Students use a pre-written Mathematica notebook or a Geogebra applet to explore how the shape of the effective potential function changes as the various parameters (angular momentum, force constant, reduced mass) are varied.
Download and run this Mathematica notebook or this Geogebra applet.
You have four different sliders that control the values of four parameters \(k\), \(\ell\), \(\mu\), and \(E\).
Answer the following questions:
- What is the physical/geometric meaning of each parameter \(k\), \(\ell\), \(\mu\), \(E\)?
- How does each parameter \(k\), \(\ell\), \(\mu\), \(E\) affect the plot?
- Which term in the effective potential (\(-k/r\) or \(\ell^2/(2\mu r^2))\) dominates for small values of r? For large values of r? Explain in terms of both the equation and the graph.
- What are the classical turning points? Under what conditions will the particle be bound? Unbound?
- How do your answers for the last question change (if at all) if you consider a repulsive potential? Hint: Figure out what you must change in this notebook and investigate.
assignment Homework
Consider a mass \(\mu\) in the potential shown in the graph below. You give the mass a push so that its initial angular momentum is \(\ell\ne 0\) for a given fixed value of \(\ell\).
assignment Homework
The function \(\theta(x)\) (the Heaviside or unit step function) is a defined as: \begin{equation} \theta(x) =\begin{cases} 1 & \textrm{for}\; x>0 \\ 0 & \textrm{for}\; x<0 \end{cases} \end{equation} This function is discontinuous at \(x=0\) and is generally taken to have a value of \(\theta(0)=1/2\).
Make sketches of the following functions, by hand, on axes with the same scale and domain. Briefly describe, using good scientific writing that includes both words and equations, the role that the number two plays in the shape of each graph: \begin{align} y &= \theta (x)\\ y &= 2+\theta (x)\\ y &= \theta(2+x)\\ y &= 2\theta (x)\\ y &= \theta (2x) \end{align}
group Small Group Activity
120 min.
assignment Homework
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}
computer Mathematica Activity
30 min.
computer Computer Simulation
30 min.
Taylor series power series approximation
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.format_list_numbered Sequence
face Lecture
120 min.
phase transformation Clausius-Clapeyron mean field theory thermodynamics
These lecture notes from the ninth week of Thermal and Statistical Physics cover phase transformations, the Clausius-Clapeyron relation, mean field theory and more. They include a number of small group activities.assignment Homework
Consider the frictionless motion of a hockey puck of mass \(m\) on a perfectly circular bowl-shaped ice rink with radius \(a\). The central region of the bowl (\(r < 0.8a\)) is perfectly flat and the sides of the ice bowl smoothly rise to a height \(h\) at \(r = a\).
assignment Homework
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.)