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Thursday
Oct282010

A Cure For Insomnia

According to quantum mechanics, everything is possible - no matter how crazy! You can walk through a solid wall... A friend of yours from across the continent can suddenly materialize next to you at a very inappropriate moment... How about jumping from Los Angeles to the moon? Sure, why not.... Everything is allowed - all Nature cares about is probability. This is the real world, not some bad science fiction movie... We know quantum mechanics is correct from an uncountable number of experiments; even from the fact we're around: atoms making us up would collapse instantly without quantum mechanics...

So, how come the last time you slammed your head on a wall, it didn't materialize on the other side? Well, it's all about probabilities. Quantum mechanics tells us that Nature itself - at a fundamental level - is not deterministic: nothing is certain, everything can happen, and the only thing that mother Nature keeps track of is the likelihood of something happening or not happening. When an event is highly unlikely or highly likely, we effectively see it in a deterministic light - like we're used to: it simply doesn't happen or happens in a predictive reproducible manner. In reality however, there is always a chance to get surprised or shocked...

So, how can you estimate the probability of an event from quantum mechanics? Here's a crude but correct method that you can use to impress people in a bar. First, identify the relevant mass, length, and time for the hypothetical event. Multiply the mass with the length times length; divide by the time. Take this resulting number and divide by ten to the power minus 34 (that's roughly Planck's constant): make sure you use kilograms for mass, meter for length, and seconds for time! You can easily convert units using google: for example, you could type "convert feet to meter". Now, take the number you got, multiply it by minus one, and raise 2.7 (that's "e" for the geeks amongst us) to the power of this number: this is an estimate of the probability of the event! That's it, you now know some quantum mechanics! 

Let's give it a try. Say I am considering slamming my head on a granite wall; it has been a particularly long morning, didn't get much sleep last night, and need to wake up somehow. I would like to know the probability that my head will go through the wall. That would certainly make the day more interesting.  So, I need a mass, a length, and a time. My head is rather substantial in weight: maybe 20 kilograms? The wall is about 1/3 of a meter in thickness. With the speed I am planning to slam my head on the wall, the time will be around one second. So, let's put things together with a calculator. 20 times 1/3 times 1/3 divided by 1 second. That's about 2. Divide by Planck's constant, that's 1E-34 on a good calculator. We get roughly 2E+34. Multiply by minus one. That's -2E+34. Final step: raise 2.7 to the power of -2E+34. That's tiny tiny tiny; your calculator will probably just say zero or just blow up... This means I would need to hit my head on the wall many many (many) times to expect to have it go through the wall once... I better start right away.

How about the probability for an electron to just appear out of thin air? Crazy, but how likely is that? You can look up the mass of an electron on google: type "mass of electron": you get 1E-30 kilograms. Say the freaky electron moves a distance of an atom's diameter - a reasonable thing to expect in the world of an electron? that's 1E-10 meters. Say it moves around a percent of the speed of light: electrons can move fast. The speed of light is around 1E+8 meters per second. A percent of that is 1E+6 meters per second. So, the time it needs to travel a distance of 1E-10 meters is 1E-10 divided by 1E+6; that's 1E-16 seconds. So, put these numbers together on a calculator: you'll get a probability of 2.7 to the power of minus one: or 0.37, that's 37% chance for an electron to appear out of thin air! Not bad at all. You now see why the microscopic world is so crazy, and our macroscopic world is so much more predictable...

And here's a simple calculator you can use to play around with this idea:

Length in meters Time in seconds Mass in kilograms Calculated probability
percent

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