physics geeking

[danaeris]

I already looked this up in Griffiths and it was not being helpful. So.

If you know the expected value for x of a one-dimensional, time-independent wave function is x', then is x' the most probable value for x?

My suckitude at probability is one of the reasons why I didn't pursue a career as a physicist. Maybe if I tried retaking probability now, things would be different, but finding out will have to wait until I have time.

Comments

[gorgo.livejournal.com]

Sadly, the answer is no. Simple counterexample: you flip a coin. If the coin comes up heads, the value of a function is 4. If the coin comes up tails, the value is 6. Assuming a fair coin, the expected value of this will be 5 (4/2 + 3/2), but 5 isn't one of the possible values of the random function.

Alternatively, think of a function whose graph is shaped like an "M". The expected value will be near the center of the "M," but the chance that you actually get the expected value as the result is low.

[angelbob.livejournal.com]

This is a better example than mine. The "M" function is pretty close to what I used, but the coin is actually a good comprehensible way to explain it :-)

[angelbob.livejournal.com]

If you're asking what I think you're asking, the answer is no. Do you mean "expected value" in a probability sense? If so, consider a function with two peaks, one at 3 and one at -3, meeting at a value of zero directly in between. Assume that they fall off asymptotically in the negative and positive infinity directions. The peak at 3 is in the positive direction, the peak at -3 is in the negative direction.

If the peaks are of identical magnitude but in opposite directions (that is, if -x(t) equals x(-t)), then the expected value, x', would be 0 because the negatives and positives would cancel out. Yet the chance of actually getting a sample with a value of zero is... nothing. There's only a single point where that's even possible. So it's not the most likely value.

[auros]

This is kinda like the physicist / engineer / statistician joke, where they go deer hunting -- the first misses by three feet to the right, the second misses by three feet to the left, and the statistician thinks that, taken jointly, they hit...