Understanding Of Quantum Mechanics Research Articles

"No Poverty, Much Comfort, Little Wealth": Bertrand Russell's 1935 Scandinavian Tour

Bertrand Russell’s Scandinavian lecture tour in October 1935 has been largely undocumented because of the longstanding embargo on the tour correspondence Russell exchanged with Marjorie (“Peter”) Spence, his lover and future third wife. These archival restrictions ended in 2009, and this paper presents annotated transcriptions of twenty letters sent by Russell to Peter during his trip to Denmark, Sweden, and Norway. The tour allowed Russell to test early versions of two important papers in his return to philosophy in the mid-193s, and his meetings with Niels Bohr and Werner Heisenberg in Copenhagen significantly enhanced his understanding of quantum mechanics. Additionally, these letters demonstrate the impact of the tour on Russell’s political thought and his evolving views on fascism and pacifism, and they provide the first intimate picture of Russell’s complex relationship with Peter.

“No Poverty, Much Comfort, Little Wealth”: Bertrand Russell’s 1935 Scandinavian Tour

Bertrand Russell’s Scandinavian lecture tour in October 1935 has been largely undocumented because of the longstanding embargo on the tour correspondence Russell exchanged with Marjorie (“Peter”) Spence, his lover and future third wife. These archival restrictions ended in 2009, and this paper presents annotated transcriptions of twenty letters sent by Russell to Peter during his trip to Denmark, Sweden, and Norway. The tour allowed Russell to test early versions of two important papers in his return to philosophy in the mid-1930s, and his meetings with Niels Bohr and Werner Heisenberg in Copenhagen significantly enhanced his understanding of quantum mechanics. Additionally, these letters demonstrate the impact of the tour on Russell’s political thought and his evolving views on fascism and pacifism, and they provide the first intimate picture of Russell’s complex relationship with Peter.

Wave-particle duality and uncertainty principle: Phenomenographic categories of description of tertiary physics students’ depictions

Quantum mechanics is often thought to be a difficult subject to understand, not only in the complexity of its mathematics but also in its conceptual foundation. In this paper we emphasize students’depictions of the uncertainty principle and wave-particle duality of quantum events, phenomena that could serve as a foundation in building an understanding of quantum mechanics. A phenomenographic study was carried out to categorize a picture of students’ descriptions of these key quantum concepts. Data for this study were obtained from a semistructured in-depth interview conducted with undergraduate physics students (N ¼ 25) from Bahir Dar, Ethiopia. The phenomenographic data analysis revealed that it is possible to construct three qualitatively different categories to map students’ depictions of the concept wave-particle duality, namely, (1) classical description, (2) mixed classical-quantum description, and (3) quasiquantum description. Similarly, it is proposed that students’ depictions of the concept uncertainty can be described with four different categories of description, which are (1) uncertainty as an extrinsic property of measurement, (2) uncertainty principle as measurement error or uncertainty, (3) uncertainty as measurement disturbance, and (4) uncertainty as a quantum mechanics uncertainty principle. Overall, we found students are more likely to prefer a classical picture of interpretations of quantum mechanics. However, few students in the quasiquantum category applied typical wave phenomena such as interference and diffraction that cannot be explained within the framework classical physics for depicting the wavelike properties of quantum entities. Despite inhospitable conceptions of the uncertainty principle and waveand particlelike properties of quantum entities in our investigation, the findings presented in this paper are highly consistent with those reported in previous studies. New findings and some implications for instruction and the curricula are discussed.

Four: A Reflection on the Wholeness of Nature, Part II

This meditation on the symbol of the number four is based on a series of personal dreams that came in August of 1999, after I completed my Ph.D. in quantum physics. Weaving around a second dream in this series, I explore the alchemical axiom of Maria Prophetissa and argue that it formulates a worldview rooted in the principle of life. After the Enlightenment, the alchemical goal transformed into what we value the most in our modern world: objectivity. Describing first the rise of mechanical objectivity in the nineteenth century, with its moral rejection of all forms of subjectivity, I show how subjectivity gradually returned to the scene of physical science by considering recent advances in the understanding of quantum mechanics. The concept of a physoid reality introduced in the first article of this series is investigated in this context of a return of the subjective.

