JPU200Y - Final Exam
May, 2000
Answers, Marking Scheme and Results

SHORT ANSWER (60 points total)

Each question is worth 4 points.

Questions 1 and 2 refer to the following figure and explanation.

Although not drawn to scale, we assume that relative to the Earth, the Planet of the Apes and Alpha Centauri are the same distance of 4.35 light years away but in opposite directions. Assume that the Planet of the Apes and Alpha Centauri are both at rest relative to the Earth. Assume that the theories of relativity are correct. Sue and Lou are twins, born at the same time and therefore are the same age. Sue blasts off for Alpha Centauri at one-half the speed of light relative to us on Earth. When she arrives at Alpha Centauri she lands on one of its planets for a rest. Lou stays on Earth for ten years. He then radios Sue that he is going to go to the Planet of the Apes, and invites Sue to join him. He blasts off for the Planet of the Apes at half the speed of light relative to us on Earth; when he arrives at the Planet of the Apes he lands. When Sue receives Lou's invitation she blasts off for the Planet of the Apes at half the speed of light relative to us on Earth; when she arrives at the Planet of the Apes she lands, and has a nice reunion with her brother Lou.

1. Draw a spacetime diagram relative to us on the Earth including Sue's worldline for her trip to Alpha Centauri, resting on a planet of Alpha Centauri, and then travelling to and landing on the Planet of the Apes. Include the worldline of Lou waiting on Earth and then travelling to and landing on the Planet of the Apes. Include the worldline of Lou's radio message to Sue inviting her to join him. Be sure to clearly label the axes and worldlines.
   
   

   Sue and Lou's worldlines not clearly for speeds < c: -1 point
   Signal from Lou to Sue not clearly at the speed of light: -1 point

2. When they rejoin on the Planet of the Apes, which twin is younger? Describe how you arrived at this answer.

   Sue is the younger twin.
   Correct answer: 2 marks only
   Any correct explanation: +2 marks. One hopefully popular answer will be "Because Sue's worldline is longer than Lou's."

3. A common form of the Second Law of Thermodynamics is: Heat never spontaneously flows from a colder body to a hotter body. However, this common form is not quite correct. What is wrong with it?

   The word "never" should be replaced by "almost never", "usually" or similar.
   Answering that the term "closed system" is misssing: 2 marks only since it is implied by the word "spontaneously."

4. For a black hole, the curvature of spacetime at the event horizon is so intense that not even light from inside the horizon can escape to outside the horizon. And yet Hawking and Bekenstein predict that black holes do radiate away their energy. What mechanism, believed to occur everywhere in the world all the time including just outside the event horizon, explains how a black hole can radiate energy?

   Virtual pair production
   "Pair production" only: 3 marks only
   "Virtual electron-positron production" only: 3 marks only

5. The Feigenbaum number is irrational and has an approximate value of 4.6692016090.... What does the number describe? Three sentences maximum please.

   It was originally discovered as the ratio of increases in the free parameter of the logistic map for successive bifurcations just before the onset of chaos.
   Identifying that it has to do with chaos: 1 mark.
   Identifying that is has to do with bifurcation: 1 mark.
   Identifying that the bifurcations are infinite: 1 mark.

6. All gases share the behavior that if a fixed quantity of the gas at constant volume has its pressure and temperature varied, the data will lie roughly on a straight line as shown. Further, if the line is extrapolated back to zero pressure, the intercept of the line with the temperature axis occurs at the same temperature value for all gases; the value of this temperature is about -273.16 Celsius. What is this value of the temperature called?

   Absolute zero
   "Absolute temperature"only : 2 marks

7. Multiple Choice: Balmer's formula for the wavelengths of the lines of the Hydrogen spectrum is shown to the right. Balmer came up with this formula by: A: Applying the Schrödinger equation to the Hydrogen atom. B: Working out the mathematics of vibrating organ pipes and applying that mathematics to the spectrum. C: An exercise in pure numerology. D: Realising that electromagnetic theory requires this equation. The correct answer will be awarded four marks. Incorrect answers will have one mark subtracted. Blank answers will be ignored.

   Answer is C: An exercise in pure numerology

8. The planetary model of the atom has the electrons in orbit around the nucleus just as the planets are in orbit around the sun. However, according to classical physics this model is impossible. Why?

   Because in this model the electrons are accelerating and therefore radiate away energy, causing them to spiral into the nucleus.
   "Radiate away energy" only: 2 marks only
   Failure to mention acceleration or radiation: 3 marks only
   Spiraling into nucleus without any mechanism: 1 mark only

9. One arrangement of the Mach-Zehnder interferometer is shown to the right. What percentage of the photons that leave the light source end up in Detector 2?

   25%
   Accept "1/4" for full marks also.

10. We have stated that no signal, information, or influence travelling faster than the speed of light and that connects two regions of spacetime, say A and B, can be causal. By this we mean that we cannot have some event at A cause an effect at B via such a superluminal connection. Why is this so?

    Because for any such superluminal signal two different inertial observers will disagree on which direction the signal is travelling, so the "cause" and "effect" cannot be distinguished.
    A mention of locality was worth 1 mark.

11. Electrons from an electron gun are passed though 10 successive spin filters, where each filter is oriented at 90o relative to the previous filter. What fraction of the electrons from the electron gun emerge from the tenth filter?

