face Lecture

10 min.

Warm-Up Powerpoint

Warm-Up

The attached powerpoint articulates the possible paths through the curriculum for new graduate students at OSU.

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Curvilinear Coordinate Sequence
The curvilinear coordinate sequence introduces cylindrical and spherical coordinates (including inconsistencies between physicists' and mathematicians' notational conventions) and the basis vectors adapted to these coordinate systems.

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Warm-Up

Warm-Up (Welcome Activity Reviewing Material from Undergraduate Physics)

This content is used in the Physics Department at OSU with incoming graduate students to remind them of undergraduate content before classes start and to help them to decide whether or not to take some Bridge Courses.

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Gradient Sequence
This sequence starts with an introduction to partial derivatives and continues through gradient. While some of the activities/problems are pure math, a number of other activities/problems are situated in the context of electrostatics. This sequence is intended to be used intermittently across multiple days or even weeks of a course or even multiple courses.

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Integration Sequence
Students learn/review how to do integrals in a multivariable context, using the vector differential \(d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}\) and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.

assignment Homework

Photo Permission
Quantum Fundamentals 2022

In the "Quizzes" section of Canvas, please fill out the "Photo Permission Form" to indicate what information you'd like me to post about you on the Physics Department Website.

Faculty & Students use this site to learn who is taking Paradigms and to network.

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Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.

assignment Homework

Exponential and Logarithm Identities
Static Fields 2022

Make sure that you have memorized the following identities and can use them in simple algebra problems: \begin{align} e^{u+v}&=e^u \, e^v\\ \ln{uv}&=\ln{u}+\ln{v}\\ u^v&=e^{v\ln{u}} \end{align}

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Ring Cycle Sequence
Students calculate electrostatic fields (\(V\), \(\vec{E}\)) and magnetostatic fields (\(\vec{A}\), \(\vec{B}\)) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.

assignment Homework

Gibbs free energy
thermodynamics Maxwell relation Energy and Entropy 2020 The Gibbs free energy, \(G\), is given by \begin{align*} G = U + pV - TS. \end{align*}
  1. Find the total differential of \(G\). As always, show your work.
  2. Interpret the coefficients of the total differential \(dG\) in order to find a derivative expression for the entropy \(S\).
  3. From the total differential \(dG\), obtain a different thermodynamic derivative that is equal to \[ \left(\frac{\partial {S}}{\partial {p}}\right)_{T} \]

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Power Series Coefficients 2
Static Fields 2022 (5 years) Use the formula for a Taylor series: \[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\] to find the first three non-zero terms of a series expansion for \(f(z)=e^{-kz}\) around \(z=3\).

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Power Series Coefficients 3
Static Fields 2022 (5 years) Use the formula for a Taylor series: \[f(z)=\sum_{n=0}^{\infty} \frac{1}{n!} \frac{d^n f(a)}{dz^n} (z-a)^n\] to find the first three non-zero terms of a series expansion for \(f(z)=\cos(kz)\) around \(z=2\).

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Arms Sequence for Complex Numbers and Quantum States
“Arms” is an engaging representation of complex numbers in which students use their left arms to geometrically represent numbers in the complex plane (an Argand diagram). The sequence starts with pure math activities in which students represent a single complex number (using prompts in both rectangular and exponential forms), demonstrate multiplication of complex numbers in exponential form, and act out a number of different linear transformation on pairs of complex numbers. Later activities, relevant to spin 1/2 systems in quantum mechanics, explore overall phases, relative phases, and time dependence. These activities can be combined and sequenced in many different ways; see the Instructor's Guide for the second activity for ideas about how to introduce the Arms representation the first time you use it.

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Volume Charge Density
Static Fields 2022 (5 years)

Sketch the volume charge density: \begin{equation} \rho (x,y,z)=c\,\delta (x-3) \end{equation}

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Gauss's Law for a Rod inside a Cube
Static Fields 2022 (3 years) Consider a thin charged rod of length \(L\) standing along the \(z\)-axis with the bottom end on the \(x,y\)-plane. The charge density \(\lambda_0\) is constant. Find the total flux of the electric field through a closed cubical surface with sides of length \(3L\) centered at the origin.

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30 min.

Magnetic Moment & Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)

Angular Momentum Spin Magnetic Moment Stern-Gerlach Experiments

Students consider the relation (1) between the angular momentum and magnetic moment for a current loop and (2) the force on a magnetic moment in an inhomogeneous magnetic field. Students make a (classical) prediction of the outcome of a Stern-Gerlach experiment.

assignment Homework

Memorize Power Series

Power Series Sequence (E&M)

Static Fields 2022 (2 years)

Look up and memorize the power series to fourth order for \(e^z\), \(\sin z\), \(\cos z\), \((1+z)^p\) and \(\ln(1+z)\). For what values of \(z\) do these series converge?

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Current in a Wire
Static Fields 2022 (3 years) The current density in a cylindrical wire of radius \(R\) is given by \(\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}\). Find the total current in the wire.

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Power Series Sequence (E&M)

The first three activities provide an active-engagement version of the canonical mathematical and geometric fundamentals for power series. The subsequent activities apply these ideas to physical situations that are appropriate for an upper-division electromagnetism course, using concepts, terminology, and techniques that are common among physicists, but not often taught in mathematics courses. In particular students use the memorized formula for the binomial expansion to evaluate various electrostatic and magnetostatic field in regions of high symmetry. By factoring out a physical quantity which is large compared to another physical quantity, they manipulate the formulas for these fields into a form where memorized formulas apply. The results for the different regions of high symmetry are compared and contrasted. A few homework problems that emphasize the meaning of series notation are included.

Note: The first two activities are also included in Power Series Sequence (Mechanics) and can be skipped in E&M if already taught in Mechanics.

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The Path

Gradient Sequence

Vector Calculus I 2022 You are climbing a hill along the steepest path, whose slope at your current location is \(1\over5\). There is another path branching off at an angle of \(30^\circ\) (\(\pi\over6\)). How steep is it?