THE SPACE AND TIME OF RELATIVITY
1. Two clocks on opposite ends of a spaceship of length 100m (proper length) are synchronised in the rest frame. The spaceship passes an
observer(or vice versa) at 0.9c. We wish to determine the amount by which the two clocks differ in the observer's frame of reference.
a) What is the ship's length in the observer's frame of reference?
b) If light from a bolt of lightning starts at a point half-way between the clocks, how long would the observer say that it would take the light to reach the 'leading" clock?
c) How long would the light take to reach the 'leading" clock?
d) So, in conclusion, the observer would say that the ___________ clock leads the other clock by__________. (Fill in the blanks.)
2. Use the Binomial Approximation to approximate: a) 983, b) 981/2, c) 311/5. No calculators, please.
3. Two astronauts, Muffy and Biff, set off from Earth in identical, lkm long, spaceships on different assignments. The two return to Earth and pass each other from opposite directions. NASA Control in Houston measures each astronaut as traveling at 0.95c. What is the length of Muffy's spaceship as measured by...
a) NASA Control?
b) Biff?
4. The tail of a spaceship traveling at 2.0´
l08m/s passes you 5m
sec after its front. What is the proper length of the ship?
RELATIVISTIC MECHANICS
1. When a neutron decays, it turns into a proton, an electron, and a massless (as far as we know) particle called an antineutrino. If the neutron began with no kinetic energy, the resulting particles will pick up some energy, because of conversion of mass to energy. The antineutrino being massless will pick up most at the energy. For argument's sake, let's say all of it.
a) Calculate the final energy of the antineutrino.
b) Calculate its momentum (Do NOT use p= g
mu!!).
c)The electron and proton must counteract this momentum. If electron and proton have the same speed, and move in the opposite direction from the antineutrino, what is their speed? (Hint: it will be highly nonrelativistic.)
2. The Milky Way galaxy is about 105 light years across. The most energetic particles in it have energies around 1020eV. Let us assume that a particle with the same mass as an electron has E=1020eV. What is its time-dilation factor, g
? What is the shortest possible half life that this particle can have if it expects to cross the galaxy?
3. The electron in a hydrogen atom is bound by 13.6eV to a proton. By what percentage is the mass of the two separate particles decreased when they join together? How fast would the electron have to travel in order to relativistically make up for that mass difference?
4. A pion is created in the laboratory with a momentum of 268MeV/c and an energy of 300MeV.
a) What is its rest mass?
b) What is g?
c) What is the speed of the pion, as a fraction of c?
5. Calculate the lowest-order correction to the classical relation K =1/2mv2.
6. A proton ('uud', composed of two up quarks and one down quark) has a mass 938.28MeV/c2 and a neutron ('udd') has a mass of 939.57MeV. What should be the mass of the 'u' and the 'd' quark (if there were no force between them). Explain briefly why this answer is no good.
7. Two weightlifters, Hans and Franz, are each lifting one-ton (m=1000kg) weights aboard their separate spacecrafts. Hans says that Franz is lifting a three-ton weight and someone on the ground watching the two says that Franz is lifting a two-ton weight. What does the Earth-bound spectator measure for the weight that Hans is lifting?
8. a) Show that for any 3-dimensional system, the most probable radius of a particle is equivalent to the solution to the following:
The partial derivative of |g(r)| with respect to r equals zero.
b) For the hydrogen atom, for n=3 and l=1, R(r)=(1/3ao)3/2 4Ö
2/3 r/ao [1-r/6ao] exp(-r/3ao).
Calculate the most probable radius. (The Bohr prediction is <r>=9ao.)
9. Imagine a particle which has spin = 2.
a) What are the possible values for the magnitude of the spin angular momentum component parallel to some magnetic field? (Warning: get those units right.)
b) What are the possible angles between the spin vector and the magnetic field?
c) If the magnitude of m is 10-22A .m2 and B is 2T. then what are the possible values of the interaction energy? (You may want to use the results of part b.)
10. Given charts of Y(q
,j) and R(r),
a) Determine y
(r, q
,j) for the 3p electronic state with the largest m, for hydrogen,
b) Find. the most probable radii, remembering that minimizing P(r) is equivalent to minimizing its square root.
c) There are three least probable radii, where P(r)=0. One is r=¥
, where are the others?
11. One can use the quantum defect model mentioned in Chapter 8 (although we didn't talk about it in class) to model the electronic energy levels in a complex, non-hydrogenic atom. In this model, the energy is given as
En=-13.6eV/[n-D(l)]2
where n and l are the quantum numbers and D(l) is a number that depends on l, but not on n. Given that the sodium yellow lines are approximately 589nm, corresponding to the 3p to 3s transition, and given that it takes 5.138eV to ionize the ground state of the outermost electron of sodium (the 3s state), calculate D(0) and D(1) for sodium, and estimate the amount of energy needed to ionize the 2s electron.
12. Given that the pseudowavefunction g(r) is equal to g(r)= r2 exp(-ar), where a= 1/2ao, and given that l=1, calculate the potential energy function and the energy in terms of a, given that the limit of U(r) is 0 as r approaches ¥
.
13. For the hydrogen atom, for n=1, the function R(r) is proportional to exp(-ar), where
a=1/2ao= mke2/(hbar)2
a) Given that r=[x2+y2+z2]1/2 show that the partial derivative of r with respect to x is x/r.
b) Give that the second partial derivative of R(r) with respect to x is -a[1/r-x2/r3-ax2/r2]exp(-ar), show that the Laplacian of R is (a2-2a/r)exp(-ar)
c) Using the Cartesian-coordinate form of Schrödinger's Equation in three dimensions, show that the appropriate wavefunction y
(x,y,z) is a proper wavefunction for U(r)= - ke2/r.
You may set. Y1,0(q
,j
)=1.
(For extra credit. you may show the result in part (b) that we took as given. )
14. Let us imagine that we want to do NMR on the cheap. To save on materials, we will use water, for which the magnetic moment of the proton is m = 2.79mn. To save money on electromagnetic radiation, let's use a 60Hz signal, easily obtained from Niagara Mohawk. What is the size of the magnetic field we would need to induce resonance absorption by the protons? (Note: the Earth's magnetic field is, on average, about 1/2G, and this 60Hz effect has been suggested as the cause of the alleged physiological effects of a.c. electromagnetic radiation.)
15. The Legendre polynomial of l=3 and m=1 is given by Y3,1= -(1/8)[21/p]1/2 sinq (5cos2q-1)eij
.
a) Show that the most probable angles at whch a particle may be found occur when the partial of Y with respect to theta is 0.
b) Solve for the most probable angles, theta. (Hint: convert all cosines into sines first.)
TUNNELING
1. Consider two step barriers, one with a height of Uo and a width of w, the other with a height of Uo/2 and a width of 2w. For an incident particle with energy E<<Uo/2, which barrier will provide a better probability of successful tunneling? Why? You may assume that this is a 'wide' barrier, such that T<<l.
2. Consider three metal cylinders aligned on the same axis. The middle cylinder, 10mm long, is held at -10V and the others are held at 0V. An electron along the axis thus sees a potential energy barrier of 10eV, 10mm long. Calculate the tunneling probability for the electron.
3. "The little train that could":
A little (15000kg) train almost makes it to the top of a hill, but ends up lm vertical distance from the top. If it could tunnel (quantum-mechanically, of course) 20m through the hill, it could continue on its way. What. is the probability of the train making it through on its first try?