11: 06 Pre-Class Assignment - Matrix Mechanics

11: 06 Pre-Class Assignment - Matrix Mechanics

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E. T. Jaynes, a promoter of the use of Bayesian probability in statistical physics, once suggested that quantum theory is "[a] peculiar mixture describing in part realities of Nature, in part incomplete human information about Nature—all scrambled up by Heisenberg and Bohr into an omelette that nobody has seen how to unscramble." [15] QBism developed out of efforts to separate these parts using the tools of quantum information theory and personalist Bayesian probability theory.

There are many interpretations of probability theory. Broadly speaking, these interpretations fall into one of three categories: those which assert that a probability is an objective property of reality (the propensity school), those who assert that probability is an objective property of the measuring process (frequentists), and those which assert that a probability is a cognitive construct which an agent may use to quantify their ignorance or degree of belief in a proposition (Bayesians). QBism begins by asserting that all probabilities, even those appearing in quantum theory, are most properly viewed as members of the latter category. Specifically, QBism adopts a personalist Bayesian interpretation along the lines of Italian mathematician Bruno de Finetti [16] and English philosopher Frank Ramsey. [17] [18]

According to QBists, the advantages of adopting this view of probability are twofold. First, for QBists the role of quantum states, such as the wavefunctions of particles, is to efficiently encode probabilities so quantum states are ultimately degrees of belief themselves. (If one considers any single measurement that is a minimal, informationally complete POVM, this is especially clear: A quantum state is mathematically equivalent to a single probability distribution, the distribution over the possible outcomes of that measurement. [19] ) Regarding quantum states as degrees of belief implies that the event of a quantum state changing when a measurement occurs—the "collapse of the wave function"—is simply the agent updating her beliefs in response to a new experience. [13] Second, it suggests that quantum mechanics can be thought of as a local theory, because the Einstein–Podolsky–Rosen (EPR) criterion of reality can be rejected. The EPR criterion states, "If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of reality corresponding to that quantity." [20] Arguments that quantum mechanics should be considered a nonlocal theory depend upon this principle, but to a QBist, it is invalid, because a personalist Bayesian considers all probabilities, even those equal to unity, to be degrees of belief. [21] [22] Therefore, while many interpretations of quantum theory conclude that quantum mechanics is a nonlocal theory, QBists do not. [23]

Fuchs introduced the term "QBism" and outlined the interpretation in more or less its present form in 2010, [24] carrying further and demanding consistency of ideas broached earlier, notably in publications from 2002. [25] [26] Several subsequent papers have expanded and elaborated upon these foundations, notably a Reviews of Modern Physics article by Fuchs and Schack [19] an American Journal of Physics article by Fuchs, Mermin, and Schack [23] and Enrico Fermi Summer School [27] lecture notes by Fuchs and Stacey. [22]

Prior to the 2010 paper, the term "quantum Bayesianism" was used to describe the developments which have since led to QBism in its present form. However, as noted above, QBism subscribes to a particular kind of Bayesianism which does not suit everyone who might apply Bayesian reasoning to quantum theory (see, for example, the Other uses of Bayesian probability in quantum physics section below). Consequently, Fuchs chose to call the interpretation "QBism," pronounced "cubism," preserving the Bayesian spirit via the CamelCase in the first two letters, but distancing it from Bayesianism more broadly. As this neologism is a homophone of Cubism the art movement, it has motivated conceptual comparisons between the two, [28] and media coverage of QBism has been illustrated with art by Picasso [7] and Gris. [29] However, QBism itself was not influenced or motivated by Cubism and has no lineage to a potential connection between Cubist art and Bohr's views on quantum theory. [30]

According to QBism, quantum theory is a tool which an agent may use to help manage his or her expectations, more like probability theory than a conventional physical theory. [13] Quantum theory, QBism claims, is fundamentally a guide for decision making which has been shaped by some aspects of physical reality. Chief among the tenets of QBism are the following: [31]

  1. All probabilities, including those equal to zero or one, are valuations that an agent ascribes to his or her degrees of belief in possible outcomes. As they define and update probabilities, quantum states (density operators), channels (completely positive trace-preserving maps), and measurements (positive operator-valued measures) are also the personal judgements of an agent.
  2. The Born rule is normative, not descriptive. It is a relation to which an agent should strive to adhere in his or her probability and quantum state assignments.
  3. Quantum measurement outcomes are personal experiences for the agent gambling on them. Different agents may confer and agree upon the consequences of a measurement, but the outcome is the experience each of them individually has.
  4. A measurement apparatus is conceptually an extension of the agent. It should be considered analogous to a sense organ or prosthetic limb—simultaneously a tool and a part of the individual.

