What is quantum tunneling?

For a rectangular barrier of height \(V_0\) and a particle with \(E<V_0\), write the decay constant inside the barrier.

Use \(T\sim e^{-2\kappa a}\) to estimate the transmission if \(\kappa=3.0\,\mathrm{nm^{-1}}\) and \(a=0.50\,\mathrm{nm}\).

If the barrier width is doubled while \(\kappa\) is unchanged, how does the tunneling estimate change?

How does increasing the particle mass affect below-barrier tunneling, all else fixed?

As \(E\) approaches \(V_0\) from below, what happens to the tunneling probability through a fixed rectangular barrier?

For a lossless one-dimensional barrier, what relation connects reflection probability \(R\) and transmission probability \(T\)?

State the matching conditions at each finite edge of a rectangular barrier.

An electron and a proton have the same energy below the same barrier. Which tunnels more easily, and why?

Write the WKB tunneling estimate for a smoothly varying barrier where \(V(x)>E\) between turning points \(x_1\) and \(x_2\).

A barrier consists of two adjacent below-energy regions with \(\kappa_1,a_1\) and \(\kappa_2,a_2\). Estimate the total exponential transmission factor.

A particle has \(E>V_0\) for a finite rectangular barrier. Is reflection necessarily zero? Explain.

Define transmission probability in terms of probability current for a scattering state.

Explain why tunneling does not violate conservation of energy.

Derive the exponential form of the wave function inside a constant below-energy barrier from the time-independent Schrodinger equation.

For a thick rectangular barrier, a more detailed estimate is \(T\approx 16E(V_0-E)V_0^{-2}e^{-2\kappa a}\). Which factor controls the strongest dependence on width?

In a scanning tunneling microscope, current is roughly proportional to \(e^{-2\kappa d}\). If the tip distance increases by \(\Delta d\), find the current ratio.

Use tunneling to explain qualitatively why alpha decay rates are extremely sensitive to nuclear energy release.

Show that the classical limit suppresses tunneling in the WKB estimate.

Prove that a single decaying exponential inside a barrier cannot by itself describe nonzero transmission through the whole barrier.