    (1/2)¹⁰
    "(1/2)⁹" : 3 marks only
    "50% of those that passed the ninth filter": 1 mark only
    

12. Stapp has characterised one particular discovery as the most important one in the entire history of science. David Harrison stated that he agrees with Stapp. To what discovery is Stapp referring?

    Bell's Theorem
    "Bell's Inequality": 4 marks

13. In our everyday life we believe that the following common sense assumptions are true:
    » There exists a reality separate from its observation.
    » No signal can travel faster than the speed of light.
    » Logic is a valid way to reason.
    However, recent experimental tests have given us further information on these assumptions. What information have the experiments given us?

    That one or more of the assumptions is wrong.
    "One of the assumptions is wrong": 2 marks only
    "All of the assumptions are wrong": 1 mark only

14. We have learned that a field, such as an electromagnetic field, has a complementary particle aspect, which is the photon in this example. The force that holds the atomic nucleus together is called the strong field. What is the name of the complementary particle aspect of the strong field?

    Meson or Pi Meson

15. The standard view of Elementary Particle Physics is that stuff is made of atoms, atoms are made of electrons and nuclei, nuclei are made of protons and neutrons, and protons and neutrons are made of quarks. In this view what are quarks made of?

    Nothing

LONG ANSWER (40 Points Total)

Choose two of the topics below and write about them in the supplied examination book. Be sure to clearly indicate which topic is being discussed. Each topic will be marked out of 20 points.

Answers will not be marked on the basis of length. In fact, excessive verbosity tends to make markers irritable. Illegibility has a similar negative effect on markers.

Note: the notes following the questions below are only partially a marking scheme. They are also some key points I looked for when marking the questions.

1. The direction of time's arrow distinguishes the past from the future. What are some possible causes that could determine the direction of time's arrow? Conclude by discussing any possible relationships between these causes.
   Thermodynamics, Expanding Universe and Consciousness. Naming and describing: up to 9 marks.
   Thermodynamics and Expanding Universe may be connected via an analysis similarly to the Hawking-Bekenstein black hole thermodynamics,
   A speculation by David Harrison may connect Consciousness to these.
2. All chaotic systems share some universal properties. Name and describe as many of these properties as you can. Conclude by discussing how the word chaos used in describing these systems compares to our everyday use of the same word.
   Sensitive Dependence on Initial Conditions; trajectory never repeats; transition preceded by infinite bifurcations; bifurcations characterised by the Feigenbaum number; non-integer fractal dimensionality: up to 10 marks.
   Discussing Lyapunov plots, return maps, etc. a bonus of up to 2 marks.
3. Heisenberg wrote about Schrödinger's formulation of Quantum Mechanics: "The more I ponder the physical part of Schrðdinger's theory, the more disgusting it appears to me." Schrödinger wrote about Heisenberg's formulation of Quantum Mechanics: "If one has to stick to this damned quantum jumping, then I regret ever having been involved in this thing." Compare and contrast these two formulations of Quantum Mechanics. Conclude by discussing how these two giants could have held such strong views when they both knew that the two formulations are formally identical.
   Schrödinger: continuous wave function; Heisenberg: discontinuous matrix purely in terms of observables: up to 12 marks
4. Popular interpretations of Quantum Mechanics include the simple probabilistic one of Born, the so-called Many-Worlds interpretation, and the Copenhagen Interpretation. Briefly describe these three interpretations. Conclude by discussing how, despite the well-known existence of these interpretations, Nobel laureate Richard Feynman (and David Harrison) can claim to not understand Quantum Mechanics.
   Born: a wave function related to but not equal to the probablility; represents partly a fact and partly our knowledge of the fact: up to 4 marks.
   Many Worlds: the "collapse" of the wave function is actually a splitting of the universe; the collapse is really an illusion; the totality is all these worlds: up to 4 marks.
   Copenhagen: from Complementarity; there is no quantum world, the observer and the observed can not be separated: up to 4 marks.
5. In exploring the idea of non-local hidden variables, David Bohm and his school proposed that there are two ordering principles in the universe: the everyday explicate one and a more unusual implicate one. Describe, compare, and contrast these two orders and how they interact with each other.
   Explicate: space, time, parts make up the whole, causal, things, describable: up to 6 marks.
   Impliciate: holographic, whole makes up the parts, acausal and non-local, interference between things and no-thing: up to 6 marks.
   The process is one of enfolding and unfolding between the orders.

Some Results of the Exam

Short Answer

Mean: 62.4%. Standard Deviation: 20.5%. Quartiles: 50%, 63%, 80%.

Long Answer

The figure to the right shows boxplots of the results of the five long answer questions. Briefly, the "waist" of the box is the median, the "shoulders" and "hips" are the first and third quartiles, and the "whiskers" extend to the highest and lowest marks. For all the long answer questions, the Mean was 49.35, the Standard Deviation was 24.2%, and the Quartiles were 35%, 50%, 65%

Total Mark

The total "raw" exam marks had a Mean of 56.8%, Standard Deviation 19.8% and Quartiles 46%, 57%, 72%. The marks were then re-scaled to U of T Marks with the following breakpoints:

The total U of T exam marks had a Mean of 68.2%, Standard Deviation 12.2% and Quartiles 63%, 68%, 76%. The results by letter grade were: