Friday, September 19, 2014

Is It The Same Temperature In Every Possible World? Part 2 - The Flavor Of Quantum Mechanics

Not This Stuff Yet

It has been decided that instead of making a strictly logical foundations post, I'm going to start by writing a post to give someone unused to quantum mechanics an idea of the general flavor. I'm going to do my best to make this maximally bland when it comes to theory, giving a couple of asides for those who already have this way of thinking in their head.

Animated Double Slit

The usual approach to quantum mechanics involves the double slit experiment. This is how Feynman approaches Quantum Mechanics in his lectures, and this is clearly how Bohm thought of the whole problem. The basic issue is this. If something is a wave, like water or possibly light, we know what will be seen on the other side. We know because Huygens taught us. If you want to know, look above. If something is a particle, then the tiny slits will only allow the particles through in one direction:

See the difference? Well, an issue comes up in interpreting the experimental results. The same things that to all - literally all - direct experiments look like a particle, move like a wave. This used to be called particle-wave duality, complementarity and other unclear names. And there is really no issue in classical quantum mechanics that can't be stated in terms of this experiment, so that's nice.

But Who Cares?

Despite the depth of this approach, I don't think it really gets at what people want to know about quantum mechanics. In addition, it doesn't obviously generalize to relativistic quantum mechanics, which you need if there are magnetic fields. So, I'm going to try a different method. If this fails to give you any intuitive view of quantum, feel free to blame me.

You And Me and Coal

Imagine that you are the blue person and I am the black person. I want to give you some carbon, and I'm holding some nice anthracite coal. Now, there are two ways that I can get this carbon to you. One is that I can move it to you on some path, perhaps by throwing it to you.

For Certain Values of "to you"

But that isn't always possible. For instance, if there is a fine grating between us, the above picture is impossible. What I can do then is burn the coal, and fan the smoke to you. Then you can reconstitute the smoke into carbon.

Less Dangerous, Less Efficient

In other words, there are two possible methods of getting the carbon to you. One is by direct movement, the other by fluctuation. In classical mechanics, fluctuation is treated as a limit of direct movement. In quantum mechanics, the two are given equal weight. It really is the case that something can get to me by traveling straightforwardly or by fluctuating over. The Schrodinger formulation emphasizes the wave like movements, the Feynman formulation the diffusive movement. The rest of the difficulties of quantum mechanics are mere physics.

Let's go through a more directly quantum version of this idea. Let's look at some particle in a box! This box is divided into three parts, outside the box there is nothing. In the first part and last part the potential is zero. In the middle there is some potential to stop particles from moving:

The Potential

If there is any direct movement path for a quantum particle to get somewhere, then you call that region classically allowed. In those place, you expect to see waves, just like classical mechanics waves. If you have to fluctuate to get there, call it classically forbidden. In those places the distribution should fall off exponentially, just like classical mechanics diffusion. This is called the WKB approximation. This idea gives the following idea about where the particle could be:

The Distribution of Where the Particle Could Be

In the classically allowed regions, the particle moves around on easy classical paths. I represent that rapid movement with a big wave. In the classical forbidden region, the particle must fluctuate into weird areas. I represent that region by a exponentially decaying curve. But this is just formulation. I must now show how this quantum view of the world differs from the classical vision. I'll do this by proposing a thought experiment. At the beginning of time, the only particle in the universe is entirely at one location. Perhaps you, a god, have some sort of sensor that lets you tell this.

In the beginning, there was the Dirac Delta

We turn off the sensor, allowing the particle to move again. At first, it mostly moves in the left classically permitted zone, moving in the quick, easy way. But soon the fluctuation type movement starts becoming visible. At this point, it starts to look like the WKB approximation above.

This again.

Over time, the particle's movement and fluctuation start to balance out. At this point, it's spread evenly in the first and third in the two classically permitted areas and to a lesser extent in the classically forbidden area. I'd make a picture, but I left all my scripts on another computer. You turn on a sensor on the right classically permitted area. You can get the expected waiting time from the probability the area under the curve over the sensor. Though this is a classically permitted area, there is no classical path that brought you here. The history of this experiment is a quantum history. This might sound silly, and I tried to make it that way. But this has real experimental and theoretical consequences.

G Gamow

This situation is analogous to an atom undergoing fusion. The left classically permitted region is like free space, the right classically permitted region is like a fused nucleus. The bump in the middle represents the region that the incoming proton is to weak to get past the nucleus's magnetic field. With only classical movement, fusion is not possible. But quantum physics permits a new type of movement, and even if one can't move past the bump, one can fluctuate past.

In the next post in this series, I'll show how this idea of movement by fluctuation leads to a path integral and maybe do some interpretation. In the mean time, I have a couple other ideas to try out. See you next time!

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