## Activity: Expectation Value and Uncertainty for the Difference of Dice

Quantum Fundamentals 2023 (3 years)

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.

1. What are the possible results of the observable for each roll?

2. 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.

3. Use your theoretical probabilities to determine a theoretical average value of the observable (the expectation value)? Indicate the expectation value on your histogram.

4. Use your theoretical probabilities to determine the standard deviation (the uncertainty) of the distribution of possible results. Indicate the uncertainty on your histogram.

5. Challenge: Use

1. Dirac bra-ket notation
2. 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 Outer Product of a Vector on Itself

group Small Group Activity

30 min.

##### Outer Product of a Vector on Itself
Quantum Fundamentals 2023 (2 years)

Completeness Relations

Students compute the outer product of a vector on itself to product a projection operator. Students discover that projection operators are idempotent (square to themselves) and that a complete set of outer products of an orthonormal basis is the identity (a completeness relation).
• group Visualization of Divergence

group Small Group Activity

30 min.

##### Visualization of Divergence
Vector Calculus II 23 (12 years) Students predict from graphs of simple 2-d vector fields whether the divergence is positive, negative, or zero in various regions of the domain using the definition of the divergence of a vector field at a point: The divergence of a vector field at a point is flux per unit volume through an infinitesimal box surrounding that point. Optionally, students can use a Mathematica notebook to verify their predictions.
• group Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute

group Small Group Activity

30 min.

##### Finding if $S_{x}, \; S_{y}, \; and \; S_{z}$ Commute
Quantum Fundamentals 2023 (3 years)
• group Operators & Functions

group Small Group Activity

30 min.

##### Operators & Functions
Quantum Fundamentals 2023 (3 years) Students are asked to:
• Test to see if one of the given functions is an eigenfunction of the given operator
• See if they can write the functions that are found not to be eigenfunctions as a linear combination of eigenfunctions.
• assignment Gibbs entropy is extensive

assignment Homework

##### Gibbs entropy is extensive
Gibbs entropy Probability Thermal and Statistical Physics 2020

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.

1. Show that the entropy of the combined system $S_{AB}$ is the sum of entropies of the two separate systems considered individually, i.e. $S_{AB} = S_A+S_B$. This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
2. Show that if you have $N$ identical non-interacting systems, their total entropy is $NS_1$ where $S_1$ is the entropy of a single system.

##### Note
In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

• group Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
Theoretical Mechanics (4 years)

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 Compare \& Contrast Kets \& Wavefunctions

face Lecture

30 min.

##### Compare & Contrast Kets & Wavefunctions

Completeness Relations

In this lecture, the instructor guides a discussion about translating between bra-ket notation and wavefunction notation for quantum systems.
• face Quantum Reference Sheet

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces 2023 (6 years)
• assignment Phase

assignment Homework

##### Phase
Complex Numbers Rectangular Form Exponential Form Square of the Norm Overall Phase Quantum Fundamentals 2023 (3 years)
1. For each of the following complex numbers $z$, find $z^2$, $\vert z\vert^2$, and rewrite $z$ in exponential form, i.e. as a magnitude times a complex exponential phase:
• $z_1=i$,

• $z_2=2+2i$,
• $z_3=3-4i$.
2. In quantum mechanics, it turns out that the overall phase for a state does not have any physical significance. Therefore, you will need to become quick at rearranging the phase of various states. For each of the vectors listed below, rewrite the vector as an overall complex phase times a new vector whose first component is real and positive. $\left|D\right\rangle\doteq \begin{pmatrix} 7e^{i\frac{\pi}{6}}\\ 3e^{i\frac{\pi}{2}}\\ -1\\ \end{pmatrix}\\ \left|E\right\rangle\doteq \begin{pmatrix} i\\ 4\\ \end{pmatrix}\\ \left|F\right\rangle\doteq \begin{pmatrix} 2+2i\\ 3-4i\\ \end{pmatrix}$
• assignment Frequency

assignment Homework

##### Frequency
Quantum Mechanics Time Evolution Spin Precession Expectation Value Bohr Frequency Quantum Fundamentals 2023 (3 years) Consider a two-state quantum system with a Hamiltonian $$\hat{H}\doteq \begin{pmatrix} E_1&0\\ 0&E_2 \end{pmatrix}$$ Another physical observable $M$ is described by the operator $$\hat{M}\doteq \begin{pmatrix} 0&c\\ c&0 \end{pmatrix}$$ where $c$ is real and positive. Let the initial state of the system be $\left|{\psi(0)}\right\rangle =\left|{m_1}\right\rangle$, where $\left|{m_1}\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $\hat{M}$. What is the frequency of oscillation of the expectation value of $M$? This frequency is the Bohr frequency.

Author Information
Paul Emigh & Liz Gire
Learning Outcomes