For fun, here's the basis of quantum tunneling. Working this out for fusion with a charged particle approaching a nucleus is hard (well, not so much "hard" as time consuming), but I'll demonstrate the simpler case of "a particle in a box", where the box is a region of zero potential energy, and the "walls" have a finite potential which is greater than the kinetic energy of the particle. Classically, this is a situation where the particle should never escape. But quantum mechanically, it can escape with some nonzero probability -- it can "tunnel out". Here's the rigor behind the why:
The time independent Schrödinger equation for the bound particle has the form
[math]-\frac{\hbar^2}{2m}\frac{d^2\psi(x)}{dx^2} + U(x)\psi(x) = E\psi(x)where
[math]\psi is the wave function,
[math]\hbar is the reduced Planck constant,
[math]m is the mass of the particle,
[math]U(x) is the potential as a function of position, and
[math]E is the particle's energy.
Suppose the region confining the particle has width "L" (from a position x=0 to x=L), with a potential
[math]U(x)=0, and the walls have potential
[math]U(x)=U_0 > EThen for the interior of the region, the wave function must satisfy
[math]\psi'' = -\frac{2m}{\hbar^2}E\psi(where double-prime is shorthand for second derivative with respect to x). This is a 2nd order differential equation, and its associated characteristic equation has complex roots:
[math]r^2 = -\frac{2m}{\hbar^2}E \Rightarrow r=\pm\sqrt{-\frac{2mE}{\hbar^2}} \Rightarrow r=\pm \beta iComplex roots means the solution is a combination of sine and cosine:
[math]\psi_{inside} = c_1 cos{\beta x} + c_2 sin{\beta x}So for the inside of the "box", the particle's wave function
[math]\psi is an oscillation. In fact, it looks like a standing wave. This is the wave-particle duality of matter.
A natural question to ask is where the particle actually is. The wave function does not say!
Instead, it tells us the
probability of finding the particle in some region. Specifically, the probability of finding the particle at any point is proportional to the square of the wave function:
[math]|\psi|^2. That means
the probability of finding the particle is higher near peaks and troughs in the standing wave, and zero at the nodes. The precise definition is that the probability of finding the particle in any given region of space is equal to the integral of the square of the wave function, or the area under the curve, and the total probability over all the space must be equal to 1 (100% chance of finding the particle if you look everywhere).Here's an image to help visualize this. This is the wave function, and the probability, for a particle in a box with an infinite energy barrier. Notice there are distinct "energy levels": n=1, n=2, and so forth. The energy of the particle in the box is quantized. Each energy level is also associated with a different length of standing wave.
But this image is for an infinite energy barrier, so that the wave function is zero outside the box. What if the barrier is not infinite?
For the region outside, we have U=U
_{0}, so the characteristic equation is
[math]r^2 = \frac{2m}{\hbar^2}\left(U_0 - E\right) = \alpha^2 \Rightarrow r=\pm\alphaIn this case the solution is
[math]\psi_{outside} = c_3 e^{\alpha x} + c_4 e^{-\alpha x}Because the total probability must be 1,
[math]c_3 must be zero to the right of the box(otherwise
[math]c_3 e^{\alpha x} will become infinite), and similarly
[math]c_4 must be zero to the left of the box.
So on either side of the box, the wave function is a simple decaying exponential
[math]\psi \propto e^{-\alpha |x|}. And the image looks like this:
Outside the box, the wave function decays by a fraction of 1/e for every distance
[math]\delta \equiv \frac{1}{\alpha} = \frac{\hbar}{\sqrt{2m(U_0 - E)}} This is often called the "penetration depth".
The penetration depth is generally an extremely small distance. If we consider an electron with an energy of 100eV, bound within a potential of 200eV, then the penetration depth is just 1.95x10
^{-11} meters! But if the width of the barrier is comparable to that distance, then the particle can tunnel through, even though the energy of the barrier is greater than the energy of the particle! To calculate exactly how probable, we need to determine the coefficients of the wave function (these are found by solving a "boundary condition problem", using the fact that the wave function must be continuous at the wall), and then compute the probability for the region outside the barrier.
This is rather tedious to work out by hand, but it can be done very quickly on a computer. In fact, the hyperphysics website has a page which will do it for you:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/barr.htmlSo that's quantum tunneling in a nutshell. As is par for the course wherever quantum mechanics is involved, it's pretty crazy!
edit: fixed a factor of 2 error in depth calculation