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Quantum Tunneling Overview
Introduction
At the heart of quantum mechanics is the idea that matter behaves as both a wave and a particle. Experimental evidence from approximately 1900 through present day shows that as we look more closely at the behavior of very small things, such as molecules, atoms, and fundamental particles, the intuitive classical predictions of how matter should behave are not adequate in predicting the results of experiments. Perhaps the most fundamental reason for this is that these small particles were cla- sically treated as point-like objects whose position and momentum can be predicted with absolute certainty. It was not until the leading physicists of the early 20th cen- tury started toying with the idea that these very small objects may behave as waves at certain times and particles other times that they began predicting the results of experiments. One of the interesting findings of quantum mechanics is that, due to the wave-like nature of matter, small particles can be found in places that would classically be forbidden. This phenomenon is called “quantum tunneling”, and it has allowed for new technologies to be developed throughout the 20th century. Such applications are: the scanning-tunneling microscope, tunneling diodes, tunneling field-effect transistors, and the understanding of radioactive decay (which, for example, powers any nuclear power plant). This phenomenon not only demonstrates the ‘strangeness’ of quantum mechanics, but also plays a fundamental role in society, and is therefore an important subject in any quantum mechanics course. In the rest of this paper, we describe in greater detail what tunneling is, and how it can be treated mathematically.
Basics of Quantum Mechanics
Energy barriers are ubiquitous in physics. A skate board ramp, an electronic circuit element, a material’s emission properties, and much more, can be described with the concept of an energy barrier. For example, a skateboard ramp provides a gravitational energy barrier for the skater, such that while the skater is on the ramp, his/her energy is constrained by the ramp. Similarly, a circuit element can provide an energy barrier for electrons, such that only electrons with a certain energy may cross the circuit element. Quantum tunneling is a problem that involves an energy barrier. Specifically, this barrier tells us about how a quantum particle (such as an electron or proton or small molecule), can be spatially and temporally located within a region of space. One barrier to consider is the one shown in Figure 1. In this figure, three ‘regions’ exist
in one dimension of space (imagine a very small wire, separated by another wire a distance L away from each other).
Figure 1: Energy Barrier for a Quantum Particle
In considering how the particle behaves near this barrier, we use the most funda- mental equation in quantum mechanics: the Schr¨odinger equation. The Schr¨odinger equation describes a quantum particle’s ‘wave function’, analagous to how Maxwell’s equations describe electric and magnetic fields.
ℏ^2
2 m
∂^2
∂x^2
ψ(x) + V (x)ψ(x) = Eψ(x) (1)
In the Schr¨odinger equation, ℏ is a constant, m is the mass of the particle under consideration, V (x) is the energy barrier that the particle will see, E is the energy of the particle, and ψ(x) is the wave function of the particle. The wave function is really what we want to figure out from this equation. If we know the mass of the particle, the energy of the particle, and the energy barrier that the particle will encounter, we can solve for ψ(x), and find what we are looking for. Once we know ψ(x), we can calculate many other properties, and essentially know ‘everything’ there is to know about the system. Physically, the wave function tells us something about the probability of finding the particle in different locations of space. We know that the particle must be located somewhere in space, and this can be represented in the following way:
∫ (^) +∞
−∞
|ψ(x)|^2 dx = 1 (2)
This equation states that by taking the absolute value of the wave function, squar- ing it, and summing that quantity over all space (or in this case, over an infinitely
and ψ′(x = L)R 2 = ψ′(x = L)R 3.^2 These two conditions provide ‘boundary conditions’ for solving the Schr¨odinger equation given in (1). In the case of both E > V 0 and E < V 0 , the solutions for ψ(x) take the same form in Region I and Region III. We skip over the details of solving this, and simply write them here:
ψR 1 (x) = Aeikx^ + Be−ikx^ (4) ψR 3 (x) = F eikx^ + Ge−ikx^ (5)
where k =
