar^2|k|^2}{2m}\right)\phi_k.$$

The transmission problem is therefore the ordinary longitudinal one at energy $E_x=E-\hbar^2|k|^2/(2m)$. At fixed $E_x$, adding the separated coordinates does not change its transmission coefficient. At fixed total $E$, changing the transverse momentum changes $E_x$, so transmission need not stay the same. For a wave packet, apply the mode decomposition and combine the outgoing fluxes for the prepared incident state.

This is an exact statement for the specified separated operator, not a derivation of the full electron–barrier Hamiltonian. Curvature, mixed kinetic terms, additional coordinate-dependent interactions, or a moving barrier can invalidate this reduction. Any claim about a measured tunnelling time also requires a defined time observable and preparation.

Verification

The spatial structure checks verify the separated differential-operator identity and show explicitly that a mixed potential destroys it.

Related

• Nested sector dynamics — states the operator conditions needed in addition to coordinate nesting.
• Nested spatial tools — companion tools for spatial evolution and normalization.