1) Write w as a function of k and differentiate. Re-write the derivative in terms of p (substituting for k), and use the relativistic form of momentum to get everything in terms of c and v. Should work itself out.
2) The dropped ball will have some random horizontal velocity due to uncertainty, which gives a spread in x. Get the momentum uncertainty from this initial uncertainty in x, which gives you the velocity uncertainty. The final uncertainty in horizontal position is then governed by normal mechanics - final position is initial position plus velocity times the time required to fall a distance H. Once you have an expression for the final spread in x, differentiate with respect to initial uncertainty to minimize. We'll go over this in class, it is sneaky.
3) Plug and chug. Momentum is (gamma)mv, plug that in de Broglie. Simplify, and define the Compton wavelength to be h/mc.
4) The uncertainty relationship gets you momentum uncertainty from a given position uncertainty. Best-case scenario: the position has its minimum uncertainty value (Delta)x. Get E in terms of x, differentiate, and plug the extremal x value back into your energy equation.
5) There is an extremely good chance that these lines correspond to the emission spectrum of a simple element. If you figure out which one, subsequently figuring out all the combination is far easier.
6) The de Broglie wavelength is h/mv. The average velocity from kinetic theory is:
\langle v \rangle = \sqrt{\frac{3k_BT}{m}}There go you. Note that the mass of a helium atom is about 4 atomic mass units. Also note that this is a problem from your text, i.e., the numerical answer is in the back of the text.7) Huh. Nearly what we did in lecture :-) You may also find the MIT course 8.04 Open Course Ware (OCW) interesting. What's different about the helium ion?
8) We'll go over this one in class. Again related to the MIT 8.04 OCW.
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