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Particles can also be called wave packets. There is some probability function that determines which part of the wave packet the mass of the particle is in. The tail of this probability function can extend into a seperate neighboring object, during which time, the particle could decide to jump to that other place and therefore reshape it's probability distribution.

An example would be a scanning tunneling microscope. It has a tiny probe-tip of conducting wire mounted on a pizeoelectric arm, which enables the tip to be scanned over the sample surface at an atomic distance. If a small voltage is applied across the tip and sample, some electrons will quantum tunnel from the tip across the gap to the sample, thus creating a measurable current. As the tip scans the atoms, the current changes, and a graphical representation of that change can be created.

Consider a small metal ball bearing put in a bowl. The ball bearing has an equilibrium position at the bottom of the bowl. Now if you were to push it a bit it would climb up the walls of the bowl, and fall back again, oscillate about the bottom and come to rest. If you were to push it hard enough however the ball would get out of the bowl. This is described by saying that the wall of the bowl acts as a potential barrier. The ball is in a potential well. For it to get out you must give it enough kinetic energy (push it hard enough) to get out.

However for very small objects things are not so simple. If the ball had been an electron and the bowl had been a quantum bowl then the ball could have got out without having enough energy to cross the potential barrier. So it is possible for the ball to simply materialize on the other side of the wall (even when it does not have enough energy to cross it) without the wall breaking or rupturing.

This is a very naive explanation of course but I hope it explains the principle behind Quantum Mechanical tunneling.

Consider a particle with energy E moving towards a potential barrier of height U0 and width a:
```                       _______
|||||||
-----------------------|||||||----------------------- E
|||||||
_______________________|||||||_______________________ x
| -a- |
|
x=0
```
Using Schrödinger's (time-independent, one-dimensional) Equation, we can solve for the wave function of the particle (using h for h-bar):
- h22ψ
--------_ + U(x)ψ = Eψ
2*m*∂x2

The potential U(x) is divided into three parts:

U(x) = { 0 : x < 0,
U0: 0 < x < a,
0 : x > a     }

In order to solve for ψ, the wave function of the particle, we also divide it into three parts: ψ0 for x < 0, ψ1 for 0 < x < a, and ψ2 for x > a. Astute readers will notice at this point that the potential is the same for ψ0 and ψ2 -- these two wave functions ought, then, to look at least somewhat similar. As we shall see, they will have the same wavelength but different amplitudes. Since U = 0 for both ψ0 and ψ2, they each take the same form as the wave function for a free particle with energy E, or:

ψ(x) = A*ei*k0x + B*e-ik0x  (where k0 = √(2*m*E/h2) )

The first portion of this equation corresponds to a wave moving rightwards while the second portion corresponds to a wave moving to the left. Or they would, had we folded in time-dependence (see note at the bottom). In order to make our lives easier, it is necessary to think a little bit about what is actually physically happening in this system. Our particle is approaching the potential barrier from the left, moving rightwards. When it hits the potential barrier, common sense says that at least some of the time, the particle will bounce off the barrier and begin moving leftwards. From this, we know that ψ0 contains both the leftward (reflected particle) and rightward (incident particle) portions of the wave function. As the other nodes in this writeup explain, when the particle hits the potential barrier, in addition to bouncing off some of the time, some of the time it will pass through. So we know that ψ2 has at least the rightward-moving component. But there is nothing in the experimental setup that would cause the particle to begin moving towards the left once it has passed through the potential barrier, so we can deduce that the leftward-moving component of ψ2 has an amplitude of zero.

Now, to deal with the particle while it is inside the barrier. Common sense would suggest that the particle can never actually exist within the barrier, (let alone cross over it). Physically, however, we know for sure that a particle can, in certain circumstances, pass through the barrier, so common sense would suggest that if it exists on both sides of the barrier, it must also exist within the barrier. But how on earth are we supposed to observe a particle while it is inside a potential barrier? The answer is that while we can't observe the particle inside the potential barrier, the mathematical properties of the wave function suggests that it does in fact exist while it is inside the barrier.

Since the only thing that matters in physics is relative potential, we can pretend like the particle, while it is inside the potential barrier, isn't in a potential of U0, but rather simply has an energy of E - U0 = - (U0 - E)  (since U0 > E). As before, then, the equation for this situation the wave equation with a wave number (k) of √(2*m*E/h2). In this case however, the particle has negative energy (tis a very good thing we can't physically observe the particle while it is inside the barrier, since negative energies can't exist), so it has an imaginary wave number, k1 = i√(2*m*(U0-E)/h2).

