Chapter 6: Quantum Tunneling

In our macroscopic world, if the block does not have enough energy in the very beginning, it will never move over the hill no matter how many times you try. We can think of this hill as a “potential energy barrier,” and if the energy required to reach a region is too high it is “classically forbidden.”

From here, let’s take a leap into the quantum world. Imagine now that an electron is moving with certain kinetic energy towards a potential barrier (e.g. due to an electric field). At first glance, this problem is identical to what we have just described: a block (electron) rolling towards a hill (potential barrier). If the kinetic energy of the electron were smaller than the potential barrier, then again, we expect the electron to bounce back.

However, when researchers attempt  this experiment, they find instead that sometimes the electrons pass through the barrier!

To develop an understanding of what is going on, we must look at the wave functions of quantum objects (like the electron). The wave function is a mathematical description of the probability of finding a particle at a particular position. (See Chapter 3: Wave Functions for more details!) 

As we sweep across the possible positions of the electron, the wave function must be continuous. In other words, the probability of finding an electron at a position x can not suddenly change if we move to a position very close by x + δx. Therefore, when an electron meets a potential barrier it can not overcome, the wave functions will not immediately go to zero. However, our classical intuition remains correct: it is highly unfavorable for the electron to occupy this classically forbidden region. Indeed, the wave function decays quickly to zero exponentially fast.

If the classically forbidden region is finite, then the exponentially decaying wave function will eventually reach the end of the forbidden region and it will re-emerge much smaller in value but no longer decaying. Since the value of the wave function after the barrier is non-zero, there is a small chance for us to observe the electron on the other side of the barrier – it has tunneled through the barrier!

So why don’t we observe quantum tunneling in everyday life? The wave function decays exponentially quickly in the classically forbidden region that on macroscopic length scales its value is effectively zero. We have effectively zero chance of observing quantum tunneling on the human scale.

[Editor’s note: How impossible is “effectively impossible”? This is a classic quantum mechanics HW problem and a potential topic for the Quantum Shorts Contest! Hint: from [1] we estimate that the probability of a 60 kg person running at a speed of 4 m/s tunneling through a wall 2 m tall and 0.5 m wide, is ~1/10^(10^36)]

Quantum tunneling has been widely observed and used in many fields. In quantum chemistry, tunneling explains chemical reactions deemed impossible by classical physics. In nuclear physics, it gives rise to the radioactive alpha decay. In microscopy, scanning tunneling microscopes employ such phenomena to realize groundbreaking resolution. Quantum tunneling of electrons also set a fundamental limit to how small we can make circuits in computer chips.

[1] 7.7: Quantum Tunneling of Particles through Potential Barriers