40.10 Quantum-Mechanical Tunneling

FIGURE 40.29a shows a ball rolling toward a hill. A ball with sufficient kinetic energy can go over the top of the hill, slowing down as it ascends and speeding up as it rolls down the other side. A ball with insufficient energy rolls partway up the hill, then reverses direction and rolls back down.

The figure labeled (“a”) illustrates a ball rolling toward a hill.

The figure labeled (b) illustrates the graph of height of energy barrier versus distance.

Full Alternative Text

We can think of the hill as an “energy barrier” of height $U_{0} = m g y_{max}$ . As FIGURE 40.29b shows, a ball incident from the left with energy $E > U_{0}$ can go over the barrier (i.e., roll over the hill), but a ball with $E < U_{0}$ will reflect from the energy barrier at the turning point. According to the laws of classical physics, a ball that is incident on the energy barrier from the left with $E < U_{0}$ will never be found on the right side of the barrier.

NOTE Figure 40.29b is not a “picture” of the energy barrier. And when we say that a ball with energy $E > U_{0}$ can go “over” the barrier, we don’t mean that the ball is thrown from a higher elevation in order to go over the top of the hill. The ball rolls on the ground the entire time, as Figure 40.29a shows, and Figure 40.29b describes the kinetic and potential energy of the ball as it rolls. A higher total energy line means a larger initial kinetic energy, not a higher elevation.

FIGURE 40.30 shows the situation from the perspective of quantum mechanics. As you’ve learned, quantum particles can penetrate with an exponentially decreasing wave function into the classically forbidden region of an energy barrier. Suppose that the barrier is very narrow. Although the wave function decreases within the barrier, starting at the classical turning point, it hasn’t vanished when it reaches the other side. In other words, there is some probability that a quantum particle will pass through the barrier and emerge on the other side!

The figure consists of two parts that illustrate a quantum particle penetration through the energy barrier.

FIGURE 40.30 Full Alternative Text

It is very much as if the ball of Figure 40.29a gets to the turning point and then, instead of reversing direction and rolling back down, tunnels its way through the hill and emerges on the other side. Although this feat is strictly forbidden in classical mechanics, it is apparently acceptable behavior for quantum particles. The process is called quantum-mechanical tunneling.

The process of tunneling through a potential-energy barrier is one of the strangest and most unexpected predictions of quantum mechanics. Yet it does happen, and you will see that it even has many practical applications.

NOTE The word “tunneling” is used as a metaphor. If a classical particle really did tunnel, it would expend energy doing so and emerge on the other side with less energy. Quantum-mechanical tunneling requires no expenditure of energy. The total energy line is at the same height on both sides of the barrier. A particle that tunnels through a barrier emerges with no loss of energy. That is why the de Broglie wavelength is the same on both sides of the potential barrier in Figure 40.30.

To simplify our analysis of tunneling, FIGURE 40.31 shows an idealized energy barrier of height $U_{0}$ and width $w$ . We’ve superimposed the wave function on top of the energy diagram so that you can see how it aligns with the potential energy. The wave function to the left of the barrier is a sinusoidal oscillation with amplitude $A_{L}$ . The wave function within the barrier is the decaying exponential we found in Equation 40.40:

ψ_{in} (0 \leq x \leq w) = ψ_{edge} e^{- x / η} = A_{L} e^{- x / η}

(40.51)

where we’ve assumed $ψ_{edge} = A_{L}$ . The penetration distance $η$ was given in Equation 40.41 as

η = \frac{ℏ}{\sqrt{2 m (U_{0} - E)}}

The figure illustrates the graph of tunneling through an idealized energy barrier.

FIGURE 40.31 Full Alternative Text

NOTE You must use SI units when calculating values of $η$ . Energies must be in J and $ℏ$ in $J s$ . The penetration distance $η$ has units of meters.

