Activities and Homework Problems

Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.

Students move their left arm in a circle to trace out the complex plane (Argand diagram). They then explore the rectangular and exponential representations of complex numbers by using their left arm to show given complex numbers on the complex plane. Finally they enact multiplication of complex numbers in exponential form and complex conjugation.

Students, working in pairs, represent two component complex vectors with their left arms. Through a short series of instructor led prompts, students move their left arms to show how various linear transformations affect each complex component.

The question is meant to get you used to using the Canvas Discussion Board. Please go to the course Canvas page and find the Discussions tab on the left hand side. Find the Discussion titled Random and add one of the following:

1. A random physics fact.
2. One thing you like about physics.
3. One question you have for Liz.

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• How to represent 3-d scalar fields in several different ways;
• The symmetries of a some simple charge distributions such as a dipole and a quadrupole.

Students use Tinker Toys to represent each component in a two-state quantum spin system in all three standard bases (\(x\), \(y\), and \(z\)). Through a short series of instructor-led prompts, students explore the difference between overall phase (which does NOT change the state of the system) and relative phase (which does change the state of the system). This activity is optional in the Arms Sequence Arms Sequence for Complex Numbers and Quantum States.

Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that \(\Delta V \approx V_g\). Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.

1. Solve for \(\frac{dp}{dT}\) in terms of the pressure of the vapor and the latent heat \(L\) and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

• Recall the relationship between the sign of the dot product and the orientation of the vectors.
• Use graphical methods to estimate the value of a vector line integral.
• Lay the groundwork for thinking about conservative and non-conservative vector fields.

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Sketch each of the vector fields below.

1. \(\boldsymbol{\vec F} =-y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)
2. \(\boldsymbol{\vec G} = x\,\boldsymbol{\hat x} + y\,\boldsymbol{\hat y}\)
3. \(\boldsymbol{\vec H} = y\,\boldsymbol{\hat x} + x\,\boldsymbol{\hat y}\)

Students use \(d\boldsymbol{\vec{A} }= d\boldsymbol{\vec{r}}_1 \times d\boldsymbol{\vec{r}}_2\) and \(d\tau=(d\boldsymbol{\vec{r}}_1\times d\boldsymbol{\vec{r}}_2)\cdot d\boldsymbol{\vec{r}}_3\) to find differential surface and volume elements for cylinders and spheres.

Use geometry to find formulas for velocity and acceleration in polar coordinates.

Students use prepared Sage code to predict the gradient from contour graphs of 2D scalar fields.

• A component of the curl of a vector field (at a point) is the circulation per unit area around an infinitesimal loop.
• How to predict the sign and relative magnitude of the curl from graphs of a vector field.
• (Optional) How to calculate the curl of a vector field using computer algebra.

• Divergence of a vector field (at a point) is the flux per unit volume through an infinitesimal box.
• How to predict the sign and relative magnitude of the divergence from graphs of a vector field.
• (Optional) How to calculate the divergence of a vector field with computer algebra.

Students use prepared Sage code or a prepared Mathematica notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.

Students use Mathematica to visualize the probability density distribution for the hydrogen atom orbitals with the option to vary the values of \(n\), \(\ell\), and \(m\).

Using either this Geogebra applet or this Mathematica notebook, explore the wave functions on a ring. (Note: The Geogebra applet may be a little easier to use and understand and is accessible if you don't have access to Mathematica, but it is more limited in the wave functions that you can represent. Also, the animation is pretty jumpy in some browsers, especially Firefox. Imagine that the motion is smooth.)

1. Look at graphs of the following states \begin{align} \Phi_1(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +\left|{-2}\right\rangle )\\ \Phi_2(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle -\left|{-2}\right\rangle )\\ \Phi_3(\phi)&=\frac{1}{\sqrt{2}}(\left|{2}\right\rangle +i\left|{-2}\right\rangle ) \end{align} Write a short description of how these states differ from each other.
2. Find a state for which the probability density does not depend on time. Write the state in both ket and wave function notation. These are called stationary states. Generalize your result to give a characterization of the set of all possible states that are stationary states.
3. Find a state that is right-moving. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are right-moving.
4. Find a state that is a standing wave. Write the state in both ket and wave function notation. Generalize your result to give a characterization of the set of all possible states that are standing waves.

