This small whiteboard question (SWBQ) serves as a quick review of the dot product. It is also an opportunity to help students see the advantages of knowing many different representations of and facts about a physical concept.
Write one thing you know about the dot product.
This swbq will not work unless most students in the class have seen and used the dot product before. Use it at the beginning of term if the students need a review. If some students in your class have not seen the dot product before, you can refer them to the section The Dot Product in our online text.
Prompt: On your small whiteboard, write one thing you know about the dot product.
(The generic term “one thing” in the prompt is important. Do not use a more specific word that cues students to give an algebraic definition or a drawing, etc.)
strategy:smallwhiteboard:dotproductclip2007.mov|example video
Walk around the room as students are answering this question and quickly pick up an example of each different representation or statement. Quickly order them in the order you would like to talk about them and prop them on the chalkboard tray. Pick up each one (or more than one if you are comparing them) and give whatever review “lecture” you would normally give. The students are far more invested if they see that you are talking about their answers and some students will even vie with each other to get you to choose their answer. You can add in any extra representations that the students haven't mentioned as you go along.
The most important thing to emphasize is that the professional physicist knows and uses all of these representations/facts.
Answers you are likely to see:
You may also want to assign the homework problem Tetrahedron which requires students to use both the algebraic and geometric definitions of the dot product to solve the problem successfully.
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A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an integer \(\ge 0\), and \(\omega\) is the classical frequency of the oscillator. We have chosen the zero of energy at the state \(n=0\) which we can get away with here, but is not actually the zero of energy! To find the true energy we would have to add a \(\frac12\hbar\omega\) for each oscillator.
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Find an expression for the free energy as a function of \(T\) of a system with two states, one at energy 0 and one at energy \(\varepsilon\).
From the free energy, find expressions for the internal energy \(U\) and entropy \(S\) of the system.
Plot the entropy versus \(T\). Explain its asymptotic behavior as the temperature becomes high.
Plot the \(S(T)\) versus \(U(T)\). Explain the maximum value of the energy \(U\).
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