Consistent treatments of quantum mechanics

The Letters reply by Yakir Aharonov, Sandu Popescu, and Jeff Tollaksen gives a misleading impression of the consistent histories interpretation of quantum mechanics. As I pointed out in the same issue, consistent histories is an alternative to their time-symmetric formulation. It resolves the “collapse problem” of quantum foundations, which Aharonov and coauthors admit they have failed to do. However, that is not its only or even its principal virtue. It also solves what I call the second quantum measurement problem: how to infer from the measurement outcome the state of the measured system at an earlier time before the measurement took place. Thus the consistent histories approach, without any fuzzy references to “ensembles,” provides a proper probabilistic framework for inferring that Sx for a spin-1⁄2 particle actually was −1⁄2 before the measurement took place. This approach evades the criticisms expressed by Michael Nauenberg and Art Hobson in their letters in the same issue.Furthermore, consistent histories, unlike certain aspects of the approach of Aharonov and coauthors, is fully consistent with the Hilbert-space structure of quantum mechanics introduced by John von Neumann and now taught to all students of the subject. Indeed, with a consistent histories analysis one can locate fairly precisely11. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 the error in reasoning that leads to claims such as that the measured spin of a spin-1⁄2 particle can equal 100.22. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 Claims about what weak measurements actually measure, when made by those who have not yet solved either the collapse problem or the second measurement problem, should be treated with great caution.The claim that the consistent histories approach is controversial was true back in the 1990s, and the controversy led to a clearer formulation of the interpretation. However, no significant problems have come to light in the decade since the publication of my book.33. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. The fact that something has been ignored does not imply it is controversial. Admirers of Richard Feynman, some of whose writings were cited by Aharonov and coauthors in their reply, may be interested to know that shortly before his death he was quite sympathetic to an early form of the consistent histories or decoherent histories approach to understanding quantum mechanics (see PHYSICS TODAY, February 1999, page 11)—an open-mindedness worth imitating.REFERENCESSection:ChooseTop of pageREFERENCES <<1. R. E. Kastner, Stud. Hist. Phil. Mod. Phys. 35, 57 (2004), http://arxiv.org/abs/quant-ph/0207182. https://doi.org/10.1016/j.shpsb.2003.02.001 , Google ScholarCrossref, ISI2. Y. Aharonov, D. Z. Albert, L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). https://doi.org/10.1103/PhysRevLett.60.1351 , Google ScholarCrossref, ISI3. R. B. Griffiths, Consistent Quantum Theory Cambridge U. Press, New York (2002), http://quantum.phys.cmu.edu/CQT/. Google Scholar© 2011 American Institute of Physics.

Double well potentials and quantum gates

A system of particles in a double well potential is a widely studied and useful example for understanding quantum mechanics. This simple system has recently been used in theoretical proposals and related experiments as a way to make quantum logic gates for ultracold atoms confined in optical lattices. Such quantum gates are the fundamental building blocks for quantum information processing; in these proposals, a regular array of cold atoms in an optical lattice serves as the quantum register. We explain how this research can be understood in terms of well-known principles for systems of identical particles.

Rovibrational eigenenergy structure of the [H,C,N] molecular system

The vibrational-rotational eigenenergy structure of the [H,N,C] molecular system is one of the key features needed for a quantum mechanical understanding of the HCN⇌HNC model reaction. The rotationless vibrational structure corresponding to the multidimensional double well potential energy surface is well established. The rotational structure of the bending vibrational states up to the isomerisation barrier is still unknown. In this work the structure of the rotational states for low and high vibrational angular momentum is described from the ground state up to the isomerisation barrier using hot gas molecular high resolution spectroscopy and rotationally assigned ab initio rovibronic states. For low vibrational angular momentum the rotational structure of the bending excitations splits in three regions. For J < 40 the structure corresponds to that of a typical linear molecule, for 40 < J < 60 has an approximate double degenerate structure and for J > 60 the splitting of the e and f components begins to decrease and the rotational constant increases. For states with high angular momentum, the rotational structure evolves into a limiting structure for v(2) > 7--the molecule is locked to the molecular axis. For states with v(2) > 11 the rotational structure already begins to accommodate to the lower rotational constants of the isomerisation states. The vibrational energy begins to accommodate to the levels above the barrier only at high vibrational excitations of v(2) > 22 just above the barrier whereas this work shows that the rotational structure is much more sensitive to the double well structure of the potential energy surface. The rotational structure already experiences the influence of the barrier at much lower energies than the vibrational one.

