Lec 10

“If I had an hour to solve a problem and my life depended on the answer, I would spend the first 55 minutes figuring out the proper questions to ask. For if I knew the proper questions, I could solve the problem in less than 5 minutes." Albert Einstein

Links for this lecture: http://www.animations.physics.unsw.edu.au/jw/phasor-addition.html

https://www.youtube.com/watch?v=k8FXF1KjzY0

https://www.youtube.com/embed/LznjC4Lo7lE

https://www.youtube.com/watch?v=r18Gi8lSkfM

https://www.youtube.com/watch?v=DT66OE2JEpU

Phasors First show the phasor circuit animation in which the phase between two phasors is varied to illustrate interference. See http://www.animations.physics.unsw.edu.au/jw/phasor-addition.html Review how phasors were used in circuits to show how Kirchhoff's law was satisfied. http://mathlets.org/mathlets/series-rlc-circuit/ https://www.geogebra.org/m/FVqeYkye

Ask the congruous question, “I know that Kirchhoff's law is used to describe a circuit and phasors apply that law for harmonic voltages but how does this work for LCR decay?” https://www.youtube.com/embed/LznjC4Lo7lE

This is Fourier analysis using phasors.

Wavefunction time average

Review the time average given by psi psi*. Show the harmonic potential animation of a coherent state. Ask what incongruous questions do you have?

Least squares and residuals

Review linear least square fits. This is emphasized in courses as an example of how calculus is applied.

https://phet.colorado.edu/sims/html/least-squares-regression/latest/least-squares-regression_en.html

http://mathworld.wolfram.com/LeastSquaresFitting.html

instead of using r^2 or chi^2 plot residuals vs dependent variable. If they oscillate about zero then the data justify the model. If there is a trend upward then not true.

Show how you would calculate the slope and intercept using the wiki text link for “Experimentation: an introduction to measurement theory and ... “ by Baird, D. C. on page 186-187. Emphasize the function form of the slope and intercept depend on the measured values xi and yi.

Show the text on how this is done and have students use this as a reference in their papers.

Linear least squares (needed for the next labs) What are the two types of statistics: descriptive (mean and standard deviation etc) and inferential (t-test, chi squared etc.). I have never asked you to apply any inferential statistics.

Least squares: minimizes the sum of squares of the model prediction minus the various data points. This is similar to chi-squared.

However, data visualization is important in recognizing patterns (remember the histogram vs scope plots of the room lights illuminating a photodiode) associated with different models. Rather than take a R^2 value as the end result you should plot the residuals. If they are random then the data support a linear model. If not, then the model is not appropriate. Examples: linearized RC data and detector response data (non-linear model).

Instead of using r^2 or chi^2, t-test, plot residuals vs dependent variable. If they oscillate about zero then the data justify the model. If there is a trend upward then not true.

http://stattrek.com/statistics/dictionary.aspx?definition=Residual%20plot

Inferential statistics don't give us much direction in formulating models. Give an example where the residual plot looks like the model should contain a quadratic component. Add a quadratic component to see if the residuals look more random. On the other hand, what would you learn from inferential statistics (see a similar discussion for the linear regression)? For the following discussion on the chi squared test see https://reference.wolfram.com/language/ref/PearsonChiSquareTest.html

"Pearson Chi Square Test” performs the Pearson goodness-of-fit test with the null hypothesis that data was drawn from a population with distribution dist, and alternative hypothesis that it was not. By default, a probability value or p-value is returned. A small p-value suggests that it is unlikely that the data came from dist."

“T-test”

http://reference.wolfram.com/language/ref/TTest.en.html

"T-test performs a hypothesis test on data with null hypothesis H=H0 that the true population mean is some value mu = mu0, and alternative hypothesis H=Ha that mu is not equal to mu0. Given data1 and data2 , T-test tests H0 that mu1 - mu2 = mu0 against Ha that mu1 - mu2 is NOT equal to mu0 . By default, a probability value or p-value is returned. A small p-value suggests that it is unlikely that the null hypothesis H0 is true." Gus Greivel (oversees the stats course at Mines) told me he worked in industry and that he often had to explain the TTest to people at his company.

“Kolmogorov-Smirnov test”

https://reference.wolfram.com/language/ref/KolmogorovSmirnovTest.html

"Kolmogorov-Smirnov Test performs the Kolmogorov-Smirnov goodness-of-fit test with null hypothesis H=H0 that data was drawn from a population with distribution dist and alternative hypothesis Ha that it was not. By default, a probability value or p-value is returned. A small p-value suggests that it is unlikely that the data came from dist."

Questions about fitting Modifying: why not use perpendicular distance rather than the vertical? Causal Creative: how does a non-linear fit work?

