Barrier Penetration

Unlike the situation in classical mechanics, quantum mechanics allows the kinetic energy $K$ to be negative. This makes the momentum $\Pi$ (equal to $(2mK)^{1/2}$ in the nonrelativistic case) imaginary, which in turn gives rise to an imaginary wavenumber.

Let us investigate the nature of a wave with an imaginary wavenumber. Let us assume that $k = i \kappa$ in a complex exponential plane wave, where $\kappa$ is real:

$$\psi = \exp [i(kx - \omega t)] = \exp (- \kappa x - i \omega t) = \exp (-\kappa x) \exp (-i\omega t) .$$ (10.22)

The wave function doesn't oscillate in space when $K = E - U < 0$, but grows or decays exponentially with $x$, depending on the sign of $\kappa$.

Figure: Real part of wave function $\mbox{Re} [\psi (x)]$ for barrier penetration. The left panel shows weak penetration occurring for a large potential energy barrier, while the right panel shows stronger penetration which occurs when the barrier is small.

For a particle moving to the right, with positive $k$ in the allowed region, $\kappa$ turns out to be positive, and the solution decays to the right. Thus, a particle impingent on a potential energy barrier from the left (i. e., while moving to the right) will have its wave amplitude decay in the classically forbidden region, as illustrated in figure 9.4. If this decay is very rapid, then the result is almost indistinguishable from the classical result -- the particle cannot penetrate into the forbidden region to any great extent. However, if the decay is slow, then there is a reasonable chance of finding the particle in the forbidden region. If the forbidden region is finite in extent, then the wave amplitude will be small, but non-zero at its right boundary, implying that the particle has a finite chance of completely passing through the classical forbidden region. This process is called barrier penetration.

The probability for a particle to penetrate a barrier is the absolute square of the amplitude after the barrier divided by the square of the amplitude before the barrier. Thus, in the case of the wave function illustrated in equation (9.22), the probability of penetration is

$$P = \vert \psi (d) \vert^2 / \vert \psi (0) \vert^2 = \exp (-2 \kappa d)$$ (10.23)

where $d$ is the thickness of the barrier.

The rate of exponential decay with $x$ in the forbidden region is related to how negative $E - U$ is in this region. Since

$$-K = U - E = -\frac{\Pi^2}{2m} = -\frac{\hbar^2 k^2}{2m} = \frac{\hbar^2 \kappa^2}{2m} ,$$ (10.24)

we find that

$$\kappa = \left( \frac{2mB}{\hbar^2} \right)^{1/2}$$ (10.25)

where the potential energy barrier is $B \equiv -K = U - E$. The smaller $B$ is, the smaller is $\kappa$, resulting in less rapid decay of the wave function with $x$. This corresponds to stronger barrier penetration. (Note that the way $B$ is defined, it is positive in forbidden regions.)

If the energy barrier is very high, then the exponential decay of the wave function is very rapid. In this case the wave function goes nearly to zero at the boundary between the allowed and forbidden regions. This is why we specify the wave function to be zero at the walls for the particle in the box. These walls act in effect as infinitely high potential barriers.

Barrier penetration is important in a number of natural phenomena. Certain types of radioactive decay and the fissioning of heavy nuclei are governed by this process. In addition, the field effect transistors used in most computer microchips control the flow of electrons by electronically altering the strength of a potential energy barrier.