PH300 Spring Homework 08

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Damian Hardy
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1 PH300 Spring 2011 Homework 08 Total Points: (1 Point) Each week you should review both your answers and the solutions for the previous week's homework to make sure that you understand all the questions and how to answer them correctly. You will receive credit for reviewing your old homework, which will be returned to you every Tuesday. Please review your homework and the solutions from last week and let me know that it was graded correctly. If it was not, state here which problems were incorrectly graded, and then contact me (via or before/after class). 2. (2 Points) Your first homework question this week is to submit one homework correction from the previous week's homework. Select one problem for which you had the wrong answer, and then: 1. Identify the question number you are correcting. 2. State (copy) your original wrong answer 3. Explain where your original reasoning was incorrect, the correct reasoning for the problem, and how it leads to the right answer. If you got all the answers correct, Great!!! Then state which was your favorite or most useful homework problem and why. 3. (0.5 Points) Which of the following is NOT a possible probability? A. 25/100 B C. 1 D (0.5 Points) Among twenty-five items, nine are defective, six having only minor defects and three having major defects. Determine the probability that an item selected at random has major defects, given that it has defects. A. 1/3 B C D. 0.08

5. (1 Point) ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN. 6 7.

2 5. (1 Point) ABCD is a square. M is the midpoint of BC and N is the midpoint of CD. A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN Two balanced dice are rolled. Let X be the sum of the two dice. Obtain the probability distribution of X (i.e. what are the possible values for X and the probability for obtaining each value?). Check that the probabilities sum to one. 6. (1 Point) What is the probability for obtaining X >= 8? 7. (1 Point) What is the average value of X? Z An atom in the state is shot into the following line of three Stern-Gerlach analyzers. Analyzer A is tilted at an angle of α from the vertical, Analyzer B at +β, and Analyzer C at γ. For these problems α = 15 0, β = 35 0 & γ = (1 Point) What is the probability the atom exits from the plus-channel of Analyzer C? 9. (1 Point) What is the probability the atom exits from the minus-channel of Analyzer C? [Hint: For Questions 8 & 9, you may find it useful to play with the Stern-Gerlach Experiment PhET sim.] 10. (1 Point) Why don t these two probabilities sum to 1?

3 A particular Stern-Gerlach analyzer has three settings, each oriented from the other. During lecture we found the probability for an atom that entered in a definite state of to leave from the plus-channel if the detector setting is random. In this same situation: Z 11. (0.5 Points) What is the probability for an atom to leave the minus-channel if the incoming atom is in the state m Z = +m B? 12. (0.5 Points) What is the probability for an atom to leave the plus-channel if the incoming atom is in the state m = +m B? 13. (0.5 Points) What is the probability for an atom to leave the minus-channel if the incoming atom is in the state m = +m B? 14. (1 Point) Consider the same situation as in Questions 11, 12 & 13, but now settings B and C are oriented at +/- 110 degrees from the vertical (instead of 120 degrees). What is the probability for an atom in the state of spin up along the +z-axis to leave from the plus-channel if the settings are random?

4 15. (0.5 Points) In the experiment depicted below, which of the following best describes the state of an atom that leaves the plus-channel of Analyzer B? A) Z B) X C) Z, X D) X, Z?? 16. (0.5 Points) The function ψ(x) is shown in the graph below: If a probability density function for the position of a particle is given by ρ(x) = ψ(x) 2 [i.e., the probability distribution is equal to the magnitude squared of the function ψ(x)], rank the probabilities of finding the particle in the regions shown. A) P[III] > P[I] > P[II] B) P[II] > P[I] > P[III] C) P[III] > P[II] > P[I] D) P[I] > P[II] > P[III] E) P[II] > P[III] > P[I]

5 17. (1 Point) Atoms leaving the plus-channel of a vertically oriented Stern-Gerlach analyzer are fed into a second analyzer oriented in the +x-direction. With Analyzer 2 oriented at 90 0 to Analyzer 1, either result or X is equally likely. X What is the average value of m X? [<m X > =?] (2 Points Total, 0.5 points each) In an Einstein-Podolsky-Rosen (EPR) experiment, an initial state of an atom pair is represented by initial and various hypothetical final states are shown below. What are the probabilities for observing each of the final states?

6 Questions 22 & 23 refer to the readings: A Quantum Threat to Special Relativity & Is the moon there when nobody looks? Available on Blackboard. 22. (4 Points 1 Point Each) What is meant by the terms realism, locality & completeness? What are some examples of hidden variables? 23. (2 Points) Does entanglement allow for faster-than-light communication? If so, what kind of information can be communicated? If not, why not? Suppose you have a classical particle in a 1-dimensional box, bouncing back and forth between the two walls without friction or other loss of energy. Since the particle is bouncing between the two walls at constant speed, there is an equal likelihood of finding it at any point in the box if we look at some random time. The probability density for the position of this classical particle is therefore constant (a flat line, meaning equal probability everywhere between the two walls): ρ ( x) = A for 0 x L ; ρ ( x) = 0 otherwise where L is the length of the box, and A is some constant with appropriate units.

7 24. (1 Point) What must A be equal to in order for ρ( x) to be normalized? In other words, for what value of A is the normalization condition ρ ( x) dx = 1 satisified? (0.5 Points) Make a sketch of the normalized ρ( x) for 0 x L. 26. (1 Point) Show (using mathematics, not symmetry arguments) that the average value of x is + equal to L/2. In other words, compute x = x ρ( x) dx Suppose instead the probability density for the position of a particle were given by: ρ π x L 2 ( x) = Asin for 0 x L ; ρ ( x) = 0 otherwise 27. (1.5 Points) What must A be equal to in order for ρ ( x) to be normalized? HINT: Use the trigonometric identity 2 1 cos(2 u) sin ( u) = (0.5 Points) Make a sketch of the normalized ρ( x) for 0 x L.

8 29. (1 Point) What is the probability of finding the particle in the left third of the box? (i.e., what P 0 x L/3?) How does this compare with finding a classical particle in the left third of is [ ] the box when the probability density is constant? 30. (2 Points) Show (using mathematics, not symmetry arguments) that the average value of x is equal to L/2. In other words, compute x = x ρ( x) dx. + HINT: Use the same trigonometric identity as in #27, and then integrate by parts: u dv = uv v du 31. (1 Point Extra-Credit) Optional Respond to the weekly feedback survey on Blackboard.

PH300 Spring Homework 07

PH300 Spring Homework 07 PH300 Spring 2011 Homework 07 Total Points: 30 1. (1 Point) Each week you should review both your answers and the solutions for the previous week's homework to make sure that you understand all the questions