http://en.wikipedia.org/wiki/Commutative_property
Scroll down for relevant discussion on the uncertainty principle.
Understanding the mathematics behind the physics involved is imperative to really understanding Heisenberg's Uncertainty Principle. We've already discussed the logic behind it and the physical consequences, but now it's time to really get down to the fundamentals. We're going to look at something we tend to take for granted in our daily interactions with math: the commutative property of mathematics.
The commutative property in math applies to addition and multiplication. It basically states that, when combining two "elements" (i.e. numbers), the order does NOT matter. This is demonstrated in the following diagram:
But we've always known this stuff, right? I mean, everybody knows that 2+3 is the same as 3+2 (at least I hope)! The same rules fly for multiplication as well.
Of course, the importance of this property goes way beyond grade school math and counting apples or orange and blue dots! Many important proofs in analysis, linear algebra and set theory are founded upon the assumption that the commutative property is legit. This stuff dates way back as well; the Ancient Egyptians were avid users of this property because, at the time, it greatly helped in simplifying some complicated multiplication problems (it's okay, we have calculators for that stuff now!).
And of course, the very idea of the combative property at a fundamental property should not be classified as something strictly in the realm of mathematics; this property manifests itself in our every day lives all of the time! For example, the process of putting on your socks is commutative. It doesn't matter if left or right goes first - the end result is exactly the same! Another example would be eating toast and drinking milk. Assuming your goal was just to finish both and acquire nourishment from them, then your stomach does not care which ones goes down your throat first (of course, some people are picky with this, but we'll disregard that for a moment).
Scroll down for relevant discussion on the uncertainty principle.
Understanding the mathematics behind the physics involved is imperative to really understanding Heisenberg's Uncertainty Principle. We've already discussed the logic behind it and the physical consequences, but now it's time to really get down to the fundamentals. We're going to look at something we tend to take for granted in our daily interactions with math: the commutative property of mathematics.
The commutative property in math applies to addition and multiplication. It basically states that, when combining two "elements" (i.e. numbers), the order does NOT matter. This is demonstrated in the following diagram:
It doesn't matter whether the blue or orange dots are counted first - same result
But we've always known this stuff, right? I mean, everybody knows that 2+3 is the same as 3+2 (at least I hope)! The same rules fly for multiplication as well.
Of course, the importance of this property goes way beyond grade school math and counting apples or orange and blue dots! Many important proofs in analysis, linear algebra and set theory are founded upon the assumption that the commutative property is legit. This stuff dates way back as well; the Ancient Egyptians were avid users of this property because, at the time, it greatly helped in simplifying some complicated multiplication problems (it's okay, we have calculators for that stuff now!).
And of course, the very idea of the combative property at a fundamental property should not be classified as something strictly in the realm of mathematics; this property manifests itself in our every day lives all of the time! For example, the process of putting on your socks is commutative. It doesn't matter if left or right goes first - the end result is exactly the same! Another example would be eating toast and drinking milk. Assuming your goal was just to finish both and acquire nourishment from them, then your stomach does not care which ones goes down your throat first (of course, some people are picky with this, but we'll disregard that for a moment).
Non-Commutors
Of course, we can't forget the other side of this coin - there are many, many examples of mathematical operations that are non-commutative (i.e. order DOES matter). The most simplest example would be division or subtraction; it is most emphatically untrue that 5/1 is equal to 1/5. We have to be careful of our order in these cases. Other examples of non-commuative operations include cross products (of vectors) and matrix multiplication.
An example of matrix multiplication
Uncertainty Principle
So, what does all of this have to do with the uncertainty principle? Well, it turns out that the commutative and non-commutative properties are also very important in quantum mechanics. According to Werner Heisenberg, if two operators (a fancy word for a symbol that tells you to do something to a number - an example would be the differentiation operator, signified by d/dx) are shown to not commute, then those two variables can NOT be measured simultaneously.
This is due to the fact that order DOES matter when dealing with non-commuters. If you were to attempt to measure the two variables simultaneously, then the order of the measurement isn't really defined - "at the same time" does not classify as an oder of operations. In quantum mechanics, the math shows us that position and linear momentum just do not commute: this is the mathematical reason for why the uncertainty principle exists and why we just have to learn to love with it - you can't beat the math!
Comments
Post a Comment