6805: 1094 Activities 4

Write your name and answers on this sheet and hand it in when requested.

Discussion questions based on previous sessions

Goal: Check and supplement your understanding of topics from in-class discussions. Ask questions!!

1. PET scan redux. Could you use ¹⁴C or ¹⁵C in a PET scanner? Explain your reasoning. [Hint: check how it decays.] Name at least one limitation to the spatial resolution of a PET scanner.
   
   
   
   
2. Based on abundance and decay information from the (alternative) online Table of Nuclei (see "in class" links), what particular isotopes do you think are causing Geiger counter activity from salt substitute (pretend we did it :) and the Fiesta-ware and what radiation do you think is being detected? [Hints: potassium chloride and uranium oxide dye.]
   
   
   
   
3. Look at the "in class" figure showing the energy of symmetric nuclear matter (SNM) and pure neutron matter (PNM) as a function of density. (Recall that these are very large collections of nucleons, either in equal numbers of neutron and protons with Coulomb turned off or all neutrons). What does it mean that SNM is negative and PNM is positive? At about what density is SNM stable? (That is, it does not tend to expand or contract.) How does this density relate to the density in the interior of a large nucleus? (Extra for experts: Explain why the Coulomb energy blows up as the nucleus gets bigger and bigger but the energy from the strong force doesn't. )
   
   
   
   
   
   
   
   
4. Why is a very weakly bound nucleus a good candidate to be a halo nucleus?
   
   
   
   
5. Because of pairing, a nucleus with an even number of neutrons is more bound than if it had an odd number of neutrons (and similarly with protons). How does this account for the pattern of bound and unbound nuclei (based on the one-neutron separation energy S_1n) from boron-15 to boron-20?
   
   
   
   

Exploring Fourier transforms as basis expansions

Goal: Gain some intuition about decomposing a function into sines and/or cosines.

1. From the 6805 home page, go to the simulations page and find the PhET simulation "Fourier: Making Waves". Start it with the right triangle ("Play"). You may get some warning dialogs; agree to everything.
2. You'll start in the Discrete tab. You can set the amplitudes of sine waves to add together by dragging the bars with your mouse or changing the numbers under A1, A2, etc. The "Sum" graph shows the net result. Try some different amplitudes to get a feel for how the waves combine and then try some of the different entries in the pull-down menu labeled Function:. Can you maintain the same shapes with different values of the amplitude?
   
   
   
   
3. Go to the "Wave Game" tab. Start at Level 1, where you have one amplitude to adjust. Then try Level 4. (Don't try Level 10 in class!) Is there more than one combination of amplitudes A_i that yield the same shape?
   
   
   
   
4. Go to the "Discrete to Continuous" tab. Now many waves of different wavelengths (or wave numbers k = 2π/λ) are added up. Notice how the sum is nonzero only in a small region as a result of many waves adding and canceling out in the other regions. The resulting wave is a "wave packet" and is the form that the probability amplitude in quantum mechanic takes for an approximately localized particle (e.g., an electron), except that it is real instead of complex. With the controls on the right, change the spread of wave numbers. What happens to the width of the wave packet? What is the product of σ_k and σ_x? If you convert the wave number to a momentum by de Broglie's relation, you have the Heisenberg Uncertainty Principle!
   
   
   
   
   
   
5. Open the Mathematica notebook fourier_scribble_mario_v2.nb, which was written by Mario Carneiro. The code is hidden by default and you won't need to modify it here; you are welcome to look at it but it may be intimidating at first. :) Hold your mouse button down and draw any continuous curve from 0 to 1; make it interesting (not just a sine wave!). Now use the "+" button to add sine waves at the proper amplitudes to reconstruct the original curve. If you hit play, it will do the additions by itself. Do you need more short-wavelength waves to reproduce smooth or sharp (rapidly changing) features of your curve?
   
   
   
   
   

Scaling of time to diagonalize a matrix

Goal: Time how long Mathematica takes to diagonalize several matrices of different size to derive a scaling law. Test it against a larger matrix.

1. Open the Mathematica notebook timing_matrix_diagonalization.nb from the "in class" link. The notebook develops a function to generate a random, real and symmetric, n-by-n matrix (so it is a Hermitian matrix), which we take as a model for a Hamiltonian of interest. Step through the notebook first. What questions do you have?
   
   
   
   
   
   
2. The "timings" table has too small values of n to clearly see the trend. Change the range from i = 1 to 8 (which means n from 2 to 2^8 = 256) to a range that goes larger and excludes the small n points (e.g., 8 to 11). You'll want at least 4 points, but it also needs to finish in a reasonable time! Now rerun the notebook. What do the graphs tell you? (I.e., which one looks like a straight line, and what does that mean for the function plotted?) What do you learn from the fit?
   
   
   
   
   
   
3. Let's make a rough "worst-case" estimate of how the time it takes to diagonalize an n-by-n matrix increases ("scales") with n. Assume that floating point multiplications (that is, multiplying two numbers with decimals, as opposed to integers) takes much more time than adding floating numbers, so we'll neglect the time of the latter. Then the most expensive operation (time-wise) will probably be a matrix-matrix multiplication (adding matrices or a matrix-vector multiplication should be quicker). So let's assume that such multiplications dominate the scaling with n. How many multiplications are there? [Hint: How many multiplications to find one entry in the product matrix? Then how many entries are there? ]
   
   
   
   
   
   
4. Is your rough estimate for the scaling with time consistent with the value you got from the Mathematica experiment?
   
   
   
   
5. How long would it take to find the eigenvalues for a matrix with dimension n = 10⁹?
   
   
   
   

Using the exponential method to find the ground state eigenvalue (and eigenvector)

Goal: Test out a method to project the lowest eigenvalue of a Hermitian matrix.

1. First a quick question: How many energy eigenvalues are there if we diagonalize a 10⁹-by-10⁹ matrix? How many do you suspect we really need (i.e., are useful)?
   
   
   
   
2. Open the Mathematica notebook exponential_matrix_lowest_eigenvalue.nb from the "in class" link. This is a test laboratory for the method described on slide 6 of the Nuclear_ab_initio_1.pdf slides. Step through sections 1 to 3 of the notebook first. What questions do you have?
   
   
   
   
   
   
   
   
3. Now do section 4. How large does tau need to be to get better than 1% accuracy?
   
   
   
   
4. Now repeat with a larger matrix (say 256). Does it still work? How do you think the timing compares to Mathematica's exact calculation?
   
   
   
   
   
5. What questions do you still have about the method used to find the lowest eigenvalue? Here's one: why to we introduce E_T?
   
   
   
   
   
   

Extra for Experts: Expanding in a harmonic oscillator basis

Goal: Understand how the harmonic oscillator basis expansion coefficients can be calculated.

1. Open the Mathematica notebook expanding_two_square_well_wfs_in_ho.nb from the "in class" link. Run the notebook to generate the pictures used to make the last part of the "Expanding in a basis" slides. Can you see how the square well is solved exactly and then the Hamiltonian diagonalized in a (truncated) basis of harmonic oscillator wave functions?