Quantum Bouncing Ball, Concluded

Well, last time we were looking about the classical probability density for a bouncing ball, and the quantum mechanical probability distribution for the same. They looked not even a little bit alike. This is kind of a problem, since we know from experience that classical physics works pretty well, and from more modern work we know quantum mechanics works pretty well also.

However, despite their apparent inconsistency it turns out that we're just not looking close enough. The quantum nature of things is usually (though not quite always) apparent when the energy level of the system in question is similar in size to the size of the quantum jumps. Most of the time these quantum jumps are very, very small in comparison to the energy of macroscopic objects such as, say, a dropped baseball. For instance, the ground state of a baseball in the 9.8 m/s^2 gravitational field of the earth is something like 0.0006 eV, which is just tiny. A baseball dropped from 1 meter will have an energy of around 1.4 joules, which is about a whopping 10^22 times larger. It's really no shock that the quantum nature of the ball isn't visible - assuming that quantum mechanics really does look like classical mechanics for large quantum numbers.

Does it? Here's the n = 1 probability distribution compared to the classical one (axis scales are arbitrary [Though see discussion in comments - Matt]):

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

They don't look much alike. In fact, there's a finite probability to find the particle actually higher than the classical limit imposed by E = mgh. Try the n = 5:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

Well, they still don't look much alike. There's still some portion of the quantum distribution above the classical turnng point, but less. And if you squint, the lumps in the quantum distribution look sort of like the smooth classical one if you take an average.

But we're at small quantum numbers yet. Try n = 50. The portion that sticks out too far is much smaller in comparison, and those bumps are looking pretty hard to distinguish from the smooth classical curve.

i-d18063683dcc0d42b9be45451a84d1e3-3.png

Extend this to n = 10^22 or so and it's starts to look pretty plausible that the classical and quantum distributions are indistinguishable to the naked eye. This is called the correspondence principle, which basically says that the quantum mechanical answer will arbitrarily approach the classical answer for large quantum numbers. As it happens, that principle doesn't quite save us all time time. There are quantum systems where large quantum numbers are not really possible - the spin of silver atoms coming out of a Stern-Gerlach device, for instance. But that's usually not to much of an issue.

More like this

All right, I'm gonna delay the next installment of the quantum bouncing ball for a brief diversion. I have a friend who's also a physics grad student, and he suggested that we along with a few other fellow students form a yet-to-be-named unofficial club whose raison d'etre is to get together every…
Right now I'm on the 4th floor of the physics building. If I walk down the hall to the balcony overlooking the atrium, I could drop a bouncy ball and watch its trajectory. It'll fall to the ground, bounce up to some fraction of its initial height, and repeat the process with a loss of energy each…
The Exponential Function. I think this is the first time we've done it here. It won't be the last. You could write a book about it, and someone probably has. Here's the usual picture: This graph isn't as pretty as the usual, because I'm at home with my old copy of Mathematica 5 instead of the…
Today we're going to have to do some groundwork to set up for a step we're going to need a post or two from now. It has to do with density in a slightly more abstract context than usual. Imagine you want to know how many people are living in a particular region. You can multiply the population…

How come you don't label your axes? I have no idea what the graphs are supposed to represent.

By Fred Cucumber (not verified) on 14 Jan 2011 #permalink

let's not point Matt's wife at xkcd...

By Nomen Nescio (not verified) on 14 Jan 2011 #permalink

Nice demonstration of why newton works at the scales of ordinary life. Or in other words why a Basketball player does not have to worry about quantum mechanics when making a shot.

They are clearly labeled just above the first graph as totally arbitrary. ;)

Actually they're not though, the x-axis is in units of (hbar^2 / (2gm^2))^(1/3). For a baseball this is about 1.4e-23 meters - less than a millionth of the diameter of a proton. Y-axis is normalized probability/length, with the same length units.

none of this makes any sense

By Marzie Cotts (not verified) on 11 Oct 2011 #permalink

can some1 help me read this??

By Marzie Cotts (not verified) on 11 Oct 2011 #permalink

ok, i'm leaving now! this is a stupid site.

By Marzie Cotts (not verified) on 11 Oct 2011 #permalink

Nice demonstration of why newton works at the scales of ordinary life. Or in other words why a Basketball player does not have to worry about quantum mechanics when making a shot.