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These are interesting questions, and to fully understand them requires a bit of math and physics. Here are some brief explanations, along with links to further reading.

1. The exact minimum value in the Heisenberg uncertainty principle depends on the
system you are studying. From what I have read, Heisenberg's original estimate (based
on crude assumptions about the position uncertainty versus wavelength) was something
more like:

"Delta(x)Delta(p) > h".

This estimate was later refined by a factor of 1/(4pi), giving the hbar/2 result in your
question. The hbar/2 value is valid if the particle wavefunction is a Gaussian curve,
otherwise the minimum uncertainty is greater than this.

There are actually multiple uncertainty relations between different pairs of observables
(momentum-position, energy-time, etc) whose operators do not *commute*
(e.g. AB not equal to BA). This takes a while to explain, so
here is a very well-written chapter (pdf) on non-commuting operators and the
uncertainty principle. It is part of the freely-available online textbook
Energy & Matter: Our
Quantum World by Robert Knop. The whole book is well worth a read.

2. The existence of zero-point energy is directly motivated by the uncertainty principle. If a particle in its lowest energy state was truly at rest, then its position and (zero) momentum would be exactly measurable, so the uncertainty principle forces it to have at least some minimum momentum so that the uncertainty principle is not violated. See section 11.4.2 in the chapter linked above, or this Wikipedia article.

3. Spin 1/2 particles *do* obey the uncertainty principle, but the value of
hbar/2 can be understood in an easier way: If you measure an electron's spin along
some given axis (let's say
*x*), then you can get one of two equal and opposite results, separated by
hbar. That gives two possible values for the spin: +hbar/2 or -hbar/2.

Section 11.3.2 in
Knop's text *does* calculate an hbar/2 uncertainty for the spin, but it is not
quite the same as the more general uncertainty relation for non-discrete quantities
covered in 11.4

I hope this is useful. The uncertainty principle and quantum mechanics are fascinating subjects, but it does take a bit of study before you can really understand what is going on. Knop's book is one of the more readable texts I have found on the subject, and I warmly recommend it. Good luck!

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