Time-symmetric quantum mechanics questioned and defended

Time-symmetric quantum mechanics questioned and defended

Improving students’ understanding of quantum mechanics via the Stern–Gerlach experiment

The Stern–Gerlach experiment can play an important role in teaching the formalism of quantum mechanics. In the context of a finite-dimensional Hilbert space students can learn how to prepare a specific quantum state starting from an arbitrary state, issues related to the time evolution of the wave function and quantum measurement. The Stern–Gerlach experiment can also be used to teach the distinction between the physical space where the experiment is performed and the Hilbert space where the state of the system lies, and how information about the state of the system in the Hilbert space can be exploited to interpret the possible outcomes of the experiment in physical space. Students can learn the advantages of choosing an appropriate basis to make predictions about the outcomes of experiments with different arrangements of Stern–Gerlach devices. The latter can also help students understand that an ensemble of identically prepared systems is not the same as a mixture. We discuss student difficulties with the Stern–Gerlach experiment based on written tests and interviews with advanced undergraduate and graduate students in quantum mechanics courses. We also discuss preliminary data which suggest that the Quantum Interactive Learning Tutorial on the Stern–Gerlach experiment is helpful in improving student understanding of these concepts.

Learning and retention of quantum concepts with different teaching methods

We measured mastery and retention of conceptual understanding of quantum mechanics in a modern physics course. This was studied for two equivalent cohorts of students taught with different pedagogical approaches using the Quantum Mechanics Conceptual Survey. We measured the impact of pedagogical approach both on the original conceptual learning and on long-term retention. The cohort of students who had a very highly rated traditional lecturer scored 19% lower than the equivalent cohort that was taught using interactive engagement methods. However, the amount of retention was very high for both cohorts, showing only a few percent decrease in scores when retested 6 and 18 months after completion of the course and with no exposure to the material in the interim period. This high level of retention is in striking contrast to the retention measured for more factual learning from university courses and argues for the value of emphasizing conceptual learning.

Nonlinear quantum mechanics, the superposition principle, and the quantum measurement problem

There are four reasons why our present knowledge and understanding of quantum mechanics can be regarded as incomplete. (1) The principle of linear superposition has not been experimentally tested for position eigenstates of objects having more than about a thousand atoms. (2) There is no universally agreed upon explanation for the process of quantum measurement. (3) There is no universally agreed upon explanation for the observed fact that macroscopic objects are not found in superposition of position eigenstates. (4) Most importantly, the concept of time is classical and hence external to quantum mechanics: there should exist an equivalent reformulation of the theory which does not refer to an external classical time. In this paper we argue that such a reformulation is the limiting case of a nonlinear quantum theory, with the nonlinearity becoming important at the Planck mass scale. Such a nonlinearity can provide insights into the aforesaid problems. We use a physically motivated model for a nonlinear Schrodinger equation to show that nonlinearity can help in understanding quantum measurement. We also show that while the principle of linear superposition holds to a very high accuracy for atomic systems, the lifetime of a quantum superposition becomes progressively smaller, as one goes from microscopic to macroscopic objects. This can explain the observed absence of position superpositions in macroscopic objects (lifetime is too small). It also suggests that ongoing laboratory experiments may be able to detect the finite superposition lifetime for mesoscopic objects in the near future.

Design and validation of the Quantum Mechanics Conceptual Survey

The Quantum Mechanics Conceptual Survey (QMCS) is a 12-question survey of students' conceptual understanding of quantum mechanics. It is intended to be used to measure the relative effectiveness of different instructional methods in modern physics courses. In this paper, we describe the design and validation of the survey, a process that included observations of students, a review of previous literature and textbooks and syllabi, faculty and student interviews, and statistical analysis. We also discuss issues in the development of specific questions, which may be useful both for instructors who wish to use the QMCS in their classes and for researchers who wish to conduct further research of student understanding of quantum mechanics. The QMCS has been most thoroughly tested in, and is most appropriate for assessment of (as a posttest only), sophomore-level modern physics courses. We also describe testing with students in junior quantum courses and graduate quantum courses, from which we conclude that the QMCS may be appropriate for assessing junior quantum courses, but is not appropriate for assessing graduate courses. One surprising result of our faculty interviews is a lack of faculty consensus on what topics should be taught in modern physics, which has made designing a test that is valued by a majority of physics faculty more difficult than expected.

