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I've been thinking lately about the many things we usually leave out of intro physics. It's hard to strike the right balance between practical and inspirational, between strong foundations and broad horizons. But I'd like to see that change, at least a little: you won't entice students to love the subject without a glimpse of what makes it beautiful.
Just for fun, let me share an example: the quantum idea that reality is a sum over all possible histories. It goes something like this. When I throw a ball and you catch it, common sense and classical physics agree that the ball follows a specific path through the air.
But quantum mechanics tells a remarkably different story. In this picture, the ball takes every possible path from me to you at once: a straight line, or a high arc, or a zig-zag, or three quick loops around my head before diving between your legs and then up over your head and back down into your hand, or anything else you can imagine. Moreover (and further straining credulity), every single possibility is equally likely! Of course, with so many options, the odds of any given path are basically nil: what we really have to compare are "neighborhoods" of almost-identical paths (those similar enough that we couldn't tell them apart anyway).
Very roughly speaking, quantum mechanics says that every possible path gets to cast an equal vote in favor of its neighborhood. But there's a catch: each path gets a specific "direction" assigned to its vote (think of this like the spinner from a game of Twister). In most neighborhoods those directions go every which way, so when you add up all the votes they pretty much cancel out (just as many lefts as rights, etc.). But there's usually one special neighborhood that votes as a block, with all its vote directions more or less the same. That overwhelmingly probable neighborhood is what we see as the classical path. Common sense emerges from chaos in a truly remarkable way.
[The deeper beauty of this story is that the math of these "vote directions" immediately explains the reason behind the equations of classical physics. The vote direction of a path is precisely its "action" (interpreted as an angle). We learn in calculus that near the minimum of any function its value is nearly constant, so the neighborhood of the path with minimum action will have nearly constant vote direction. And indeed, the "principle of least action" was the basis of the Lagrangian formulation of mechanics long before quantum mechanics was ever imagined, but nobody knew why. Good stuff!]
Just for fun, let me share an example: the quantum idea that reality is a sum over all possible histories. It goes something like this. When I throw a ball and you catch it, common sense and classical physics agree that the ball follows a specific path through the air.
But quantum mechanics tells a remarkably different story. In this picture, the ball takes every possible path from me to you at once: a straight line, or a high arc, or a zig-zag, or three quick loops around my head before diving between your legs and then up over your head and back down into your hand, or anything else you can imagine. Moreover (and further straining credulity), every single possibility is equally likely! Of course, with so many options, the odds of any given path are basically nil: what we really have to compare are "neighborhoods" of almost-identical paths (those similar enough that we couldn't tell them apart anyway).
Very roughly speaking, quantum mechanics says that every possible path gets to cast an equal vote in favor of its neighborhood. But there's a catch: each path gets a specific "direction" assigned to its vote (think of this like the spinner from a game of Twister). In most neighborhoods those directions go every which way, so when you add up all the votes they pretty much cancel out (just as many lefts as rights, etc.). But there's usually one special neighborhood that votes as a block, with all its vote directions more or less the same. That overwhelmingly probable neighborhood is what we see as the classical path. Common sense emerges from chaos in a truly remarkable way.
[The deeper beauty of this story is that the math of these "vote directions" immediately explains the reason behind the equations of classical physics. The vote direction of a path is precisely its "action" (interpreted as an angle). We learn in calculus that near the minimum of any function its value is nearly constant, so the neighborhood of the path with minimum action will have nearly constant vote direction. And indeed, the "principle of least action" was the basis of the Lagrangian formulation of mechanics long before quantum mechanics was ever imagined, but nobody knew why. Good stuff!]
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