Students observe three different plots of linear combinations of spherical combinations with probability density represented by color on the sphere, distance from the origin (polar plot), and distance from the surface of the sphere.

Students explore the effects of putting a point charge at various places inside, outside, and on the surface of a cubical Gaussian surface. The Mathematica worksheet or Sage activity shows the electric field due to the charge, then plots the the flux integrand on the top surface of the box, calculates the flux through the top of the box, and the value of the flux through the whole cube.

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector \(\vec{k}\) and is asked to calculate the value of \(\vec{k} \cdot \vec {r}\) for each point on the grid and to draw the set of points with constant value of \(\vec{k} \cdot \vec{r}\) using rainbow colors to indicate increasing value.

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The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU. Make sure to update this powerpoint yearly to reflect current course offerings and sequencing. It was partially, but not completely edited in fall 2022.

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• How to translate a complicated wavefunction into eigenstates.
• Refresher on how to find expectation values and probabilities in a region.
• How to use the symmetry of the wavefunction to tell you something about measurements.

This very short lecture introduces Wein's displacement law.

• Vectors and their magnitudes are geometric quantities, independent of coordinates and choice of basis

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.

These lecture notes covering week 8 of https://paradigms.oregonstate.edu/courses/ph441 include a small group activity in which students derive the Carnot efficiency.

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• How to form a state as a column vector in matrix representation.
• How to do probability calculations on all three representations used for quantum systems in PH426.
• How to find probabilities for and the resultant state after measuring degenerate eigenvalues.

Basic format

• Blank white paper (8.5” x 11”) is the best paper for scanning into Gradescope. It is also OK to use ruled paper. Do not use grid paper because scanned images of grid paper are hard to read in Gradescope.
• Write with dark colors that are easy to scan/digitize. Do not use red ink. Do not use faint-grey pencil.
• Leave space between the lines of math to make your answers easy to read.
• Leave enough empty space on the page for the grader's comments.
• Leave space between problems; No more than two problems per page.
• Very big and very small numbers must be expressed in scientific notation (for example, \(1.2\times 10^6\)). You will lose points if you use decimal notation to express numbers that are greater than \(10^6\) or less than \(10^{-3}\). Do not use E-notation (for example, do not write 1.2E6).
• Circle your final answer.
• If you are typing your answers in LaTeX or other editing software, read the 9 Rules of Typography in Physics.
• When uploading your answer to Gradescope, follow the instructions to create pointers that link specific question numbers to specific pages of your scan.

Mathematical communication

Part of your grade for all homework and exams will be based on the quality of your mathematical communication. Effective mathematical communication enables others to read and evaluate (and learn from) your work. Effective communication aids your own thinking, and supports your collaborations with other scientists.

Mathematical communication aids thinking by facilitating external cognition. Our brains are powerful but have limitations. To solve challenging physics problems we need to offload our ideas and store them for later use. Learning to organize your reasoning clearly on paper will enable you to more easily and accurately solve more challenging problems. You can read and evaluate your own work, and streamline the process of finding errors that you may have made.

Diagrams

Diagrams are a powerful tool for visualizing a physical system and connecting its details to your mathematical analysis. A well-drawn diagram can clarify relationships, streamline problem-solving, and often replace lengthy explanations. Make it a habit to sketch diagrams; they can save you a thousand words!

Defining algebraic variables

Any variables you use must be defined. This can happen early in the problem, or in a figure, or right after an equation with a phrase like “where \(m\) is the mass of the particle”.

Declaring numerical quantities

Option 1: You could say “Assume that the specific energy density of gasoline is 40 MJ/kg”.

Option 2: You could declare your assumptions in within an equality. In the example below, I've written enough to communicate my assumptions about the battery size, and the type of charger: \begin{align} \text{time to charge the battery} = \frac{\text{storage capacity of battery}}{\text{charging rate}} = \frac{5\times 10^7\text{ J}}{5\times 10^{3}\text{ J/s}}&= 10^4\text{ s} \end{align}

Units

When a physicist writes down the numerical value of a quantity, they write the units next to the number (unless it's a dimensionless quantity). See the examples above.

