Expert Answers

How does \psi(x,t) evolve in time given \psi(x,0) = sqrt(1/6)*\phi_0(x) + sqrt(1/3)*\phi_1(x) + sqrt(1/2)*\phi_2(x) where \phi_i(x) are the i^th eigenstates of a 1D infinite square well that spans for x=0 to x=L

Consider the smooth function \psi(x) = N/2 * ( \tanh( (x + a/2)/b ) - \tanh( (x - a/2)/b ) ), Sketch \psi(x) for various values of b to convince yourself that ?7b(x) well approximates ?7(x) as b ? 0. Argue that, for b « a, N is roughly independent of b. Compute N when b ? 0.

In an experiment performed by Jonsson in 1961 (!!!), electrons were accelerated through a 50kV potential towards two slits separated by a distance d = 2 10?4cm, then detected on a screen D = 35cm beyond the slits. Calculate the electron’s de Broglie wavelength, ?, and the fringe spacing, w

Electrons of momentum p fall normally on a pair of slits separated by a distance d. What is the distance, w, between adjacent maxima of the interference fringe pattern formed on a screen a distance D beyond the slits? note: You may assume that the width of the slits is much less than the electron de Broglie wavelength.

On any given flight, an airline’s goal is to have the plane be as full as possible, without overbooking. If, on average, 10 cancel their tickets, all independently of each other, what is the probability that a particular flight with maximum capacity 300 people will be overbooked if the airline sells 320 tickets? Find an approximate answer.

Your friend challenges you to a coin flipping contest. However you know this friend to be of questionable moral character. In fact you know that he usually caries a weighted coin that comes up heads with probability 0.55, along with a fair coin. You demand that he flip one coin 1000 times, and if it comes up heads more than 525 times, then you will play using the other coin. Assuming he uses the fair coin, find an approximation for the probability that you will think it is the biased coin.

Sam and Pat are playing foosball. When they begin, the score is 0­:0. To make things interesting, if the score ever becomes tied, it is instantly reset to 0:­0. Starting from any score, the probability that Sam gets the next point is 1/3 . The game stops when one player’s score reaches 2. What is the probability that Pat wins?

Fred is giving out samples of dog food. He makes calls door to door, but he leaves a sample (one can) only on those calls for which the door is answered and a dog is in residence. On any call the probability of the door being answered is 3/4, and the probability that any household has a dog is 2/3. Assume that the events “Door answered” and “A dog lives here” are independent and also that the outcomes of all calls are independent. Determine the probability that Fred gives away his first sample on his third call.

Sam and Pat are dating. All of their dates are scheduled to start at 9pm at the Kendall/MIT T stop. Sam always arrives promptly at 9pm. Pat is highly unorganized, and arrives at a time that is uniformly distributed between 8pm and 10pm. Sam gets irritated when Pat is late. Let X be the time in hours between 8pm and the time when Pat arrives. When Pat arrives after 9pm, their date will last for a time that is uniformly distributed between 0 and (3 ? X) hours. When Pat arrives before 9pm, their date will last exactly 3 hours. The date starts at the time that they meet. Sam will end the relationship after the second date on which Pat is late by more than 45 minutes. All dates are independent of any other dates. What is the expected number of hours Pat waits for Sam to arrive?

Four fair 6-sided dice are rolled independently of each other. Let X1 be the sum of the numbers on the first and second dice, and X2 be the sum of the numbers on the third and fourth dice. Convolve the PMFs of the random variables X1 and X2 to find the probability that the outcomes of the four dice rolls sum to 8.