The wave function decreases exponentially within the barrier, but before it can decay to zero, it emerges again on the right side $(x > w)$ as an oscillation with amplitude

A_{R} = ψ_{in} (a t x = w) = A_{L} e^{- w / η}

(40.52)

The probability that the particle is to the left of the barrier is proportional to $| A_{L} |^{2}$ , and the probability of finding it to the right of the barrier is proportional to $| A_{R} |^{2}$ . Thus the probability that a particle striking the barrier from the left will emerge on the right is

P_{tunnel} = \frac{| A_{R} |^{2}}{| A_{L} |^{2}} = {(e^{- w / η})}^{2} = e^{- 2 w / η}

(40.53)

This is the probability that a particle will tunnel through the energy barrier.

Now, our analysis, we have to say, has not been terribly rigorous. For example, we assumed that the oscillatory wave functions on the left and the right were exactly at a maximum where they reached the barrier at $x = 0$ and $x = w$ . There is no reason this has to be the case. We have taken other liberties, which experts will spot, but—fortunately—it really makes no difference. Our result, Equation 40.53, turns out to be perfectly adequate for most applications of tunneling.

Because the tunneling probability is an exponential function, it is very sensitive to the values of $w$ and $η$ . The tunneling probability can be substantially reduced by even a small increase in the thickness of the barrier. The parameter $η$ , which measures how far the particle can penetrate into the barrier, depends both on the particle’s mass and on $U_{0} - E$ . A particle with $E$ only slightly less than $U_{0}$ will have a larger value of $η$ and thus a larger tunneling probability than will an identical particle with less energy.

EXAMPLE 40.11 Electron tunneling

Find the probability that an electron will tunnel through a 1.0-nm-wide energy barrier if the electron’s energy is 0.10 eV less than the height of the barrier.
Find the tunneling probability if the barrier in part a is widened to 3.0 nm.
Find the tunneling probability if the electron in part a is replaced by a proton with the same energy.

SOLVE

An electron with energy 0.10 eV less than the height of the barrier has $U_{0} - E = 0.10 eV = 1.60 \times 10^{- 20} J$ . Thus its penetration distance is

$\begin{matrix} η & = & \frac{ℏ}{\sqrt{2 m (U_{0} - E)}} \\ = & \frac{1.05 \times 10^{- 34} J s}{\sqrt{2 (9.11 \times 10^{- 31} kg) (1.60 \times 10^{- 20} J)}} \\ = & 6.18 \times 10^{- 10} m = 0.618 nm \end{matrix}$

The probability that this electron will tunnel through a barrier of width $w = 1.0 nm$ is

$P_{tunnel} = e^{- 2 w / η} = e^{- 2 (1.0 nm) / (0.618 nm)} = 0.039 = 3.9 %$
Changing the width to $w = 3.0 nm$ has no effect on $η$ . The new tunneling probability is

$\begin{matrix} P_{tunnel} & = & e^{- 2 w / η} = e^{- 2 (3.0 nm) / (0.618 nm)} = 6.0 \times 10^{- 5} \\ = & 0.006 % \end{matrix}$

Increasing the width by a factor of 3 decreases the tunneling probability by a factor of 650!
A proton is more massive than an electron. Thus a proton with $U_{0} - E = 0.10 eV$ has $η = 0.014 nm$ . Its probability of tunneling through a 1.0-nm-wide barrier is

$P_{tunnel} = e^{- 2 w / η} = e^{- 2 (1.0 nm) / (0.014 nm)} \approx 1 \times 10^{- 64}$

For practical purposes, the probability that a proton will tunnel through this barrier is zero.

REVIEW If the probability of a proton tunneling through a mere 1 nm is only $10^{- 64}$ , you can see that a macroscopic object will “never” tunnel through a macroscopic distance!

Quantum-mechanical tunneling seems so obscure that it is hard to imagine practical applications. Surprisingly, it is the physics behind one of today’s most important technical tools, as we describe in the next section.

40.10 Quantum-Mechanical Tunneling

FIGURE 40.29 A hill is an energy barrier to a rolling ball.

FIGURE 40.30 A quantum particle can penetrate through the energy barrier.

FIGURE 40.31 Tunneling through an idealized energy barrier.

EXAMPLE 40.11 Electron tunneling