Overview#

• A potential barrier is a region of width \(L\) where \(U = U_b\). For \(E < U_b\), the barrier is classically forbidden—the particle would need negative kinetic energy inside.

• Quantum tunneling: A matter wave can “leak” through the barrier. The wave function decays exponentially inside the barrier but is nonzero beyond it—the electron has a finite probability of appearing on the other side.

• The transmission coefficient \(T \approx e^{-2\beta L}\) is exponentially sensitive to barrier thickness, particle mass, and energy deficit.

• The scanning tunneling microscope uses electron tunneling between a sharp tip and a surface to image atoms.

Potential barrier and classical forbidden region#

Replace the potential step with a potential barrier: a region \(0 < x < L\) where \(U = U_b\), with \(U = 0\) on both sides. An electron approaches from the left with energy \(E\).

If \(E > U_b\), the electron classically slows in the barrier but passes through (section 38-8: some reflect at each boundary).

If \(E < U_b\), the kinetic energy in the barrier would be \(E - U_b < 0\)—impossible. Region 2 is classically forbidden. A classical particle would always reflect at \(x = 0\).

Quantum tunneling#

Because the electron is a matter wave, the wave function does not vanish inside the barrier. Schrödinger’s equation there yields solutions that decay exponentially with \(x\). If \(L\) is small enough, the wave function is still nonzero at \(x = L\)—the electron can “tunnel” through and emerge on the right with its original energy \(E\).

Transmission coefficient \(T\): the probability that an incident electron tunnels through. For a barrier of height \(U_b\), thickness \(L\), and particle mass \(m\) with \(E < U_b\):

\(T\) is very sensitive to:

• Barrier thickness \(L\): doubling \(L\) roughly squares \(T\) (e.g., \(T\) drops from \(10^{-2}\) to \(10^{-4}\)).

• Mass \(m\): heavier particles tunnel less readily.

• Energy deficit \(U_b - E\): larger deficit \(\Rightarrow\) smaller \(T\).

Probability density in the three regions#

| Region | \(|\psi|^2\) | |——–|————-| | Left (\(x < 0\)) | Oscillating—incident and reflected waves interfere (standing-wave pattern). | | Inside (\(0 < x < L\)) | Exponential decay with \(x\). | | Right (\(x > L\)) | Constant (small)—transmitted wave. |

The electron can be detected beyond the barrier with probability proportional to \(T\). The transmitted wave has the same wavelength as the incident wave (same energy \(E\)).

Scanning tunneling microscope#

The STM images surfaces at atomic resolution using tunneling. A sharp conducting tip is brought very close to the surface (gap \(\sim\) 1 nm). The gap acts as a potential barrier; electrons tunnel from surface to tip (or vice versa), producing a tunneling current.

A small voltage (e.g., 10 mV) is applied between tip and surface. Piezoelectric rods scan the tip in \(x\) and \(y\) and adjust its height \(z\). A feedback loop keeps the tunneling current constant by varying the tip height—so the tip follows the surface contour. The recorded \(z(x,y)\) yields an atomic-scale image.

Summary#

• Potential barrier: \(U = U_b\) for \(0 < x < L\); \(E < U_b\) \(\Rightarrow\) classically forbidden.

• Tunneling: Matter wave decays inside barrier but can emerge on the far side; \(T \approx e^{-2\beta L}\).

• \(\beta = \sqrt{8\pi^2 m(U_b - E)/h^2}\); \(T\) is exponentially sensitive to \(L\), \(m\), and \(U_b - E\).

• STM: Tunneling current between tip and surface; feedback maintains constant current while scanning \(\Rightarrow\) atomic-resolution surface image.