Significant figures

Use an appropriate number of significant figures. If you give an answer 6.70931309283 kg, you are probably claiming a level of precision that is impossible/unrealistic. If the problem is a zeroth-order estimate, use 1 or 2 sig. fig. to convey a precision of 10 to 20%. If you have a more refined answer, use 2 or 3 sig. fig. to indicate the appropriate level of precision.

Starting equation

A starting equation is an equation that is not deduced from equations you already have written in your answer. You may have several starting equations in a given problem. Often, starting equations will be physical laws. In some cases your starting equations may simply be conversions. When the source of a starting equation is not obvious, you must describe where it came from (or why it is true) in words. For example, “From the Stefan-Boltzmann law...”.

If you find yourself writing a lot more words than math, please check in with the instructor.

Following equation

A following equation is an equation that can be mathematically shown to be true based on equations already present. In many cases, following equations do not require words to explain them. However, if it is not obvious what you did, then some concise words of explanation are important (like a comment in a computer code). For example, “Change the integration variable to \(u \equiv 1/r\)” or “Assume that all heat entering the water came from the rock”.

For algebraic manipulations, it is sufficient to simple write the lines of algebra. It is unnecessary (and a nuisance) to write words such as "We now divide both sides of the equation by the \(m\)."

If you find yourself writing a lot more words than math, please check in with the instructor.

Equation sequences

The equation sequence is a common idiom in mathematical communication. One side of the equation does not change: \begin{align} \vec{F} &= m\vec{a} \\ &= m \frac{d^2\vec r}{dt^2} \\ &= -m x_0\omega^2 \sin\omega t \end{align} In this idiom, we do not rewrite the left-hand side of the equation, but it is taken to be identical. Each expression on the right-hand side in this example is equal to the force.

Equate the leftside to the rightside

Variables and numerical quantities must always be part of either a sentence or a mathematical eqution. For example, do not write: \begin{align} \frac{5\times 10^3\text{ J}}{5\times 10^{7}\text{ J/s}} \text{(no!)} \end{align}

In the example above, there is no equals sign (it is not an equation). It is not a statement that conveys meaning. It is analogous to a sentence with no verb.

Always check that the leftside and rightside of an equation have the same dimensions. For example, if the leftside has dimensions of length, the rightside must also have dimensions of length.

Writing a model answer for your term project

When you create your model answer for the term project, it should look like the answers to example exercises that you find in the PH315 textbooks (the Sustainable Energy textbook, and the Six Ideas textbooks). These model answers give the reader detailed explanations and guidence. Additionally, they often include sensemaking even if the question didn't explicitly ask for sensemaking.

Remind the reader of your goal

At the start, write a sentence (or two) reminding the reader of the goal. For example, “We want to find how much heat is leaking out of a typical house.” Use pictures/diagrams to help define the system and define what will be calculated.

Describe your assumptions

When you construct a mathematical model of the physical system, you will likely need to make some assumptions. Clearly describe these assumptions and provide justification for each assumption, explaining why it is reasonable in the given context.

Layout the calculation

The calculation is the heart of your answer. Show all quantities you used, and the mathematical steps you took to get to the final answer. Insert sentences/comments to explain significant steps in the calculation. These sentences/comments should be concise and useful. If you find yourself writing a lot more words than math, please check in with the instructor.

Sensemaking

At the end of your answer, you might include a comment such as “An energy leak of 1000 J/s would correspond to 24 kWh per day. This is consistent with my expectation that heating is a large fraction of the the typical energy consumption of a household, and I know that an average household uses  40 kWh/day)”.

The type of sensemaking will differ depending on the question. A differnt example of a sensemaking comment is: “This solution approaches the classical limit [fill in the details] when the number of photons is large”.

In a solid, a free electron doesn't see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be screened” from far away. A common model for such problems is described by the Yukawa or screened potential: \begin{equation} U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}} \end{equation}

1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter \(\alpha\).
2. Draw the effective potential for the two choices \(\alpha=10\) and \(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of \(\alpha\) is there the possibility of stable circular orbits?

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