Quantum logic as a dynamic logic

We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others (Among whom we may count or not E. W. Beth, depending on how we interpret some of his statements), our answer is a clear “no”. Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth’s proposal of applying Tarski’s semantical methods to the analysis of physical theories, with an empirical–experimental approach to Logic, as advocated by both Beth and Putnam, but understood by us in the view of the operational- realistic tradition of Jauch and Piron, i.e. as an investigation of “the logic of yes–no experiments” (or “questions”). Technically, we use the recently-developed setting of Quantum Dynamic Logic (Baltag and Smets 2005, 2008) to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results (Baltag and Smets 2005), we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the “non-classicality” of quantum behavior than any perspective based on static Propositional Logic.

A tutorial on the precessional behaviour of hydrogen nuclei in external magnetic fields

The purpose of this tutorial is to derive the precessional characteristics of the magnetic moments of hydrogen nuclei in the presence of a constant external magnetic field using the Dirac bra-ket formulation of quantum mechanics (QM). This behaviour has many applications, most notably in nuclear magnetic resonance (NMR) and magnetic resonance (MR) imaging. Many NMR and MR imaging textbooks claim that the QM expectation value of the magnetic moment of a proton in a magnetic field reduces to the classical picture of a precessing magnetic dipole. This paper validates this conclusion by reducing the cumbersome QM integrals using Dirac notation and matrix algebra, and then comparing this result with the classical picture. It illustrates the connections between the quantum and classical pictures and demonstrates quantitatively how they differ and how they are similar. This tutorial targets students and researchers interested in the fundamental physics behind the MR phenomenon and assumes that the reader has a basic understanding of QM. This may be helpful for undergraduate QM students learning about spin angular momentum and Dirac notation.

Approximate quantum mechanical method for describing excitations and related properties of finite single-walled carbon nanotubes

Optical properties of three kinds of zigzag (5,0), (13,0), and (9,0) single-walled carbon nanotubes (SWCNTs) are studied using an approximate quantum mechanical method named complete neglect of differential overlap, which distinguishes basis atomic orbitals with different azimuthal $l$ quantum numbers (CNDOL). This method models the electron energy transitions and excited state charge distributions through a configuration interaction of singly (CIS) excited determinants allowing the direct understanding of properties related with the total electronic wave function of nanoscopic systems, projecting a reliable quantum mechanical understanding to real life objects. The finite SWCNT's structures were obtained by replicating the unit cells of periodic SWCNTs and saturating the edge dangling bonds with hydrogens. The unit cell was previously relaxed using standard density functional theory methods. The behavior of these SWCNTs were interpreted in the framework of the CNDOL scheme by increasing the lengths of the tubes above 3 nm. As the nanotubes grow in length, the position of excited states for each SWCNT evolve differently: in contrast with (9,0) SWCNT, which exhibits favorable conditions for photoexcitation, the (13,0) and (5,0) SWCNTs do not show a lowering of the lowest excited states. This behavior is discussed by taking into account electron---electron interactions as considered in the framework of the CIS procedure. Furthermore, the (13,0) and (5,0) SWCNTs present forbidden transitions for the lowest excitations and its first dipole-allowed transitions are at 0.9--1.0 and 1.4--1.6 eV, respectively. In contrast, (9,0) SWCNT allows excitations by photon at less than 0.4 eV as the length of the nanotube tends to infinite. Excitons appear more bounded, energetically and spatially, in the (13,0) than in the (9,0) and (5,0) SWCNTs.

