Finite Barrier Tunneling and the Uncertainty Principle

When a quantum particle encounters a discontinuity in the form of a finite barrier, there is a nonzero probability that it will be transmitted to the other side of the barrier. In fact, as the graphic at right shows, the wavefunction bifurcates into a reflected as well as a transmitted part, a decidedly nonclassical and headscratching result (remember the Copenhagen interpretation) -- the three curves, by the way, show the real and imaginary parts of the wavefunction as well as the probability density.

Such tunneling phenomena are found throughout physics, chemistry and molecular biology, from explanations of alpha and beta decay to kinetic isotope effects and electron and proton transport in enzymes. It is likely that all redox chemistry has a tunneling component, which becomes even more prominent as the temperature falls.

Before moving onto the next quantum system, the particle in a finite box, we take a moment to consider one of the most important discoveries in all of quantum mechanics, the uncertainty principle. Heisenberg, in his development of matrix mechanics (an alternate description of quantum behavior, using matrices instead of differential operators), he discovered that some matrices did not commute, particularly those for momentum and position. After a little work he was able to demonstrate that this placed limits on the knowledge of the corresponding observable and we were doomed to indeterminacy. In particular, he found that ∆x∆p ≥ hbar/2, where ∆x and ∆p are the standard deviations of position and momentum.

Classically these values are zero: we can always know the position of, say, a marble and its velocity (and hence its momentum). At the quantum scale, these values are nonzero and, more baffling, are connected. The more precise we can nail down the position, for example, the less we will know about its momentum (and vice versa). Clearly we have to give up the Newtonian idea of knowing the trajectory of any quantum particle. This is not really an issue with measurement itself but rather a fundamental description of the quantum nature of the universe.

As we will see there are a number of uncertainty principles and they arise whenever we have noncommutative operators. In other words, whenever [a,b] ≠ 0, we will have an uncertainty principle in the corresponding observables: ∆a∆b ≥ hbar/2.