ST Probability in classical and quantum mechanics HW-2 c. Suppose that the lengt

Question

ST Probability in classical and quantum mechanics HW-2 c. Suppose that the length of region c were increased. Regions A and Bare unchanged. Would the total amount of time it takes a block to traverse the entire track increase, decrease, or remain the same? If there is not enough information, state so explicitly. Briefly explain. Would the probability density at the center of region A increase, decrease, or remain the same? If there is not enough information state so explicitly. Explain. Would the probability density at the center of region C increase, decrease, or remain the same? If there is not enough information, state so explicitly. (Hint: Consider a small region of unit length near the center of region C) Explain. 2. The graph at right shows the probability density p versus position x for a classical particle. Region A lies between 0 cm and r 3 cm. Region B lies between x 3 cm and x 5 cm, as shown. a. On the graph at right, label the vertical axis the appropriate values. Include units. Explain how you 0 1 2 3 4 5 6 7 determined your answer. Position (cm) b. What is the probability of the particle being in region A? Write your answer in decimal format. Explain and show any work.

Explanation / Answer

Note area under the probability distribution curve (here probability density curve) is always equal to 1.
Which means probability of occurance of any one event out of infinite events should be one

Thus area under the curve is 0.5 * 5 * height of triangle = 1 (area of triangle = 1)
therefore height = 2/5 = 0.4

Probability does not have any unit. So on the curve the top most point of y axis is 0.4

probability of particle in region A = 0.5 * base of triangle * (y cordinate on line at x = 3)
base of triangle = 3
y cordinate on line = 0.24 = 0.4*3/5

So, probability of region A = 0.5*3*0.24 = 0.36

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