Homeopathy: Ex nihilo fit nihil

Homeopathy is an eloquent example of the natural tendency of human beings to delude themselves. It is a cure that does not make sense in the light of science and whose usefulness has not been firmly demonstrated in any disease. At best it probably is a waste of resources. Yet still the popularity of this practice seems to be intact if not on the increase. In Europe, pharmacies continue to sell questionable products like Oscillococcinum alongside “real” medicines, many doctors practice homeopathy full or part-time, medical faculties hold courses and confer Master qualifications in homeopathy, sympathizers react angrily to any factual criticism. Why? Saying that homeopathy cannot possibly work because it has no scientifically plausible grounds has proven useless. Strangely enough, a clear-cut argument like the one stating that a molecule cannot have any biological effect after it has been totally removed from a given remedy has failed to dissuade patients and many health operators. Denounciations of the improbable water chemistry of homeopathy, the gratuitousness of the “Law of Infinitesimals” and of the principle of “similia similibus curantur” have proved equally ineffective. And finding faults in homeopathy – as well as in any other alternative medicine – is now seen in a broad sense as unbecoming. As the American mathematician Norman Levitt says, “we have moved from the concept of equality of individuals to equality of ideas and beliefs. Today it is politically incorrect to call something a dumb idea.” Supporters of homeopathy have answered accusations of groundlessness with an array of doubtful though ingenious explanations which have apparently been successful. They concede – and how could they do otherwise? – that the scientific foundations of homeopathy are not strictly proven but they maintain that the practice is plausible in the light of modern developments of science like quantum mechanics. Here are a few aspects of this type of argumentation. According to Richard Feynman nobody understands quantummechanics. However, quantum mechanics “works”, and according to supporters of homeopathy this shows that ignoring the mechanism of a physical action (such as the supposed transfer of information from the solute to the solvent) is not a sufficient reason to deny its existence. Similarly, Heisenberg's “uncertainty principle” is often quoted as a statement in support of the fact that the world is unpredictable and, therefore, that in nature nothing can be ruled out a priori. Nonintuitive quantistic concepts like “entanglement” and “collapse of the wave

H. Weyl, Mind and Nature: Selected Writings on Philosophy, Mathematics and Physics; edited and with an introduction by P. Pesic; Princeton: Princeton University Press, pp. 261, £19.95 HB

One of the greatest mathematicians who ever lived, Hermann Weyl also made significant contributions to physics: applying group theory to quantum mechanics, introducing the concept of a gauge field and discovering new solutions to Einstein’s equations. His 1918 book on general relativity, Space-Time-Matter, was described by Einstein as a ‘symphonic masterpiece’, and by Husserl, with whom Weyl had studied, as the book he would have written if only he’d known more physics! Even for a product of his time, Weyl was astonishingly reflective about the philosophical foundations of his work, as is evident in his more accessible books, Symmetry and Philosophy of Mathematics and Natural Science. Pesic has collected here a selection of Weyl’s writings that beautifully present the underlying synthesis of philosophy, mathematics and physics in his work, and in particular those that have remained unpublished, untranslated or that have simply fallen out if print. Beginning with the gauge-oriented essay on ‘Electricity and Gravitation’ the volume also includes an exchange of letters between Weyl and Einstein in which Weyl identifies the statistical nature of quantum physics as effecting a fundamental change in our world-view, as well as essays on the nature of time, ‘Mind and Nature’ and the unity of knowledge. The final piece, ‘Insight and Reflection’, is particularly rich, reflecting as it does on Weyl’s own philosophical reflectiveness. Veering off from biographical recollections to describe the ‘reasonable attitude on the problem of space’ and the ‘Pythagorean’ nature of the metric, Weyl describes how it was Husserl who lead him away from positivism and how, of all his spiritual experiences, the ones that brought him the greatest happiness were his readings of Hilbert in 1905 and Eckhart in 1922. But although he touches on his profound group-theoretic understanding of quantum mechanics in his 1946 address to the Princeton Bicentennial Conference, one is left with a sense that a truly integrative

Electron diffraction chez Thomson: early responses to quantum physics in Britain

AbstractIn 1927, George Paget Thomson, professor at the University of Aberdeen, obtained photographs that he interpreted as evidence for electron diffraction. These photographs were in total agreement with de Broglie's principle of wave–particle duality, a basic tenet of the new quantum wave mechanics. His experiments were an initially unforeseen spin-off from a project he had started in Cambridge with his father, Joseph John Thomson, on the study of positive rays. This paper addresses the scientific relationship between the Thomsons, father and son, as well as the influence that the institutional milieu of Cambridge had on the early work of the latter. Both Thomsons were trained in the pedagogical tradition of classical physics in the Cambridge Mathematical Tripos, and this certainly influenced their understanding of quantum physics and early quantum mechanics. In this paper, I analyse the responses of both father and son to the photographs of electron diffraction: a confirmation of the existence of the ether in the former, and a partial embrace of some ideas of the new quantum mechanics in the latter.

