1. (2)(a) Write down the Doppler formula, explaining briefly what it means.

 (1)(b) A spaceship with a clock C on its front is approaching you at a speed 0.8c. If you observe the clock through binoculars, how fast does it appear to be running? (That is, how fast do you see the hands to be moving compared to an identical clock at rest in your frame?) Hint: use (a).

(2)(c) Suppose the approaching spaceship passes in succession two space stations with clocks C₁, C₂ fixed in your frame of reference and 10 light seconds apart. These two clocks are synchronized in your frame. As the spaceship passes C₁, the ship clock C and C₁ both read zero.

As you watch the spaceship through binoculars, what time interval elapses (according to your own watch) between the time you see C to pass C₁ and the time you see C to pass C₂? (Don’t forget the light reaching you from C₁, C₂ takes different times!)

 (1)(d) Using your results from (b), (c) what will clock C actually read as it passes C₂?

 (2)(e) What will C₂ read as C passes it?

(2)(f) Using the above results, explain how, although the clock C appears to be running fast when observed through the binoculars, it is in fact running slow, like all moving clocks. Use (d), (e) to find just how slow.

2. l è ç l

Two masses, each of rest mass m₀, are moving at relativistic speed +v, -v respectively along the x-axis, and crash into each other at the origin, where they stick together forming a single mass. Assume no energy is radiated away in this inelastic collision.

(2)(a) In the center of mass frame S, find the total mass and total momentum both before and after the collision.

(3)(b) Now consider the collision in the frame S' in which the left-hand particle is initially at rest. What is the total mass and total momentum in this frame before the collision?

 (3)(c) Use your result from part (a) to find the total mass and total momentum in frame S' after the collision.

 (2)(d) Is momentum conserved in both frames in this inelastic collision? Explain.

 3. (3)(a) Explain briefly why Rutherford concluded from his a-scattering experiment that atoms have nuclei. What were the essential properties of the nucleus needed to explain his results?

 (3)(b) He used a-particles from polonium, the a's had a speed of 1.6 x 10⁷ meters per second (but ignore relativistic corrections in this question). Assuming the scattering pattern agreed with that predicted by using the Coulomb force, what can you conclude about the size of the gold nucleus (charge = 79)? (Hint: find how close one of these a's could get to that nucleus. The force between two point charges q₁, q₂ a distance r apart is 9x10⁹q₁q₂/r²)

 (2)(c) Would you expect the experiment to work if b rays had been used instead of a's? Explain why or why not.

 (2)(d) Would you expect the experiment to work if g rays had been used instead of a's? Explain why or why not.

 4. In Thomson's model atom, assuming the electron stays inside the atom, it feels a force of magnitude Kr directed towards the center of the atom.

(2)(a) Write down the equation of motion for an electron in a circular orbit inside a Thomson atom.

(3)(b) Using Bohr's quantization condition, find the radii of the allowed circular orbits (in terms of the constant K, etc.).

(2)(c) Find the frequency of an emitted photon (in terms of K) for the n = 2 to n = 1 and the n = 3 to n = 2 transitions.

(3)(d) Assume now that Thomson's model represents a hydrogen atom, and has a diameter 10^-10 meters. Evaluate K, and hence of the size of the predicted smallest Bohr orbit. (Note: the force between two point charges q₁, q₂ a distance r apart is 9x10⁹q₁q₂/r²). Comment on your result—is the model consistent?

5. (4)(a) Draw a potential well which has an infinitely high wall at x = 0 and a wall of height D at x = L. Assume D is such that there are only two bound states in the well.

Sketch the wavefunctions for the two bound states. Do you expect any differences in how far outside the well the two wavefunctions will tunnel?

(2)(b) Sketch a wavefunction for a particle in this potential with energy greater than D. Make clear how the wavelength and amplitude vary from inside the well to outside the well.

(4)(c) Assume this potential is a model for binding an electron to a surface (so this is the potential the electron sees in the direction perpendicular to the surface). If we take L to be 4 x 10^-10 meters, come up with some estimate of D given that there are only two bound states.

6. Give a brief explanation of each of the following:

(2)(a) What is the relevance of Planck's constant to the operation of an x-ray tube?

 (2)(b) Elementary particles with the same quantum numbers are indistinguishable. How is that related to the exclusion principle?

 (2)(c) Why do we never find elements with really large nuclei, such as Z = 120, in nature?

 (2)(d) For nuclei that decay by emitting a - particles, state very approximately the lifetimes of the longest lived and the shortest lived. Why is the range so great?

 (2)(e) Why do specific heats of solids decrease at low temperatures?