Quantum tunneling is the phenomenon in which a particle crosses a barrier even when its classical energy is smaller
than the barrier height. In classical mechanics, a particle with E < V0 cannot pass through a rectangular
barrier. In quantum mechanics, however, the wave function penetrates into the classically forbidden region and can
emerge on the other side with a nonzero probability.
Rectangular barrier setup
For a rectangular barrier of height V0 and width L, the particle energy is assumed to satisfy
E < V0. Inside the barrier, the wave function does not oscillate in the same way as it does in a
classically allowed region. Instead, it decays exponentially. The key quantity governing that decay is the barrier
constant κ:
κ = √(2m(V0 − E)) / ħ
Here, m is the particle mass and ħ is the reduced Planck constant. A larger value of κ means the wave decays more
rapidly inside the barrier. This is why heavier particles, taller barriers, wider barriers, or lower incident energies
all tend to suppress transmission.
Exact transmission probability
For a rectangular barrier, the exact transmission coefficient for E < V0 can be written as:
Texact = 1 / [1 + V02 sinh2(κL) / (4E(V0 − E))]
This formula comes from matching the wave function and its derivative at both barrier boundaries. The hyperbolic sine
factor encodes the exponential behavior within the forbidden region. When κL becomes large, the transmission becomes
very small.
Thick-barrier approximation
When the barrier is thick enough that κL is clearly larger than 1, the exact formula simplifies to a widely used
approximation:
Tapprox ≈ 16E(V0 − E) / V02 · exp(−2κL)
The dominant factor here is exp(−2κL), which shows directly why tunneling can decrease so dramatically with barrier
width. Even a modest increase in L can change T by many orders of magnitude. This exponential sensitivity is one of the
defining features of tunneling.
Energy dependence
As the particle energy E rises toward V0 from below, the difference V0 − E becomes smaller, so
κ decreases. That makes the decay inside the barrier weaker and increases the transmission probability. This is why the
tunneling curve rises sharply as E approaches the barrier top. On a logarithmic graph, this trend is much easier to see
because T often spans many orders of magnitude.
Conversely, when E is far below V0, κ becomes large and the tunneling probability can become extremely
small. This is the regime in which the approximate exponential expression is especially useful for physical intuition.
Physical meaning and applications
Tunneling is central to many real quantum phenomena. In alpha decay, a nucleus emits an alpha particle by tunneling
through the nuclear barrier. In scanning tunneling microscopy, the measurable current depends exponentially on the
distance between tip and surface because electrons tunnel across the gap. Tunneling also appears in semiconductor
junctions, Josephson devices, quantum transport, and field emission.
The rectangular barrier is an idealized model, but it captures the main lesson clearly: the barrier does not have to be
overcome classically for transmission to occur quantum mechanically. Instead, the particle’s wave nature allows a finite
amplitude beyond the barrier, and that amplitude leads to a measurable probability of transmission.
| Concept |
Main relation |
Meaning |
| Barrier regime |
E < V0 |
Classically forbidden region |
| Decay constant |
κ = √(2m(V0 − E)) / ħ |
Controls exponential decay inside the barrier |
| Exact transmission |
Texact = 1 / [1 + V02sinh2(κL)/(4E(V0−E))] |
Exact rectangular-barrier result |
| Approximate transmission |
Tapprox ≈ 16E(V0−E)/V02 · exp(−2κL) |
Thick-barrier approximation |
| Barrier exponent |
2κL |
Main suppression strength |
| Physical trend |
Larger L or larger V0 − E ⇒ smaller T |
Explains the strong exponential sensitivity |