## Activity: Time Dilation Light Clock Skit

Students act out the classic light clock scenario for deriving time dilation.
What students learn
• The path length difference of the photon in the light leads to disagreements about time intervals. Assists in a deriving an algebraic expression for time dilation.
• Media

## Activity Structure

This is a kinesthetic activity to help derive the expression for time dilation.

2 people are needed for this skit. It helpts to have each person toss a small ball in the air to represent a photon in the light clock.

## Prompt

Have one person walking with their light clock and one person standing still with their light clock.

“For each light clock, consider two events: the photon is emitted and the photon is detected. How would each person describe these events for a tick of my light clock? For hers?

How do the time intervals for your light clock in (1) your reference frame $\Delta t$ and (2) your friend's reference frame $\Delta t'$ compare?

## Student Conversations

• Symmetry: "My photon is emitted and detected at the same horizontal coordinate in my reference frame. From her reference frame, my light clock travels a distance between emission and detection of the photon. Similarly, her photon is emitted and detected at the same horizontal coordinate in her reference frame. From my reference frame, her light clock travels a distance between emission and detection of the photon. They situations are exactly symmetric."
• When 2 events happen at the same spatial coordinate (colocated) My clock measures the proper time interval because the emission and detection events happen at the same spatial ($x$) coordinate in my reference frame. Since $\gamma \gt 1$, that the time interval I observe for my own clock is shorter than the time interval for my clock measured in any other reference frame. I measure a special time interval, the proper time. The proper time is the time measured by a clock moving with the events. That clock measures the shortest time between the events.
• assignment_ind Time Dilation

assignment_ind Small White Board Question

10 min.

##### Time Dilation
Theoretical Mechanics (4 years)

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face Lecture

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##### Statistical Analysis of Stern-Gerlach Experiments
• group Mass is not Conserved

group Small Group Activity

30 min.

##### Mass is not Conserved
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Groups are asked to analyze the following standard problem:

Two identical lumps of clay of (rest) mass m collide head on, with each moving at 3/5 the speed of light. What is the mass of the resulting lump of clay?

• assignment Light bulb in a refrigerator

assignment Homework

##### Light bulb in a refrigerator
Carnot refridgerator Work Entropy Thermal and Statistical Physics 2020 A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?
• computer Blackbody PhET

computer Computer Simulation

30 min.

##### Blackbody PhET
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• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
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• assignment Matrix Elements and Completeness Relations

assignment Homework

##### Matrix Elements and Completeness Relations

Completeness Relations

Quantum Fundamentals 2023 (3 years)

Writing an operator in matrix notation in its own basis is easy: it is diagonal with the eigenvalues on the diagonal.

What if I want to calculate the matrix elements using a different basis??

The eigenvalue equation tells me what happens when an operator acts on its own eigenstate. For example: $\hat{S}_y\left|{\pm}\right\rangle _y=\pm\frac{\hbar}{2}\left|{\pm}\right\rangle _y$

In Dirac bra-ket notation, to know what an operator does to a ket, I needs to write the ket in the basis that is the eigenstates of the operator (in order to use the eigenvalue equation.)

One way to do this to stick completeness relationships into the braket: \begin{eqnarray*} \left\langle {+}\right|\hat{S_y}\left|{+}\right\rangle = \left\langle {+}\right|(I)\hat{S_y}(I)\left|{+}\right\rangle \end{eqnarray*}

where $I$ is the identity operator: $I=\color{blue}{\left|{+}\right\rangle _{yy}\left\langle {+}\right|}\;+\;\color{blue}{\left|{-}\right\rangle _{yy}\left\langle {-}\right|}$. This effectively rewrite the $\left|{+}\right\rangle$ in the $\left|{\pm}\right\rangle _y$ basis.

Find the top row matrix elements of the operator $\hat{S}_y$ in the $S_z$ basis by inserting completeness relations into the brakets. (The answer is already on the Spins Reference Sheet, but I want you do demonstrate the calculation.)

• group Optical depth of atmosphere

group Small Group Activity

30 min.

##### Optical depth of atmosphere
Contemporary Challenges 2021 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
• group Hydrogen emission

group Small Group Activity

30 min.

##### Hydrogen emission
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• assignment_ind Possible Worldlines

assignment_ind Small White Board Question

10 min.

##### Possible Worldlines
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Student consider several curves on a spacetime diagram and have to judge which curves could be worldlines for an object.

Learning Outcomes