Reactions to the QBist interpretation have ranged from enthusiastic [13] [28] to strongly negative. [32] Some who have criticized QBism claim that it fails to meet the goal of resolving paradoxes in quantum theory. Bacciagaluppi argues that QBism's treatment of measurement outcomes does not ultimately resolve the issue of nonlocality, [33] and Jaeger finds QBism's supposition that the interpretation of probability is key for the resolution to be unnatural and unconvincing. [12] Norsen [34] has accused QBism of solipsism, and Wallace [35] identifies QBism as an instance of instrumentalism QBists have argued insistently that these characterizations are misunderstandings, and that QBism is neither solipsist nor instrumentalist. [17] [36] A critical article by Nauenberg [32] in the American Journal of Physics prompted a reply by Fuchs, Mermin, and Schack. [37] Some assert that there may be inconsistencies for example, Stairs argues that when a probability assignment equals one, it cannot be a degree of belief as QBists say. [38] Further, while also raising concerns about the treatment of probability-one assignments, Timpson suggests that QBism may result in a reduction of explanatory power as compared to other interpretations. [1] Fuchs and Schack replied to these concerns in a later article. [39] Mermin advocated QBism in a 2012 Physics Today article, [2] which prompted considerable discussion. Several further critiques of QBism which arose in response to Mermin's article, and Mermin's replies to these comments, may be found in the Physics Today readers' forum. [40] [41] Section 2 of the Stanford Encyclopedia of Philosophy entry on QBism also contains a summary of objections to the interpretation, and some replies. [42] Others are opposed to QBism on more general philosophical grounds for example, Mohrhoff criticizes QBism from the standpoint of Kantian philosophy. [43]

Certain authors find QBism internally self-consistent, but do not subscribe to the interpretation. [44] For example, Marchildon finds QBism well-defined in a way that, to him, many-worlds interpretations are not, but he ultimately prefers a Bohmian interpretation. [45] Similarly, Schlosshauer and Claringbold state that QBism is a consistent interpretation of quantum mechanics, but do not offer a verdict on whether it should be preferred. [46] In addition, some agree with most, but perhaps not all, of the core tenets of QBism Barnum's position, [47] as well as Appleby's, [48] are examples.

Popularized or semi-popularized media coverage of QBism has appeared in New Scientist, [49] Scientific American, [50] Nature, [51] Science News, [52] the FQXi Community, [53] the Frankfurter Allgemeine Zeitung, [29] Quanta Magazine, [16] Aeon, [54] and Discover. [55] In 2018, two popular-science books about the interpretation of quantum mechanics, Ball's Beyond Weird and Ananthaswamy's Through Two Doors at Once, devoted sections to QBism. [56] [57] Furthermore, Harvard University Press published a popularized treatment of the subject, QBism: The Future of Quantum Physics, in 2016. [13]

The philosophy literature has also discussed QBism from the viewpoints of structural realism and of phenomenology. [58] [59] [60]

Copenhagen interpretations Edit

The views of many physicists (Bohr, Heisenberg, Rosenfeld, von Weizsäcker, Peres, etc.) are often grouped together as the "Copenhagen interpretation" of quantum mechanics. Several authors have deprecated this terminology, claiming that it is historically misleading and obscures differences between physicists that are as important as their similarities. [14] [61] QBism shares many characteristics in common with the ideas often labeled as "the Copenhagen interpretation", but the differences are important to conflate them or to regard QBism as a minor modification of the points of view of Bohr or Heisenberg, for instance, would be a substantial misrepresentation. [10] [31]

