## Lines in Polar Coordinates

• assignment Ring Function

assignment Homework

##### Ring Function
Central Forces 2023 (3 years) 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: $$\Phi\left(\phi\right)= \frac{N}{2+\cos(3\phi)}$$ 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 Working with Representations on the Ring

group Small Group Activity

30 min.

##### Working with Representations on the Ring
Central Forces 2023 (3 years)
• assignment Scattering

assignment Homework

##### Scattering
Central Forces 2023 (3 years)

Consider a very light particle of mass $\mu$ scattering from a very heavy, stationary particle of mass $M$. The force between the two particles is a repulsive Coulomb force $\frac{k}{r^2}$. The impact parameter $b$ in a scattering problem is defined to be the distance which would be the closest approach if there were no interaction (See Figure). The initial velocity (far from the scattering event) of the mass $\mu$ is $\vec v_0$. Answer the following questions about this situation in terms of $k$, $M$, $\mu$, $\vec v_0$, and $b$. ()It is not necessarily wise to answer these questions in order.)

1. What is the initial angular momentum of the system?
2. What is the initial total energy of the system?
3. What is the distance of closest approach $r_{\rm{min}}$ with the interaction?
4. Sketch the effective potential.
5. What is the angular momentum at $r_{\rm{min}}$?
6. What is the total energy of the system at $r_{\rm{min}}$?
7. What is the radial component of the velocity at $r_{\rm{min}}$?
8. What is the tangential component of the velocity at $r_{\rm{min}}$?
9. What is the value of the effective potential at $r_{\rm{min}}$?
10. For what values of the initial total energy are there bound orbits?
11. Using your results above, write a short essay describing this type of scattering problem, at a level appropriate to share with another Paradigm student.

• group Equipotential Surfaces

group Small Group Activity

120 min.

##### Equipotential Surfaces

Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
• assignment Central Force Definition

assignment Homework

##### Central Force Definition
Central Forces 2023 (3 years)

Which of the following forces can be central forces? which cannot? If the force CAN be a central force, explain the circumstances that would allow it to be a central force.

1. The force on a test mass $m$ in a gravitational field $\vec{g~}$, i.e. $m\vec g$
2. The force on a test charge $q$ in an electric field $\vec E$, i.e. $q\vec E$
3. The force on a test charge $q$ moving at velocity $\vec{v~}$ in a magnetic field $\vec B$, i.e. $q\vec v \times \vec B$

• group Wavefunctions on a Quantum Ring

group Small Group Activity

30 min.

##### Wavefunctions on a Quantum Ring
Central Forces 2023 (2 years)
• assignment Visualization of Wave Functions on a Ring

assignment Homework

##### Visualization of Wave Functions on a Ring
Central Forces 2023 (3 years) 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.
• computer Visualization of Power Series Approximations

computer Computer Simulation

30 min.

##### Visualization of Power Series Approximations
Theoretical Mechanics (13 years)

Power Series Sequence (E&M)

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.
• assignment Polar vs. Spherical Coordinates

assignment Homework

##### Polar vs. Spherical Coordinates
Central Forces 2023 (3 years)

Show that the plane polar coordinates we have chosen are equivalent to spherical coordinates if we make the choices:

1. The direction of $z$ in spherical coordinates is the same as the direction of $\vec L$.
2. The $\theta$ of spherical coordinates is chosen to be $\pi/2$, so that the orbit is in the equatorial plane of spherical coordinates.

• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• Central Forces 2023 (3 years)

The general equation for a straight line in polar coordinates is given by: $$r(\phi)=\frac{r_0}{\cos(\phi-\delta)}$$ where $r_0$ and $\delta$ are constant parameters. Find the polar equation for the straight lines below. You do NOT need to evaluate any complicated trig or inverse trig functions. You may want to try plotting the general polar equation to figure out the roles of the parameters.

1. $y=3$
2. $x=3$
3. $y=-3x+2$