Students take the inner product of vectors that lie on the spacetime axis to show that they are orthogonal. To do the inner product, students much use the Minkowski metric.
- Draw two spacetime vectors, one on the \(ct'\) axis and one on the \(x'\) axis.
- Write these vectors are column matrices with the primed coordinates. (You can choose numbers or do it generally with variables.)
- Take the dot product between these two vectors. What is the result? What does it mean?
- How would an observer in the unprimed frame write these columns? (Hint: Lorentz Transformation.)
- Take the dot product between these two vectors. What is the result? What does it mean?
face Lecture
30 min.
assignment Homework
Which pairs of events (if any) are simultaneous in the unprimed frame?
Which pairs of events (if any) are simultaneous in the primed frame?
Which pairs of events (if any) are colocated in the unprimed frame?
Which pairs of events (if any) are colocated in the primed frame?
Which event occurs first in the unprimed frame?
Which event occurs first in the primed frame?
group Small Group Activity
5 min.
Special Relativity Spacetime Diagrams Simultaneity Colocation
Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.face Lecture
120 min.
Planck distribution blackbody radiation photon statistical mechanics
These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.keyboard Computational Activity
120 min.
electrostatic potential python
Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python usingnumpy
and matplotlib
.
keyboard Computational Activity
120 min.
quantum mechanics operator matrix element particle in a box eigenfunction
Students find matrix elements of the position operator \(\hat x\) in a sinusoidal basis. This allows them to express this operator as a matrix, which they can then numerically diagonalize and visualize the eigenfunctions.face Lecture
30 min.
assignment Homework
Consider a system which has an internal energy \(U\) defined by: \begin{align} U &= \gamma V^\alpha S^\beta \end{align} where \(\alpha\), \(\beta\) and \(\gamma\) are constants. The internal energy is an extensive quantity. What constraint does this place on the values \(\alpha\) and \(\beta\) may have?
assignment Homework
Consider the bottle in a bottle problem in a previous problem set, summarized here.
A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.The volume of the small bottle is 0.001 m^{23} and the volume of the big bottle is 0.01 m^{3}. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).
How many molecules of gas does the large bottle contain? What is the final temperature of the gas?
Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).