QBism takes probabilities to be personal judgments of the individual agent who is using quantum mechanics. This contrasts with older Copenhagen-type views, which hold that probabilities are given by quantum states that are in turn fixed by objective facts about preparation procedures. [13] [62] QBism considers a measurement to be any action that an agent takes to elicit a response from the world and the outcome of that measurement to be the experience the world's response induces back on that agent. As a consequence, communication between agents is the only means by which different agents can attempt to compare their internal experiences. Most variants of the Copenhagen interpretation, however, hold that the outcomes of experiments are agent-independent pieces of reality for anyone to access. [10] QBism claims that these points on which it differs from previous Copenhagen-type interpretations resolve the obscurities that many critics have found in the latter, by changing the role that quantum theory plays (even though QBism does not yet provide a specific underlying ontology). Specifically, QBism posits that quantum theory is a normative tool which an agent may use to better navigate reality, rather than a set of mechanics governing it. [22] [42]

Other epistemic interpretations Edit

Approaches to quantum theory, like QBism, [63] which treat quantum states as expressions of information, knowledge, belief, or expectation are called "epistemic" interpretations. [6] These approaches differ from each other in what they consider quantum states to be information or expectations "about", as well as in the technical features of the mathematics they employ. Furthermore, not all authors who advocate views of this type propose an answer to the question of what the information represented in quantum states concerns. In the words of the paper that introduced the Spekkens Toy Model,

if a quantum state is a state of knowledge, and it is not knowledge of local and noncontextual hidden variables, then what is it knowledge about? We do not at present have a good answer to this question. We shall therefore remain completely agnostic about the nature of the reality to which the knowledge represented by quantum states pertains. This is not to say that the question is not important. Rather, we see the epistemic approach as an unfinished project, and this question as the central obstacle to its completion. Nonetheless, we argue that even in the absence of an answer to this question, a case can be made for the epistemic view. The key is that one can hope to identify phenomena that are characteristic of states of incomplete knowledge regardless of what this knowledge is about. [64]

Leifer and Spekkens propose a way of treating quantum probabilities as Bayesian probabilities, thereby considering quantum states as epistemic, which they state is "closely aligned in its philosophical starting point" with QBism. [65] However, they remain deliberately agnostic about what physical properties or entities quantum states are information (or beliefs) about, as opposed to QBism, which offers an answer to that question. [65] Another approach, advocated by Bub and Pitowsky, argues that quantum states are information about propositions within event spaces that form non-Boolean lattices. [66] On occasion, the proposals of Bub and Pitowsky are also called "quantum Bayesianism". [67]

Zeilinger and Brukner have also proposed an interpretation of quantum mechanics in which "information" is a fundamental concept, and in which quantum states are epistemic quantities. [68] Unlike QBism, the Brukner–Zeilinger interpretation treats some probabilities as objectively fixed. In the Brukner–Zeilinger interpretation, a quantum state represents the information that a hypothetical observer in possession of all possible data would have. Put another way, a quantum state belongs in their interpretation to an optimally-informed agent, whereas in QBism, any agent can formulate a state to encode her own expectations. [69] Despite this difference, in Cabello's classification, the proposals of Zeilinger and Brukner are also designated as "participatory realism," as QBism and the Copenhagen-type interpretations are. [6]

Bayesian, or epistemic, interpretations of quantum probabilities were proposed in the early 1990s by Baez and Youssef. [70] [71]

Von Neumann's views Edit

R. F. Streater argued that "[t]he first quantum Bayesian was von Neumann," basing that claim on von Neumann's textbook The Mathematical Foundations of Quantum Mechanics. [72] Blake Stacey disagrees, arguing that the views expressed in that book on the nature of quantum states and the interpretation of probability are not compatible with QBism, or indeed, with any position that might be called quantum Bayesianism. [14]

Relational quantum mechanics Edit

Comparisons have also been made between QBism and the relational quantum mechanics (RQM) espoused by Carlo Rovelli and others. [73] In both QBism and RQM, quantum states are not intrinsic properties of physical systems. [74] Both QBism and RQM deny the existence of an absolute, universal wavefunction. Furthermore, both QBism and RQM insist that quantum mechanics is a fundamentally local theory. [23] [75] In addition, Rovelli, like several QBist authors, advocates reconstructing quantum theory from physical principles in order to bring clarity to the subject of quantum foundations. [76] (The QBist approaches to doing so are different from Rovelli's, and are described below.) One important distinction between the two interpretations is their philosophy of probability: RQM does not adopt the Ramsey–de Finetti school of personalist Bayesianism. [6] [17] Moreover, RQM does not insist that a measurement outcome is necessarily an agent's experience. [17]

