Final Examination

 

Physics 2424

 

23 April 2003

 

 

Instructions

 

 

 

 

§         Please answer all the questions on this test.

 

§         All questions are worth 10 marks each except where noted.

 

§         Show all your work.

 

§         You may use your textbook for reference.

 

§         You may use your prepared spreadsheets

 

§         Time to do the test is three hours maximum.

 

§         If you have any questions, remain seated and raise your hand.

 

§         Please start each question on a new page in your booklet.

 

1.         What would the Periodic Table look like if the energies levels shown in Fig. 7-19 of the text followed the simpler pattern 1s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5s, … Include a sketch of your periodic table with the columns labelled as in the normal table on the back cover of your text.

 

2.         Neutrons are 1839 times as massive as an electron. Consider a 1D finite square well with V0 = 2.00 eV and L = 0.05 nm. What are the lowest and highest energy levels in this well for neutrons? How many neutrons could be held in this well? Explain.

 

3.         Consider a cube of material. It has mass m and side length L. Derive a formula for its density as it moves away with speed v if;

(a) one face is perpendicular to the motion?

(b) one edge is perpendicular to the motion?

The diagram shows the two cases as if one is looking down from above. Recall that the area of a triangle is A = ½bh.

 

 

 

4.         We followed a simple procedure for getting the wavefunction in potential wells, barriers, and steps. Consider the potential well in the diagram below. It has four regions.

a) Write down the general expression for y(x) in each region.

b) Write down the set of constraint equations that we get from continuity.

You do not need to solve these!

 

 

 

5.         Consider the hydrogen atom wavefunction y200(r, q, f). Find <r>. The integral identity  is helpful.

 

6.         Consider the hydrogen-like atom formed by a proton and the negatively charged particle the muon. The muon is much like an electron in that it has the same charge but the mass of the muon is 105.45 MeV/c2.

            a)   What is the reduced mass m?

            b)   What will be the energy of the first level of this atom? Recall E1 = 13.6 eV for hydrogen.

            c)   When a hydrogen in placed in a magnetic field B, the observed splitting between the m = +1 and m = -1 orbitals due to the Zeeman Effect is 4.0 ´ 10-5 eV. What splitting would you expect for a muonic atom placed in the same field?

 

7.         Consider the nuclear reaction where a neutron hits a 24Mg nucleus producing

 

n + 24Mg → 23Na + nX .

 

            (a) What is nX?

(b) Can this reaction occur if the neutron is moving very slowly and the 24Mg is at rest? Explain.

(c) If the neutron needs to be moving for this reaction to occur, how fast must it be moving?

 

8.         An astronaut shines a beam of light through a solid transparent tube of proper length L. The tube is made of a material with index of refraction n, meaning that the speed of light in the material is v = c/n.

            (a) How long will it take the light to get through the tube in the astronaut’s frame of reference?

            Now the tube and astronaut are in a spaceship moving forward with speed bc in your frame of reference. At the instant the light enters the tube, the back of the tube is at the origin in both frames.

            (b) Carefully sketch a spacetime diagram showing the wordline of the light while in the tube and of the ends of the tube itself.

            (c) Determine the speed of the light while in the tube according to your frame of reference.

            (d)  Determine a formula for the index of refraction of light in a moving tube.

 

9.         A spaceship A leaves earth at vA = 0.4 c to the right. At the same instant, in earth’s frame, a second spaceship B leaves a spacestation and heads to earth at vB = 0.6 c. The spacestation is 100 lighthours away to the left. Every hour, in its frame, spaceship B send a radio signal toward earth and the departing spaceship.

            (a) Determine the frequency of received signals as seen by an earth observer.

            (b) Determine the frequency of received signals as seen by an observer on spaceship A.

            (c) Determine the frequency of received signals at spaceship A as seen by an earth observer.

            (c) Produce an accurate* spacetime diagram for this problem including several of the radio signals.

            *You may use your spreadsheet to produce the diagram and add extra lines or comments by hand. Make sure your name is on the printout!

 

 

10.       Sketch the n = 15 wavefunction for the potential well below. Explain the features of your sketch.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Useful Constants

 

1 u = 931.49401 MeV/c2         c = 2.99792458 ´ 108 m/s       h = 4.135667 ´ 10-15 eV×s.

 

me = 0.5119989 MeV/c2