«STUDENT RESEARCH PAPERS SUMMER 2011 VOLUME 22 REU DIRECTOR UMESH GARG, PH.D. REU Student Research Papers – Summer 2011 University of Notre Dame – ...»
outside of the box. Finally, we specify an initial condition φ(x, 0) = f (x).
With these conditions it is quite simple to formulate a discrete approximation
to our ﬁeld. To evaluate the n+1-th time step at the k-th spacial step:
φ(kh, (n+1)h) = (φ((k+1)h, nh)+φ((k−1)h, nh)(1−h2 /2))−φ(kh, (n−1)h) This diﬀerence equation came out nicely because we assumed the step size for the time and spacial steps were equal. In general, we also need to know the value at the k-th and n-th step as well. This formulation’s error is second order in both time and space, so by taking smaller step sizes, the error decreases quadratically.
However, there is a glaring issue with this approach, we treated the equation classically and non-relativistically. Even for the simple case of the KleinGordon Field, we must aﬃx harmonic oscillators at every point in space. By specifying a ﬁnite region, we ensured that our eigenfunctions would have a discrete spectrum, but we did nothing to ensure these eigenfunctions satisfy relativistic constraints, nor did we consider the fact that there is a countably inﬁnite basis of eigenfunctions. We also failed to treat φ(x,t) as an operator. It should be obvious then that more care is needed to apply numerical methods for quantum ﬁelds.
5 A Second Attempt From the last section, it is now obvious that the equation of motion of a ﬁeld is not the correct thing to consider. This is obvious when the importance of the Lagrangian in Quantum Field Theory is recognized. We are ultimately interested in describing interactions among particles and calculating scattering cross sections and probability amplitudes. To return to the Klein-Gordon
equation, lets consider the true form of our ﬁeld:
From here it is obvious that φ should act on state kets. If we want to calculate the amplitude for a particle to propagate between two points we
We now begin to touch on the heart of the matter. Calculations in Quantum Field Theory are not diﬀerential equations, they are integral equations. Attempting to apply numerical techniques to the equations of motion for the Klein-Gordon and Dirac ﬁelds will give solutions that are diﬃcult to interpret and possibly nonphysical and probably nonsense.
6 A Solution and Applications We have avoided the functional integral until now, because it is not the most obvious approach. Of course, by prioritizing the Lagrangian over the Hamiltonian, we move away from the more familiar grounds of Quantum Mechanics, and so the picture becomes less algebraic and more analytic. However, it is undeniable that the path integral formulation makes concerns about preserving symmetry and Lorentz invariance much less of an issue. The reason this method was not considered from the start stems from a numerical standpoint. Solving an integral equation is more involved than a diﬀerential equation and is substantially more computationally intensive. This approach does allow for many diﬀerent techniques to be applied and allows us to delay the need to discretized any equations, or at least to work with semi-discrete equations. Additionally, by working with integral equations we no longer work with point by point evaluation, but with the construction of explicit functions that both well approximate the equation and are easier to work with. In short, this approach trades ease of computation for ﬂexibility.
The ﬂexibility of integral equations comes from the two radically diﬀerent methods of evaluating the integral. Using the power of Stoke’s Theorem, we can change the type of integral from surface to volume or vice versa, depending on the nature of the integrand. We could also construct a weak solution using functional analysis. This variational method has been successfully applied to Atomic Physics and could be quite useful for examining the bound
states of elementary particles.
7 Conclusion This paper represents the preliminary investigations into how systems in Quantum Field Theory can be numerically evaluated. It is evident that the restrictions quantization and relativity put on the equations make a consistent and accurate numerical approach complex. Further investigation will be made into how successfully numerical schemes can be constructed for the path integral formulation of Quantum Field Theory. In doing so, methods of transforming continuous symmetries into discrete ones will be explored to see if the approximations made for the integral equations retain any sort of symmetry.
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W.R. Johnson, S.A. Blundell, and J. Sapirstein, Phys. Rev. A 37, 2 (1988).
M. Peskin and D. Schroeder, An Introduction to Quantum Field Theory (Westview Press, 1995).
L. Schumaker, Spline Functions: Basic Theory (Cambridge University Press, Cambridge, 2007).
J.J. Sakurai and S.F Tuan, Modern Quantum Mechanics (Addison-Wesley, 1994).
Superconductors have gained signicant attention in the last decade. Understanding the properties of superconducting materials is a critical step both in the eld of condensed matter physics and the next advances in engineering. Our group sought to investigate the topic of magnetic vortices in Type II superconductors through low-temperature, ultra high vacuum scanning tunneling microscopy. An important component of a successful STM image is the quality of the tips used in the microscope. Two platinum iridium tip-making methods were tested: the Argonne method and a variation on a method introduced in a paper by Lindahl et al. Sample images taken on an open-air STM by Professor Kandel's group in chemistry, as well as electron microscope pictures of the products of the two tip-making procedures are included and discussed.