Other uses of Bayesian probability in quantum physics Edit

QBism should be distinguished from other applications of Bayesian inference in quantum physics, and from quantum analogues of Bayesian inference. [19] [70] For example, some in the field of computer science have introduced a kind of quantum Bayesian network, which they argue could have applications in "medical diagnosis, monitoring of processes, and genetics". [77] [78] Bayesian inference has also been applied in quantum theory for updating probability densities over quantum states, [79] and MaxEnt methods have been used in similar ways. [70] [80] Bayesian methods for quantum state and process tomography are an active area of research. [81]

Conceptual concerns about the interpretation of quantum mechanics and the meaning of probability have motivated technical work. A quantum version of the de Finetti theorem, introduced by Caves, Fuchs, and Schack (independently reproving a result found using different means by Størmer [82] ) to provide a Bayesian understanding of the idea of an "unknown quantum state", [83] [84] has found application elsewhere, in topics like quantum key distribution [85] and entanglement detection. [86]

Adherents of several interpretations of quantum mechanics, QBism included, have been motivated to reconstruct quantum theory. The goal of these research efforts has been to identify a new set of axioms or postulates from which the mathematical structure of quantum theory can be derived, in the hope that with such a reformulation, the features of nature which made quantum theory the way it is might be more easily identified. [51] [87] Although the core tenets of QBism do not demand such a reconstruction, some QBists—Fuchs, [26] in particular—have argued that the task should be pursued.

One topic prominent in the reconstruction effort is the set of mathematical structures known as symmetric, informationally-complete, positive operator-valued measures (SIC-POVMs). QBist foundational research stimulated interest in these structures, which now have applications in quantum theory outside of foundational studies [88] and in pure mathematics. [89]

The most extensively explored QBist reformulation of quantum theory involves the use of SIC-POVMs to rewrite quantum states (either pure or mixed) as a set of probabilities defined over the outcomes of a "Bureau of Standards" measurement. [90] [91] That is, if one expresses a density matrix as a probability distribution over the outcomes of a SIC-POVM experiment, one can reproduce all the statistical predictions implied by the density matrix from the SIC-POVM probabilities instead. [92] The Born rule then takes the role of relating one valid probability distribution to another, rather than of deriving probabilities from something apparently more fundamental. Fuchs, Schack, and others have taken to calling this restatement of the Born rule the urgleichung, from the German for "primal equation" (see Ur- prefix), because of the central role it plays in their reconstruction of quantum theory. [19] [93] [94]

The following discussion presumes some familiarity with the mathematics of quantum information theory, and in particular, the modeling of measurement procedures by POVMs. Consider a quantum system to which is associated a d < extstyle d>-dimensional Hilbert space. If a set of d 2 < extstyle d^<2>> rank-1 projectors Π ^ i >_> satisfying

Note that the urgleichung is structurally very similar to the law of total probability, which is the expression

The second equality is written in the Heisenberg picture of quantum dynamics, with respect to which the time evolution of a quantum system is captured by the probabilities associated with a rotated SIC measurement < D j >= < U ^ † H ^ j U ^ >< extstyle <>>=<>^>_>>> of the original quantum state ρ ^ >> . Then the Schrödinger equation is completely captured in the urgleichung for this measurement:

Those QBists who find this approach promising are pursuing a complete reconstruction of quantum theory featuring the urgleichung as the key postulate. [93] (The urgleichung has also been discussed in the context of category theory. [96] ) Comparisons between this approach and others not associated with QBism (or indeed with any particular interpretation) can be found in a book chapter by Fuchs and Stacey [97] and an article by Appleby et al. [93] As of 2017, alternative QBist reconstruction efforts are in the beginning stages. [98]

The primary textbook for this course (both semesters) is by Michael Peskin and Daniel Schroeder. To a large extent, the course is based on this book and should follow it fairly closely, but don't expect a 100% match.

