In Part 1 we looked at classical point particles interacting gravitationally. We saw they could convert an infinite amount of potential energy into kinetic energy in a finite time! Then we switched to electromagnetism, and went a bit beyond traditional Newtonian mechanics: in Part 2 we threw quantum mechanics into the mix, and in Part 3 we threw in special relativity. Briefly, quantum mechanics made things better, but special relativity made things worse.

Now let’s throw in *both!*

Great. If you reread a bit with this in mind, it’ll make more sense.

Classically a positive and negative charged particle can be stationary and as close as you like. Their kinetic energy will be zero and their potential energy will be whatever negative number you desire.

Quantum mechanically, they can’t be very close (with a high probability) while still moving very slow (with a high probability).

]]>Aha. Thanks much John, this helps a lot.

]]>arch1 wrote:

The first of those sentences concerns only the

potentialenergy part, right?

No.

In the classical theory of particles interacting by an attractive 1/r potential, the total energy—potential plus kinetic—is not bounded below. In the quantum theory it is, thanks to the uncertainty principle.

This makes a big difference! This is how the quantum theory avoids the horrible problems that afflict the classical theory.

]]>The first of those sentences concerns only the *potential* energy part, right? Which suggests that the second sentence does, too. But based on the rest of your discussion, I don’t see anything implying that in the quantum case, the (expected value of) *potential* energy is in fact bounded below.

Rather, it seems that in both the classical and quantum cases, the PE is *not* bounded below, but it *is* balanced by the KE (which arises from motion in the classical case and from momentum uncertainty in the quantum case). Which would seem to imply that in the quantum case, as in the classical case, the (expected value of) KE could get arbitrarily large.

Where did I go off the tracks?

]]>Wouldn’t it be more accurate to replace the word “known” with “defined”?”

Both are well defined, but the information about their simultaneous position and momentum is only approximate, within the limits set by the Uncertainty Principle. “Known” here refers to experimental observation of the particle or system, “defined” describes the modelling of that system using theoretical descriptions. At least that’s how I’ve understood it. And “known” is limited by the accuracy of the observation – instrumentation and the like, whereas “defined” seems far more concrete.

]]>(To be clear, I always realized you couldn’t possibly have meant “known by a subsystem”; I was just quibbling about how best to describe the uncertainty principle, and about that I do think “know” suggests incorrectly that there is “a way things really are” which is not subject to the same tradeoff in precision. So for *me*, defined is “blander”, but I have no idea whether that’s true for most readers.)

(At least to me, “known” makes me imagine that there is one portion of the system which knows something about another portion — not what you meant! But I doubt I’m very representative.)

]]>You can take either an ontic (“is”) or epistemic (“is known”) stance to the wavefunction. As a quantum subjective Bayesian, I tend to think of the state of a quantum system as serving the same basic role as a probability distribution in probability theory, namely as a summary of our beliefs about what’s going on. But I don’t think there’s a way things “really are”, standing over and above what we know.

Needless to say, these issues don’t matter for what I’m talking about here. So, I should try to use whatever language is the least likely to get people interested in the issue you just raised!

Thus, accuracy is less important to me here than blandness. It’s possible that “defined” will be better in this respect, since it doesn’t bring up any issues of “subjectivity”. Thanks. I’m going to publish this series of blog articles as a paper, but there’s still time to make changes.

]]>In particular, this difference must be accurately known! Thus, by the uncertainty principle, their difference in momentum must be very poorly known ….

Wouldn’t it be more accurate to replace the word “known” with “defined”?

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