“One of the most difficult topics in all of data analysis in the physical sciences is fitting data to nonlinear models.” from

https://reference.wolfram.com/applications/eda/FittingDataToNonlinearModels.html

-try to cast the model and data in linear form. For exponential decay take the log

Answers to worksheet:

1.) Please write one question only in the incongruous category about the harmonic potential animation of a coherent state. Incongruous: The “unknown unknowns are the ones we don’t know we don’t know.”

How can there be any time dependance if the wavefunction times it complex conjugate yields a time average since any time average is an integral over time so time is eliminated in the integration?

“The fatal pedagogical error is to throw answers like stones at the heads of those who have not yet asked the questions.”

Paul Tillich Philosopher and Theologian 20th century

Having asked the question, consider the answer. The wavefunction for a superposition of two such states is Psi = Fn[x,wn]Exp[-I wn t]+Fm[x,wm] Exp[-I wm t] where F[x,wn] is proportional to the nth Hermite polynomial and wn is the frequency of that state. Note that time is only found in the complex exponential. Now, PDF = Psi ComplexConjgate[Psi] = Fn[x,wn]^2 +Fm[x,wm]^2 + 2 Fn[x,wn] Fm[x,wm] Cos[(wn-wm) t]. So what time average is happening? Consider not using Eulers theorem for the wavefunction: Psi = Fn[x,wn] Cos[ wn t]+Fm[x,wm] Cos[wm t] so Psi Psi = Fn[x,wn]^2 Cos[ wn t]^2 +Fm[x,wm]^2 Cos[ wm t]^2 + 2 Fn[x,wn] Fm[x,wm] Cos[ wn t] Cos[ wm t] To understand the time dependent terms, use trig identities

Cos[u]^2 = (1+Cos[2u)/2 and Cos[u] Cos[v] = (Cos[u-v] +Cos[u+v])/2

The first two terms of Psi Psi become Fn[x,wn]^2 (1+Cos[2 wn t)/2 +Fm[x,wm]^2 (1+Cos[2 wm t)/2 While the cross term becomes 2 Fn[x,wn] Fm[x,wm] (Cos[wn t-wm t] +Cos[wn t+wm t])/2

Compare this with the QM result PDF = Psi ComplexConjgate[Psi] = Fn[x,wn]^2 +Fm[x,wm]^2 + 2 Fn[x,wn] Fm[x,wm] Cos[(wn-wm) t] It time averages the sum frequencies but not the difference frequencies? How does this happen? Like a lot of math it is not obvious. If you find this discussed in any textbook I would really like to see it.

“The test of a good teacher is not how many questions he can ask his pupils that they will answer readily, but how many questions he inspires them to ask him which he finds it hard to answer.” Alice Wellington Rollins (American writer 1847-1897)

Before we talk about time averaging what are the frequencies wn and wm? This frequency is about 10^14 Hz. Show the animations here (look light the weight room?) https://en.wikipedia.org/wiki/Molecular_vibration

The Cos[wn t-wm t] and Cos[wn t+wm t] are not that much different for low energy states. However this is not the case for high energy states.

Note this result is applicable to the cart velocity interferometer where two EM waves of different frequency are superposed. The incident frequency is 5 10^14 and the reflected is 11 orders of magnitude smaller. Assume that the detector measures the electric field. No detector is not fast enough to measure the frequency of this wave. It measures a time average field or the beat. Therefore using the complex exponential notation is a simple way to do this time averaging to get the detector output.

When you cover this next semester ask your professor, “How can there be any time dependance if the wavefunction times it complex conjugate yields a time average since any time average is an integral over time so time is eliminated in the integration?”

2.) Outline how you would calculate the error in the slope and intercept for a linear least square fit, given the functional form for the slope and intercept.

From page 86 of “Experimentation: an introduction to measurement theory and ... “ by Baird, D. C. the error in an expression due to errors in the variable it is a function of is given as the Sqrt of the sum of the squares of the derivatives times the error in each variable. In this case the slope is a function of the data values xi and yi. Assume that there is no error in xi. Then the error in the slope is the Sqrt of the sum of derivatives wrt yi times the error in yi.

3.) A Michelson interferometer in which one mirror moves for a quantum particle yields the wavefunction at the output port as Psi = Psi0 Exp[I (k0 x - w0 t)] + Psi0 Exp[I (k1 x - w1 t)] where k0, k1, and w0, w2 are the wavevectors and frequencies for reflection from the static and moving mirrors, respectively. Answer on the back of this paper please.

Psi ComplexConjugate[Psi] = ( Psi0 Exp[I (k0 x - w0 t)] + Psi0 Exp[I (k1 x - w1 t)] ) ( Psi0 Exp[-I (k0 x - w0 t)] + Psi0 Exp[-I (k1 x - w1 t)] ). Multiplying out the terms we get

Psi ComplexConjugate[Psi] = Psi0^2 +Psi0^2+2 Psi0^2 Cos[(k0-k1)x-(w0-w1)t]