Since both the course and the main textbook are introductory in nature, many questions would be left an-answered. The best reference book for finding the answers is by Steven Weinberg. The first two volumes of this three-volume series are based on a two-year course Dr. Weinberg used to teach here at UT &mdash but of course they also contains much additional material. To a first approximation, Dr. Weinberg's book teaches you everything you ever wanted to know about QFT and more &mdash which is unfortunately way too much for a one-year intoductory course. (Weinberg's volume 3 is about supersymmetry, a fascinating subject I would not be able to cover at all in this course.)

I have told the campus bookstore that I use Peskin's book as a textbook for both 396 K and 396 L (Fall 2015, Fall 2016, and Spring 2017), Weinberg's vol.1 as a supplementary texbook for the 396 K (Fall 2015 and Fall 2016) and vol.2 as a supplementary textbook for the 396 L (Spring 2017). I hope the store have stocked the books accordingly, but you should buy them while the supply lasts.

Start reading Chapter 3 in Fetter & Walecka.

Continue reading Chapter 3 in Fetter & Walecka.

  1. The figure above shows a box of mass im sliding on the frictionless surface of an inclined plane (angle θ). The inclined plane itself has a mass M and is supported on a horizontal frictionless surface. Write down the Lagrangian for this system in terms of the generalized coordinates X and s and solve for the equations of motion, assuming that the system is initially at rest.

Introduction to Finite Element Methods

The idea for an online version of Finite Element Methods first came a little more than a year ago. Articles about Massively Open Online Classes (MOOCs) had been rocking the academic world (at least gently), and it seemed that your writer had scarcely experimented with teaching methods. Particularly compelling was the fact that there already had been some successes reported with computer programming classes in the online format, especially as MOOCs. Finite Element Methods, with the centrality that computer programming has to the teaching of this topic, seemed an obvious candidate for experimentation in the online format. From there to the video lectures that you are about to view took nearly a year. I first had to take a detour through another subject, Continuum Physics, for which video lectures also are available, and whose recording in this format served as a trial run for the present series of lectures on Finite Element Methods.

Here they are then, about 50 hours of lectures covering the material I normally teach in an introductory graduate class at University of Michigan. The treatment is mathematical, which is natural for a topic whose roots lie deep in functional analysis and variational calculus. It is not formal, however, because the main goal of these lectures is to turn the viewer into a competent developer of finite element code. We do spend time in rudimentary functional analysis, and variational calculus, but this is only to highlight the mathematical basis for the methods, which in turn explains why they work so well. Much of the success of the Finite Element Method as a computational framework lies in the rigor of its mathematical foundation, and this needs to be appreciated, even if only in the elementary manner presented here. A background in PDEs and, more importantly, linear algebra, is assumed, although the viewer will find that we develop all the relevant ideas that are needed.

The development itself focuses on the classical forms of partial differential equations (PDEs): elliptic, parabolic and hyperbolic. At each stage, however, we make numerous connections to the physical phenomena represented by the PDEs. For clarity we begin with elliptic PDEs in one dimension (linearized elasticity, steady state heat conduction and mass diffusion). We then move on to three dimensional elliptic PDEs in scalar unknowns (heat conduction and mass diffusion), before ending the treatment of elliptic PDEs with three dimensional problems in vector unknowns (linearized elasticity). Parabolic PDEs in three dimensions come next (unsteady heat conduction and mass diffusion), and the lectures end with hyperbolic PDEs in three dimensions (linear elastodynamics). Interspersed among the lectures are responses to questions that arose from a small group of graduate students and post-doctoral scholars who followed the lectures live. At suitable points in the lectures, we interrupt the mathematical development to lay out the code framework, which is entirely open source, and C++ based.

It is hoped that these lectures on Finite Element Methods will complement the series on Continuum Physics to provide a point of departure from which the seasoned researcher or advanced graduate student can embark on work in (continuum) computational physics.

There are a number of people that I need to thank: Shiva Rudraraju and Greg Teichert for their work on the coding framework, Tim O'Brien for organizing the recordings, Walter Lin and Alex Hancook for their camera work and post-production editing, and Scott Mahler for making the studios available.

