Lecture20 - Physics 344 Foundations of 21st Century...

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Physics 344 Foundations of 21 st Century Physics: Relativistic and Quantum Systems Instructor: Dr. Mark Haugan Office: PHYS 282 haugan@purdue.edu TA: Dan Hartzler Office: PHYS 7 dhartzle@purdue.edu Grader: Fan Chen Office: PHYS 222 chen926@purdue.edu Office Hours: If you have questions, just email us to make an appointment. We enjoy talking about physics! Help Session: Thursdays 2:00 – 4:00 in PHYS 154 Exam 1 average score 70
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The Decay of Unstable Quantum Systems We’ve been considering processes like the beta decay of a radioactive nucleus as a first example of a quantum mechanical process. At a more fundamental level, this process is due to the decay of a neutron in the nucleus via the reaction, e n p e ν + + In the decay of a radioactive nucleus by this process, the electron and the anti-neutrino escape while the proton remains in the nucleus, now that of a different element than before the decay. The realization that chemical elements are not immutable was one of the many striking consequences of the discovery of radioactivity. Question for later – Why do all of the neutrons in every nucleus not decay in this way? Since we realized that we could explain the exponential decay of the average activity of a radioactive source by modeling such decays as independent random processes, like the flip of a coin, we’ve been discussing additional ideas about probability and statistics so that we can understand the activity fluctuations predicted by this model as well as the behavior of the average activity.
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The Central Limit Theorem: Sample Average and Sample Variance The flip of a coin is a familiar process that has two possible outcomes, like the decay or not of an unstable system. So, we chose to use the context of coin flipping to illustrate relevant aspects of probability theory in ways that we can easily apply to quantum systems. We discussed a statistic s N , the number of heads to come up when we flip N coins. [ This is analogous to the number of decays, n N , during a fixed time period, t , when we have a source consisting of N active nuclei. Each passing of a time period t is the analogue of a flip. In our decay model, t is short enough that the probability of any single nucleus decaying is p = λ ∆ t << 1. ] The individual random process is a single coin flip. When we flip N coins we have N independent trials (samples) of the process. Here are a couple of samples of s 16 , from flipping 16 coins.
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In terms of s N , our intuition is that if we flip more coins, i.e., increase, N , we are more likely to have nearly equal numbers of heads and tails and, thus, a sample value of s N close to the value N /2. This means that if we divide the sample value of s N by N to obtain a sample average, it is more likely to be close to the single-flip expected value, < s 1 >. Our definition of s 1 is the number of heads that come up when we flip one coin, so, < s 1 > is simply the probability of getting heads, p H = p 1 . The sample averages of our
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This note was uploaded on 12/09/2011 for the course PHYS 344 taught by Professor Garfinkel during the Fall '08 term at Purdue University-West Lafayette.

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Lecture20 - Physics 344 Foundations of 21st Century...

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