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It was mentioned in lecture that the Free Particle is an interesting problem that was ignored for a long time by physicists (if I heard correctly); does this have a conceptual basis aside from the odd mathematics involved? I'm curious because there really can't be a “free particle” anywhere in the universe; wherever it is, it is still under the influence of all kinds of potentials (though they may be very small).

I think the idea of the “free particle” is that we are assuming (allowing; for theoretical purposes) that is is in fact “free” from all external forces. We are only assuming this so we can analyze such a case (whether or not it is possible).

Just curious as to why, for the harmonic oscillator, we have decided to make it customary to number the states beginning with n = 0 instead of n = 1?

As the professor said during class, <math>E=\(n + \frac{1}{2}\)\hbar\omega</math> starting from 0 makes it more clear what the ground state energy is than <math>E=\(n - \frac{1}{2}\)\hbar\omega</math> starting at n = 1.

This may be a little ahead, but on pg 98, the book discusses the indeterminacy of quantum mechanics. It says, “Ordinarily, when you measure an observable Q on an ensemble of identically prepared systems,all in the same state Psi, you do not get the same result each time. This is the indeterminacy of quantum mechanics.” Is it really valid scientific reasoning to assume that the cause of the indeterminacy is unknowable? Of course we have to operate within the boundaries of available knowledge, but doesn't labeling a certain aspect of quantum mechanical behavior “indeterminable” block or attempt to block further inquiry? Why even do this? If one studied humans a particles in this way, one could come to a similar conclusion, and might not ever find the source of behavior: the brain and its chemical reactions. Why not just say “we don't know what causes this uncertainty yet, but we will find out!

“Indeterminable” and “indeterminate” are two very different things. If something is indeterminable, it has a correct answer, but we can't figure it out. If something is indeterminate, there is no correct answer. Pages 3 and 4 talk about this some. Scientists have done what you describe; experiments involving the spins of particles generated from the decay of a 0-spin particle suggest that particles in the realm of quantum mechanics have “indeterminate” properties, not “indeterminable.” There are still competing hypotheses (such as the “global hidden variable” hypothesis referred to in the footnote on page 4).

I don't think anyone is suggesting that humanity give up on finding some reason for characteristics of particles to be indeterminate (maybe look into the source of the Schrodinger equation?), but (a) to make progress, we have to accept propositions at least as suppositions, and (b) no matter how much we ask “why are things like this?”, eventually, the answer will have to be “they just are.” Eventually, there won't be another answer. This isn't to suggest that we should stop searching; I mean that a statement that it looks like things “just are” has never stopped scientists from continuing to ask “why;” things have always seemed like they “just were.” The fact that it seems like particles' location and momentum are indeterminate doesn't mean we won't keep exploring; people all over the world conduct research in the field of quantum mechanics.

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