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6.837 Intro to Computer Graphics Assignment 5: Voxel Rendering

In this assignment and the next, you will make your ray tracer faster using a spatial-acceleration data structure. This week we will focus on the implementation of a grid data structure and on fast ray grid intersection. Next week you will use the grid to accelerate your ray tracer.

To test your grid structure, you will implement the grid as a modeling primitive. Volumetric modeling can be implemented by affecting a binary opaqueness value for each grid cell. This is the equivalent of the discrete pixel representation of 2D images. Each volume element (or voxel) will be rendered as a solid cube. You can easily rasterize simple primitives in a grid (similar to pixel rasterization). For example, to rasterize a sphere, simply compare the distance between the center of a voxel and the sphere center to the sphere radius. You will use the grid to store the objects of your scene. In order to test your object insertion code, you will render cells that contain one or more objects as opaque and use color to visualize the density.

Initially, you may assume that no transformations are used. This way you may effectively ignore the group hierarchy and insert all primitives by scanning the scene in a depth-first manner. For the later test cases you will need to correctly transform the bounding boxes of each primitive before rasterizing it to the grid.


To avoid creating an infinite bounding box, don't try to compute the bounding box of a Plane (the BoundingBox of this primitive should be NULL). Don't include this infinite primitive when computing the bounding box of a group. Test your bounding box code by printing out the intermediate and scene bounding boxes for various test cases from previous assignments. Make sure your code handles scenes whose bounding box does not include the origin.

Internal to your grid class, an array of nx x ny x nz boolean values stores whether each voxel is opaque or transparent. Later we'll replace this array of booleans with an array of arrays of Objects3D.

Note that this function is not pure virtual. Initially this function will do nothing. Override this function for spheres by implementing the corresponding method for the Sphere class. This method sets the opaqueness of each voxel which may intersect the sphere. You can do this by comparing the center of voxel to the sphere center and radius. Your computation should be conservative: you must label cells which might overlap the sphere, but it's also ok to label cells opaque which do not overlap the sphere. You may ignore the 2nd argument (Matrix *m) for this part, it will be used later in the assignment.

You will need a place to store the information for the current ray and the current grid cell. Implement a MarchingInfo class that stores the current value of tmin the grid indices i, j, and k for the current grid cell the next values of intersection along each axis (t_next_x, t_next_y, and t_next_z) the marching increments along each axis (d_tx, d_ty, d_tz), and the sign of the direction vector components (sign_x, sign_y, and sign_z). To render the occupied grid cells for visualization you will also need to store the surface normal of the cell face which was crossed to enter the current cell. Write the appropriate accessors and modifiers.

This function computes the marching increments and the information for the first cell traversed by the ray. Make sure to treat all three intersection cases: when the origin is inside the grid, when it is outside and the ray hits the grid, and when it is outside and it misses the grid. Test your initializeRayMarch routine by manually casting simple axis-aligned rays into a low-resolution grid. Also test more general cases. Make sure to test ray origins which are inside and outside the scene bounding box. Next, implement:

This update routine choose the smallest of the next t values (t_next_x, t_next_y, and t_next_z), and updates the corresponding cell index. Test your nextCell ray marching code using the same strategy as for initialization. Manually compute the marching sequence corresponding to a particular ray and then print the steps taken by your code. Try other origins and directions to make sure that your code works for all orientations (in particular, test both positive and negative components of the direction).

GLCanvas::initialize method has been modified to take in two additional parameters, the Grid, and a boolean indicating whether to visualize the grid or not. Insert the following commands within your Grid intersection routines as appropriate:

To specify an entire cell, call AddHitCellFace 6 times. In the examples below, a color gradient has been used to show the order in which the cells are traversed (white, purple, . orange, red). This gradient is optional, but can be helpful in debugging. To see the ray intersections more clearly, you may wish to turn off the transparent ray rendering in RayTree::paint()

In the examples below, a color gradient has been used to visualize the number of primitives which overlap each grid cell. Cells colored white have just 1 primitive, purple have 2 primitives, . and cells colored red have many more. You should implement a similar visualization, but you may use a different color scheme.