2 mE/ℏ^2 , and we give different amplitudes (A, B, F and G) for the two regions to indicate that the amplitudes may be different across the different regions. Note that these solutions are just sine waves, with one sine wave (the +ikx term) traveling towards positive x and the other (the −ikx term) traveling towards negative x. A sum of sine waves simply adds to another sine wave, so the wave function in both Region I and Region III looks like a sinusoidal wave. If we consider the case of E > V 0 , the solution to the Schr¨odinger equation in Region II takes the same form as (4), but the constant k is slightly different. We can write this solution as:
ψR 2 (x) = Ceik^2 x^ + De−ik^2 x^ (6) where k 2 =
2 m(E − V 0 )/ℏ^2. The difference between k 2 and k is important, because it tells us something about how the wavelengths in Region I and III compare to the wavelength in region II. Recall that k = 2π/λ, where λ is the wavelength of the sine wave. Because k 2 < k, we should expect that λ 2 > λ. Therefore the wave function has a larger wavelength in Region II, so the solution looks slightly different. In the case of E < V 0 , the solution to the Schr¨odinger equation in Region II no longer looks like a sine wave. The reason for this is that because V 0 is larger than E, the only acceptable solution in that region takes the form of real exponentials. Thus, we can write the solution in Region II for E < V 0 as:
ψR 2 (x) = Ceκx^ + De−κx^ (7) where κ =
2 m(V 0 − E)/ℏ^2. Here, we have a term with exponential decay in x added to a term with exponential increase in x. At this point, we have solved the solutions to Schr¨odinger’s equation in all of the three regions for the case of E > V 0 and the case of E < V 0. Here, we summarize our solutions: (^2) These two constraints are postulates of quantum mechanics; there is really no other way to
explain why we use this.
For E > V 0 :
ψ(x) =
Aeikx^ + Be−ikx, for x < 0 Ceik^2 x^ + De−ik^2 x, for 0 ≤ x ≤ L F eikx^ + Ge−ikx, for x > L
For E < V 0 :
ψ(x) =
Aeikx^ + Be−ikx, for x < 0 Ceκx^ + De−κx, for 0 ≤ x ≤ L F eikx^ + Ge−ikx, for x > L
As is the process with solving any differential equation, after solving for the general solutions (as we have now done), we must plug in boundary conditions to solve for the unknown quantities. In our situation, we now have 6 unknown quantities (A, B, C, D, F , and G) for both cases of E.^3 So far, we have discussed four boundary conditions: the condition that ψ(x) must be continuous at both x = 0 and x = L (this gives us two), and the fact that ψ′(x) must be continuous at both x = 0 and x = L. However, we have 6 unknown quantities, so it seems we need 2 more boundary conditions. To determine the remaining boundary conditions, we must refer to the physical situation that we are dealing with. We are interested in a particle approaching the barrier from either the left side (Region I, moving in the direction of +x) or a particle approaching from the right side (Region III, moving in the direction of -x). In the first case, we can consider what happens to the particle as it approaches the energy barrier. Starting with the change in V (x) at x = 0, we can have some transmission of the wave function, as well as some reflection of the wave function. Recall that the top equation in (8) and (9) describes the sum of a right-going wave (corresponding to the coefficient A) and a left-going wave (corresponding to B). By saying that the wave is coming from Region I, we are essentially saying that at x → −∞, we are setting the amplitude of A at some constant value. On the other hand, if we were talking about case 2 (the left-going wave), we would be saying that at distance x → +∞, we are fixing G to be a constant value. Therefore, by giving information about where the particle is coming from, we have provided ourselves one more boundary condition, and know everything there is to know about A or G.^4 If we continue to consider the first case of the wave approaching from Region I, the barrier at x = 0 can allow for some reflection of the wave, and some transmission
(^3) Note that by unknowns, we are not referring to the energy (E), mass (m), or potential (V (x)),
nor anything that depends on those quantities (such as k, k 2 , or κ). We assume that those quantities are chosen for our given physical situation, and that we now want to watch what happens. (^4) It may seem strange to say that A or G is now known, when we haven’t set A or G equal to
some quantity. But remember that A and G are just constants; we could arbitrarily rename them something else, but it won’t make a difference in our calculation at this point.
References
[1] Taylor, J.R., Zafiratos, M.D., and Dubson, M.A. (2003). Modern Physics for Sci- entists and Engineers. Addison-Wesley. ISBN-10: 013805715X
[2] Krane, K.S. (2012). Modern Physics, 3rd Ed. Wiley. ISBN-10: 1118061144.
[3] Griffiths, D.J. (2004). Introduction to Quantum Mechanics, 2nd Ed. Pearson Pren- tice Hall. ISBN-10: 0131118927.
