Find Force Law

    • assignment Differential Form of Gauss's Law

      assignment Homework

      Differential Form of Gauss's Law
      Static Fields 2022 (4 years)

      For an infinitesimally thin cylindrical shell of radius \(b\) with uniform surface charge density \(\sigma\), the electric field is zero for \(s<b\) and \(\vec{E}= \frac{\sigma b}{\epsilon_0 s}\, \hat s\) for \(s > b\). Use the differential form of Gauss' Law to find the charge density everywhere in space.

    • accessibility_new Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)

      accessibility_new Kinesthetic

      30 min.

      Using Arms to Visualize Transformations of Complex Two-Component Vectors (MathBits)
      Quantum Fundamentals 2021

      arms complex numbers phase rotation reflection math

      Arms Sequence for Complex Numbers and Quantum States

      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.
    • assignment Ring Function

      assignment Homework

      Ring Function
      Central Forces 2021 Consider the normalized wavefunction \(\Phi\left(\phi\right)\) for a quantum mechanical particle of mass \(\mu\) constrained to move on a circle of radius \(r_0\), given by: \begin{equation} \Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)} \end{equation} where \(N\) is the normalization constant.
      1. Find \(N\).

      2. Plot this wave function.
      3. Plot the probability density.
      4. Find the probability that if you measured \(L_z\) you would get \(3\hbar\).
      5. What is the expectation value of \(L_z\) in this state?
    • group Box Sliding Down Frictionless Wedge

      group Small Group Activity

      120 min.

      Box Sliding Down Frictionless Wedge
      Theoretical Mechanics 2021 (2 years)

      Lagrangian Mechanics Generalized Coordinates Special Cases

      Students solve for the equations of motion of a box sliding down (frictionlessly) a wedge, which itself slides on a horizontal surface, in order to answer the question "how much time does it take for the box to slide a distance \(d\) down the wedge?". This activities highlights finding kinetic energies when the coordinate system is not orthonormal and checking special cases, functional behavior, and dimensions.
    • assignment Isolength and Isoforce Stretchability

      assignment Homework

      Isolength and Isoforce Stretchability
      Energy and Entropy 2021 (2 years)

      In class, you measured the isolength stretchability and the isoforce stretchability of your systems in the PDM. We found that for some systems these were very different, while for others they were identical.

      Show with algebra (NOT experiment) that the ratio of isolength stretchability to isoforce stretchability is the same for both the left-hand side of the system and the right-hand side of the system. i.e.: \begin{align} \frac{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{x_R}}{\left(\frac{\partial {x_L}}{\partial {F_L}}\right)_{F_R}} &= \frac{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{x_L}}{\left(\frac{\partial {x_R}}{\partial {F_R}}\right)_{F_L}} \label{eq:ratios} \end{align}

      Hint
      You will need to make use of the cyclic chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{C} = -\left(\frac{\partial {A}}{\partial {C}}\right)_{B}\left(\frac{\partial {C}}{\partial {B}}\right)_{A} \end{align}
      Hint
      You will also need the ordinary chain rule: \begin{align} \left(\frac{\partial {A}}{\partial {B}}\right)_{D} = \left(\frac{\partial {A}}{\partial {C}}\right)_{D}\left(\frac{\partial {C}}{\partial {B}}\right)_{D} \end{align}

    • assignment Visualization of Wave Functions on a Ring

      assignment Homework

      Visualization of Wave Functions on a Ring
      Central Forces 2021 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.
    • assignment Energy, Entropy, and Probabilities

      assignment Homework

      Energy, Entropy, and Probabilities
      Energy Entropy Probabilities Thermodynamic identity

      The goal of this problem is to show that once we have maximized the entropy and found the microstate probabilities in terms of a Lagrange multiplier \(\beta\), we can prove that \(\beta=\frac1{kT}\) based on the statistical definitions of energy and entropy and the thermodynamic definition of temperature embodied in the thermodynamic identity.

      The internal energy and entropy are each defined as a weighted average over microstates: \begin{align} U &= \sum_i E_i P_i & S &= -k_B\sum_i P_i \ln P_i \end{align}: We saw in clase that the probability of each microstate can be given in terms of a Lagrange multiplier \(\beta\) as \begin{align} P_i &= \frac{e^{-\beta E_i}}{Z} & Z &= \sum_i e^{-\beta E_i} \end{align} Put these probabilities into the above weighted averages in order to relate \(U\) and \(S\) to \(\beta\). Then make use of the thermodynamic identity \begin{align} dU = TdS - pdV \end{align} to show that \(\beta = \frac1{kT}\).

    • face Time Evolution Refresher (Mini-Lecture)

      face Lecture

      30 min.

      Time Evolution Refresher (Mini-Lecture)
      Central Forces 2021

      schrodinger equation time dependence stationary states

      Quantum Ring Sequence

      The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
    • face Fermi and Bose gases

      face Lecture

      120 min.

      Fermi and Bose gases
      Thermal and Statistical Physics 2020

      Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition

      These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.
    • face Entropy and Temperature

      face Lecture

      120 min.

      Entropy and Temperature
      Thermal and Statistical Physics 2020

      paramagnet entropy temperature statistical mechanics

      These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.
  • Central Forces 2021

    Find the force law for a central-force field that allows a particle to move in a spiral orbit given by \(r=k\phi^2\), where \(k\) is a constant.