You are not required to handle transformations in your Sphere::insertIntoGrid method. If m &ne NULL, it's ok to fall back on the Object3D::insertIntoGrid implementation. To explicitly call a parent class method, pre-pend the method with the class name:

Ideas for Extra Credit

  • Implement a special case for transformations of triangle primitives to get a tighter bounding box
  • Test if the plane of the triangles intersects the grid cells (less useful for small triangles)
  • Volumetric rasterization of other fun implicit objects

Input Files

New Triangle Models

Sample Results

Visualize Hit Cells Visualize Entered Faces

Note: the grid voxelization of the green triangle uses an optional special case for transformed triangles and will look different if you have not implemented this option.

Note: the grid voxelization of the blue rhombus uses an optional special case for transformed triangles and will look different if you have not implemented this option.

Energetic Materials

5.1 Computational model

To directly simulate the condensed-phase chemical reactivity of HMX, we use the SCC-DFTB method to determine the interatomic forces and simulate the decomposition at constant-volume and temperature conditions. The initial condition of the simulation included six HMX molecules in a cell, corresponding to the unit cell of the δ phase of HMX ( Fig. 10 ) with a total of 168 atoms. It is well known [76] that HMX undergoes a phase transition at 436 K from the β phase (two molecules per unit cell with a chair molecular conformation, density = 1.89 g/cm 3 ) to the δ phase (with boat molecular conformation, density = 1.50 g/cm 3 ). We thus chose the δ phase as the initial starting structure so as to include all the relevant physical attributes of the system prior to chemical decomposition. The calculation started with the experimental unit cell parameters and atomic positions of δ HMX. The atomic positions were then relaxed in an energy minimization procedure. The resulting atomic positions were verified to be close to the experimental positions.

The volume of the cell was then reduced to the final density of the simulation. The atomic structure was subsequently fully optimised at the corresponding cell volume. Our intention is to study the high- pressure and high- temperature chemistry of HMX in general, so the exact density and temperature used in our simulation is somewhat arbitrary. We used a density of 1.9 g/cm 3 and a temperature of 3500 K. This state is in the neighborhood of the Chapman-Jouget state of β-HMX (3500 K, 2.1 g/ cm 3 ) as predicted through thermochemical calculations described later. The closest experimental condition corresponding to our simulation would be a sample of HMX, which is suddenly heated under constant volume conditions, such as in a diamond anvil cell.

The molecular dynamics simulation was conducted at constant volume and constant temperature. Periodic boundary conditions, whereby a particle exiting the cell on one side is reintroduced on the opposing side with the same velocity were imposed. Constant temperature conditions were implemented through simple velocity rescaling. The probability to rescale atom velocities was chosen to be 0.1 per time step. A dynamic time-step of 0.5 fs was used, and snapshots at 2.5 fs steps were collected.

A procedure was implemented to identify the product molecules of interest: H2O, N2, CO2, and CO. Covalent bonds were identified according to bond distance. This is motivated by the difference between covalent (1 - 1.7 Å) and van der Waals bond distances (

3 Å). We chose maximum bond distances of: R (O-H) = 1.3, R (CO) = 1.7, and R(N-N) = 1.5 Å in the molecule identification procedure. The results of the molecular identification procedure were confirmed through visual examination of representative simulation steps. We note that the procedure may incorrectly identify transition states as being molecular species. Since transition states are short lived, and since we apply the procedure to small molecules, this problem should not significantly affect the time averaged concentrations reported here.

11: 06 Pre-Class Assignment - Matrix Mechanics

We have discussed assignment operator overloading for dynamically allocated resources here . This is a an extension of the previous post. In the previous post, we discussed that when we don’t write our own assignment operator, compiler created assignment operator does shallow copy and that cause problems. What happens when we have references in our class and there is no user defined assignment operator. For example, predict the output of following program.

Compiler doesn’t creates default assignment operator in following cases

1. Class has a nonstatic data member of a const type or a reference type
2. Class has a nonstatic data member of a type which has an inaccessible copy assignment operator
3. Class is derived from a base class with an inaccessible copy assignment operator

When any of the above conditions is true, user must define assignment operator. For example, if we add an assignment operator to the above code, the code works fine without any error.

Watch the video: Οδηγίες για εξ αποστάσεως εκπαίδευση για τον καθηγητή - (October 2021).