Lec 8

“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

Links for this lecture: https://en.wikipedia.org/wiki/Shot_noise https://en.wikipedia.org/wiki/Mode-locking http://physics.weber.edu/schroeder/software/HarmonicOscillator.html https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

Lecture overview for today: We will review two topics then cover two more topics about which you will complete a worksheet. The topics are: asking pertinent questions for a particle in a well and devising a better way to measure e/m. Answers to this worksheet are discussed after the worksheet is completed.

Lab overview for today: In the lab you will complete a worksheet, get approval for your model and procedure sections, and collect final data.

Review photodetector: The signal from the star increases with the number of photoelectrons collected by the capacitor. In a 1 ns period this number with spread is (n_ave +- sigma). Collecting photons for N ns yields a mean photon number for 1 ns of n_ave with spread in the mean of +- sigma/Sqrt[N]. However, the total number of photons collected (with spread) for N ns is Nx (n_ave+- sigma/Sqrt[N]) = N x n_ave +- Sqrt[N] sigma. The signal is proportional to N while the variation in photon number or noise is proportional to Sqrt[N]. Therefore the signal to noise ratio is proportional to Sqrt[N}. The signal increases compared to the noise! Notice that this shown in the link https://en.wikipedia.org/wiki/Shot_noise Notice also how the graininess of the pictures changes due to the percent noise decreasing even though the total noise increases. It is the percent change that you notice. With a small number of photons you get a 10% variation in intensity which is much easier to notice than a 1% variation. Photons and their photoelectrons come in discrete quantities. This is the reason why integrating for 1 ns yields variations and therefore noise.

New content: Asking questions about a particle in a well or harmonic oscillator. In class worksheet question: Write at least one question only in the incongruous category on the quantum particle in a square well or harmonic potential.

Show applet of particle in a well https://en.wikipedia.org/wiki/Mode-locking Relate this to the video of the pendulums swinging in synchrony. https://www.youtube.com/watch?v=yVkdfJ9PkRQ

You could have asked a modifying question about what happens if the pendulums are not released at the same time. A sum of the amplitudes would then not generate a huge constructive peak at certain times but rather small incoherence oscillations. Click on the phasors in the harmonic well applet to change their phase in the coherent state of the particle moving back and forth.

This is just what would happen if the harmonic standing waves had an arbitrary phase phi_K added to the K waves oscillating at different frequencies: psi=Sum_{K=1}^{N}[Cos[K w0 t+phi_K]. If they all have the phi_K = 0 then a pulse bounces back and forth. If they have a particular phase relationship at the time of release then they will come back together with this same phase relationship after the same time in the video. They will never all be at the same position at the same time however.

What does this mean in quantum mechanics for the square well? It represents the probability density for a particle bouncing back and forth in a potential well. What do the individual harmonic waves correspond to? How are the frequencies of these wave related? How is the frequency related to the particles kinetic energy and momentum?

Show the harmonic oscillator. Ask incongruous questions (think about what you don’t understand. http://physics.weber.edu/schroeder/software/HarmonicOscillator.html https://en.wikipedia.org/wiki/Quantum_harmonic_oscillator

“As we known, there are known knowns; . . . . We also know there are known unknowns; . . . But there are also unknown unknowns-the ones we don’t know we don’t know.” Donald Rumsfeld.

I would like you to search for the latter by asking incongruous questions.

New content: Antiproton charge to mass measurements:

-e/m is a classic experiment. -causal/creative question: How would you measure the e/m ratio more accurately? -informational: What is the most precise measurement technique possible?

Time and freq measurements are the most accurate. It is possible to measure a single electron going around in a circle using the changing magnetic field it generated through a stationary coil. Measure the time to go around one cycle in the e/m apparatus. https://en.wikipedia.org/wiki/Penning_trap http://titan.triumf.ca/equipment/penning_trap/ https://en.wikipedia.org/wiki/Antiproton_Decelerator#ATRAP http://www.nature.com/nature/journal/v524/n7564/full/nature14861.html

From the nature paper point out: Author contributions: Abstract: model described (standard model of particle physics), how many measurements were made (repeatability), error describe cyclotron frequency and relate to the e/m expt they did. Frequency (or time) measurements are the most accurate. https://en.wikipedia.org/wiki/Cyclotron_resonance

Describe how an rf field with E polarized along the direction of motion can give the charged particle a kick to go in a circle.

Equations which are often dimensionless and define variables after they are introduced. Note energy over mass is dimensionless in these eqns.

ANSWERS TO WORKSHEET QUESTIONS:

1.) What is the direction of the force on a dipole when it is near a point charge.

You are taught free body diagrams in PH100. The training involves sketching the free body diagram, setting up a coordinate system, summing the forces along these coordinates and setting them equal to mass time acceleration in this direction. Do this also for the torques. The dipole experiences an upward force and a torque. It moves up and rotates. “To innovate you need more than rote knowledge. You need a trained imagination.” Philosopher Martha Nussbaum By having a knowledge of these diagrams you are gaining a trained imagination to help you do more than rote work.

2.) Quantum mechanics: localized particle in well .

How can energy be conserved if the particle is in a state of different energies? Does this mean that you could have perpetual motion? How can the mass be related to the wavelength?

A final comment on the questioning procedure: you were taught the free body diagram procedure yet when I asked you to determine the force on a dipole few of you applied the procedure of drawing a free body diagram. The questioning procedure is the same idea. Apply or practice it in your classes next semester (and beyond) to improve your understanding.

You don’t really understand quantum unless you ask questions first such as how is energy conserved.