1 Argonne Method The idea behind the Argonne method is to use a two-step chemical etching setup (coarse etch then ne etch/micropolishing) in order to shape the tips. The advantage of this two step method is the short preparation time (less than 20 minutes per tip) and decent control over tip shaping. Some papers advocate a one-step coarse etch instead of the additional micropolishing step. However, various factors in the coarse etch step make any ne control or observation of the tip dicult, including the rapidity of the reaction and wide range of etching times that vary with voltage, length of wire in solution, and even the chemical batch of calcium chloride used to make the solution. The opacity of the precipitate that is produced also makes it dicult to judge the progress of the etching. It is these uncertainities that make single-step coarse etching of tips usually unreliable, especially because the end of the wire breaking o tends to produce tips with ball-shaped ends. This feature is necessary for a sharp result from ne-etching but detrimental to immidiate use as a STM tip.
A note on the cut-o time (the time between when the end of the wire drops o and the time when the power to the etching steup is shut o); it is advantageous to minimize this time as much as possible. A longer cut-o time will result in the tip being etched after the desired shape has been obtained. This leads to blunt tips and destroys the ball at the end of the tip. Some groups propose building an automatic cut-o circuit in order to minimize the cut-o time, but it was found that these cut-o circuits still had a signicant delay (500 ns at minimum, but usually on the order of miliseconds). This is an improvement on manual shut-o times but the desire to nely control the last stages of this etching is what motivated the alternate approach in section 2 of this paper. If a manual shut o is used, the sound that the reaction produces can be used as a rough guide to determine when etching may be close to completion, with a noticeable decrease in volume when the end of the wire is about to break o.
1. Cut a clean piece of 0.25mm diameter Pt-Ir wire with wire cutters about 12-15mm long.
2. Insert the wire into the copper clips attached to the coarse etching setup. Make sure the wire protruding is as close to vertical as possible in order to ensure that the etched mass on the end falls o cleanly and without bending the tip.
7. Secure the tip in the ne etching setup and wet the tungsten loop with calcium chloride solution. The solution will need to be replaced frequently, as precipitate builds up rapidly and the available chloride ions are depleted.
8. Set up the electronics for ne etching. This setup is described in detail in the Argonne procedure paper, but it consists most basically of an AC power supply set to about 2.5 V AC3, a switch box with a button to apply current, and a multimeter to verify the voltage. A microscope stage was modied to hold a copper clip to hold the tip in place, a platform to mount the tungsten loop and an additional two-way adjustable stage in order to easier position the loop in relation to the xed position of the clip.
1 It was found that less than 1.3mm led to the entire end of the wire being etched o, with no 'neck' that was thinned and eventually broken o. More than 2mm was found to deform the tip after the drop o. This is speculated to be due to the increased weight and bulk of the end that forces the drop o to occur sooner, when the neck of the wire is not as thin, and strains the wire. Around 1.5mm was found to be a good length for this step.
2 AC power is absolutely necessary, as DC power will not circulate the chloride ions around the wire as much as AC power will. This will result with an etching time on the order of hours instead of minutes.
3 It may be tempting to use voltages of 10 V or above in order to speed the ne etch. However, around 10 V or higher, the small amount of solution in the loop is used up very quickly and must be replaced a few times every minute. Voltages higher than 15 V were found to lead to rapid bubble formation in the loop and give blunt tips. It is therefore advisable to restrict voltages for this step below 10 V.
To Table of ContentsFigure 2: Fine etching setup, showing voltmeter, AC power supply, modied microscope, stage, tungsten loop, and copper clip.
9. Align the tip with the center of the loop. Position the loop behind the ball on the end of the tip, and pull the loop back slowly while pressing the switch box. It is important to apply the power in ONE direction only for proper tip shaping.
10. This process is repeated (changing solution as necessary) until the section of wire just behind the ball is thinned enough to be etched away, leaving a very sharp tip with no ball or other tip anomalies.
11. The tip is the cleaned again, using the same procedure as outlined in step 6 and placed under an optical microscope for characterization and cataloging.
#5/Chris/Octanediol on gold #8/Guido/Octanediol on gold #9/Chris/Octanediol on gold
1.3 Discussion One immidiately obvious feature of this experiment is that tip grading by an optical microscope is very dicult. No good measurement of sharpness can be made at 400x magnication and all three tips look roughly the same in terms of apex shape. The optical microscope pictures are best used as a method to check tips for obvious deformities or crashes. However, tip #5 produced the best images by far and no anomalies occured in either stage of its preparation. Tip #8 had a average quality ne etch and produced a good image, despite the discontinuity in the middle third of the picture. Tip #9 had a slightly longer cut-o time than recommended but still managed to produce a reasonably coherent image at 60.9nm. It can be concluded that the Argonne method produces good tips when the procedure works and has some resilience against minor errors in tip preparation.
To Table of Contents2 Experimental Method The prime motivation behind the experimental method was inspired by a paper published by Lindahl et al.
In it, a method of etching is outlined that stops the coarse etch just before the end of the wire drops o, bends the wire 90◦ and then continues the etch in sulfuric acid and then does some amount of micropolishing.
Their method was found to be unworkable for our group (the bending of the wire did not seem to have a purpose and the entire process was very labor intensive) but the idea of stopping the coarse etch earlier was incorporated into our experimental tip procedure. The idea behind this inclusion was to stop the etching when the neck of the wire is thin and close to dropping o, but nish the etching in the micropolishing setup under a microscope.