Simulation of n-qubit quantum systems. V. Quantum measurements

Abstract The Feynman program has been developed during the last years to support case studies on the dynamics and entanglement of n-qubit quantum registers. Apart from basic transformations and (gate) operations, it currently supports a good number of separability criteria and entanglement measures, quantum channels as well as the parametrizations of various frequently applied objects in quantum information theory, such as (pure and mixed) quantum states, hermitian and unitary matrices or classical probability distributions. With the present update of the Feynman program, we provide a simple access to (the simulation of) quantum measurements. This includes not only the widely-applied projective measurements upon the eigenspaces of some given operator but also single-qubit measurements in various pre- and user-defined bases as well as the support for two-qubit Bell measurements. In addition, we help perform generalized and POVM measurements. Knowing the importance of measurements for many quantum information protocols, e.g., one-way computing, we hope that this update makes the Feynman code an attractive and versatile tool for both, research and education. New version program summary Program title: FEYNMAN Catalogue identifier: ADWE_v5_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADWE_v5_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 27 210 No. of bytes in distributed program, including test data, etc.: 1 960 471 Distribution format: tar.gz Programming language: Maple 12 Computer: Any computer with Maple software installed Operating system: Any system that supports Maple; the program has been tested under Microsoft Windows XP and Linux Classification: 4.15 Catalogue identifier of previous version: ADWE_v4_0 Journal reference of previous version: Comput. Phys. Commun. 179 (2008) 647 Does the new version supersede the previous version?: Yes Nature of problem: During the last decade, the field of quantum information science has largely contributed to our understanding of quantum mechanics, and has provided also new and efficient protocols that are used on quantum entanglement. To further analyze the amount and transfer of entanglement in n-qubit quantum protocols, symbolic and numerical simulations need to be handled efficiently. Solution method: Using the computer algebra system Maple, we developed a set of procedures in order to support the definition, manipulation and analysis of n-qubit quantum registers. These procedures also help to deal with (unitary) logic gates and (nonunitary) quantum operations and measurements that act upon the quantum registers. All commands are organized in a hierarchical order and can be used interactively in order to simulate and analyze the evolution of n-qubit quantum systems, both in ideal and noisy quantum circuits. Reasons for new version: Until the present, the FEYNMAN program supported the basic data structures and operations of n-qubit quantum registers [1], a good number of separability and entanglement measures [2], quantum operations (noisy channels) [3] as well as the parametrizations of various frequently applied objects, such as (pure and mixed) quantum states, hermitian and unitary matrices or classical probability distributions [4]. With the current extension, we here add all necessary features to simulate quantum measurements, including the projective measurements in various single-qubit and the two-qubit Bell basis, and POVM measurements. Together with the previously implemented functionality, this greatly enhances the possibilities of analyzing quantum information protocols in which measurements play a central role, e.g., one-way computation. Running time: Most commands require ⩽10 seconds of processor time on a Pentium 4 processor with ⩾ 2 GHz RAM or newer, if they work with quantum registers with five or less qubits. Moreover, about 5–20 MB of working memory is typically needed (in addition to the memory for the Maple environment itself). However, especially when working with symbolic expressions, the requirements on the CPU time and memory critically depend on the size of the quantum registers owing to the exponential growth of the dimension of the associated Hilbert space. For example, complex (symbolic) noise models, i.e. with several Kraus operators, may result in very large expressions that dramatically slow down the evaluation of e.g. distance measures or the final-state entropy, etc. In these cases, Maple's assume facility sometimes helps to reduce the complexity of the symbolic expressions, but more often than not only a numerical evaluation is feasible. Since the various commands can be applied to quite different scenarios, no general scaling rule can be given for the CPU time or the request of memory.

Graduate quantum mechanics reform

We address four main areas in which graduate quantum mechanics education can be improved: course content, textbook, teaching methods, and assessment tools. We report on a three year longitudinal study at the Colorado School of Mines using innovations in all these areas. In particular, we have modified the content of the course to reflect progress in the field in the last 50 years, used textbooks that include such content, incorporated a variety of teaching techniques based on physics education research, and used a variety of assessment tools to study the effectiveness of these reforms. We present a new assessment tool, the Graduate Quantum Mechanics Conceptual Survey, and further testing of a previously developed assessment tool, the Quantum Mechanics Conceptual Survey. We find that graduate students respond well to research-based techniques that have been tested mainly in introductory courses, and that they learn much of the new content introduced in each version of the course. We also find that students' ability to answer conceptual questions about graduate quantum mechanics is highly correlated with their ability to solve calculational problems on the same topics. In contrast, we find that students' understanding of basic undergraduate quantum mechanics concepts at the modern physics level is not improved by instruction at the graduate level.