You have a system that consists of two identical (fair) six-sided dice. Imagine that you will perform an experiment where you roll the pair of dice together and record the observable: the norm of the difference between the values displayed by the two dice.
What are the possible results of the observable for each roll?
What is the theoretical probability of measuring each of those results? Assume the results are fair.
Plot a probability histogram. Use your histogram to make a guess about where the average value is and the standard deviation.
Use your theoretical probabilities to determine a theoretical average value of the observable (the expectation value)? Indicate the expectation value on your histogram.
Use your theoretical probabilities to determine the standard deviation (the uncertainty) of the distribution of possible results. Indicate the uncertainty on your histogram.
Challenge: Use
- Dirac bra-ket notation
- matrices
to represent:
the possible states of the dice after a measurement is made;
the state of the dice when you're shaking them up in your hand;
an operator that represents the norm of the difference of the dice.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.
group Small Group Activity
30 min.
energy conservation mass conservation collision
Groups are asked to analyze the following standard problem:
Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?
face Lecture
30 min.
Bra-Ket Notations Wavefunction Notation Completeness Relations Probability Probability Density
In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.assignment Homework
\(z_1=i\),
assignment Homework