We now know enough to write out all three parts of the wave equation:

ψ(x) = {
A*ei*k0x + B*e-ik0x : x < 0
C*e-ik1x + D*eik1x   : 0 < x < a
E*eik0x           : x > a
}

The wave function and its first derivative have to continuous over all x ∈ R. We can use these boundary conditions to get four relationships among the constants (ψ0(0) = ψ1(0), ψ0'(0) = ψ1'(0), ψ1(a) = ψ2(a), and ψ1'(a) = ψ2'(a)). Actually solving for the constants is impossible given just these conditions (five unknowns but only four equations), but we can find the probability that the particle reflects off the barrier, and the probability that it tunnels through the barrier. Recall that the probability function of a particle with wave function ψ is

P(x) = |ψ(x)|2

Since we know that the first portion of ψ0 (with amplitude A) represents the incident particle, and the second portion (with amplitude B) represents the reflected particle, the ratio of the two wave functions |B|2/|A|2 is the fraction of the time that the incident particle will reflect off the barrier. Similarly, the ratio |E|2/|A|2 is the fraction of the time that the particle will tunnel through the barrier. After a bit of extraordinarily ugly algebra (don't try this at home), we find that:

```                                  1
|E|2/|A|2  =    -----------------------
U02sinh2(k1*a)
1 + 1/4 ---------------
E (U0 - E)
```

It shouldn't be too hard to convince yourself that since the particle has to do something after hitting the barrier, the probability that it will reflect off is just 1 - |E|2/|A|2. This probability decreases exponentially with a (since sinh(x) = (ex - e-x)/2), so the largest factor in determining tunneling probability is the width of the potential barrier. tdent notes that since the probability also depends exponentially on k1, there's a large dependence on the difference between the barrier height and the energy of the particle, but since the dependence on (U0 - E) is under a square root, this still has less of an effect than a.

Note: For time-independent potentials (∂U/∂t = 0), the time-dependent solution to the Schrödinger equation is just ψ(x)*e-iωt, where ψ is the time-independent wave function and ω = E/h. So, the time-dependent form of the solution ends up looking like:

A*ei(kx-ωt)+B*ei(-kx-ωt)

As t increases then, for the first part of the function to remain constant x must increase and for the second part to remain constant x must decrease. So the first portion of the equation represents a wave travelling towards increasing x (the right), and the second portion represents a wave travelling towards decreasing x (the left).

From personal notes, Modern Physics by Kenneth Krane, and http://uw.physics.wisc.edu/~bruch/p03sl19.pdf (for the solution to |E|2/|A|2).

Potential Barriers and Quantum Tunneling - A Layman's Introduction

Note: This is a layman's introduction to quantum tunneling only. For a general introduction to quantum mechanics, please see Mauler's Layman's Guide to Quantum Mechanics

Quantum tunneling is a concept from quantum mechanics, a branch of modern physics. The concept is explained using the following anecdote.

Suppose there is a hill, a real-world hill which you might walk up, if you were so inclined (no pun intended). Also suppose that three identical balls are rolling at different speeds towards the hill*. Due to this speed difference, each ball has a different energy of motion to the others. As the balls begin to roll up the hill, they also begin to slow down. The slowest ball does not have enough energy of motion to make it up the hill. It slows and slows, and eventually stops somewhere below the top for an instant in time, before rolling back down the hill. The second ball has enough energy to make it to the top of the hill, but no more. It comes to a stop on top of the hill. The last ball has more energy of motion than it actually needs to make it to the top of the hill. So when it makes it to the top, it still has some motion energy, and it rolls over the top, and down the other side.

This is all perfectly normal behaviour for balls on hills - nothing new there. However scientists (more specifically quantum physicists) discovered earlier last century, that when the balls are very very small, something very strange happens.

In the world of the very very small, balls usually behave in the same well-known manner described in the anecdote above. However, sometimes they don't. Sometimes balls which DO have enough energy to roll right up that hill and keep going down the other side, don't make it up the hill. That's weird. Imagine taking a bowling ball, and hurling it with all your might up a gentle hill. You know it's got enough energy to go over the top, but you blink, and when you open your eyes again, the bowling ball is rolling back down the hill towards you.

What's even stranger though, is that in this world of the very very small (and it is the REAL world, inhabited by you and I), sometimes balls which DON'T have enough energy to get up the hill, still do so (and continue down the other side). So it's like your bowling ball comes back out of the return shute, and you take it and roll it ever so gently up that same hill. You know it doesn't have enough energy to make it to the top, but then you blink, and when you open your eyes, there it is, rolling down the other side.

This puzzling behaviour has actually been observed to happen, many many times, by scientists. The phenomenon has been given the name "tunneling", for it is as if the ball (or 'particle' as we call it) digs a tunnel through that hill, to get to the other side. In such quantum experiments, scientists fire very small bullets at very small walls, and sometimes those bullets which do not have enough energy to break through the wall, are observed a short time later, on the other side (where it would seem, they have no right to be!).

Regarding this strange behaviour, I stress that THIS IS A REAL PHENOMENON. It actually applies to everything in the universe, but the chance of it happening to something as large as an elephant, or even a baseball, or a marble, is very small indeed. So small in fact, that it will probably never be seen to happen by a human on this planet. The smaller a thing is, the greater the chance of quantum tunneling occuring to it. Things that you can see with the naked eye are far too big. The kinds of particles to which tunneling commonly occurs can only be seen with special microscopes**.

As a final point, please note that it is probably a good thing that quantum tunneling is almost never observed to happen to everyday objects. It would not be too much fun if that butchers' knife you just placed safely on the table, suddenly tunneled through and found its way into the top of your foot. Of course it might tunnel through your foot as well, but.....well......if you ever see that happen, please let me know.

--------
* In quantum physics, the hill is known as a 'potential barrier'

** The kind of microscopes necessary to see the particles to which tunneling routinely occurs, are know as Scanning Tunneling Microscopes (S.T.M.). In an ironic twist, the technology which drives the S.T.M., itself relies on the principle of quantum tunneling to operate.

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