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Title: causality and a possible interpretation of quantum mechanics.
Abstract: From the ancient Einstein-Podolsky-Rosen paradox to the recent Sorkin-type impossible measurements problem, the contradictions between relativistic causality, quantum non-locality, and quantum measurement have persisted. Our work provides a framework based on quantum field theory to harmoniously integrate these three aspects. This framework consists of causality expressed by reduced density matrices and an interpretation of quantum mechanics that considers quantum mechanics to be complete. Specifically, we utilize reduced density matrices to characterize the local information of the quantum state and demonstrate that they cannot evolve superluminally. Unlike recent approaches focusing on causality, we do not introduce new operators or fields specifically to describe detectors; instead, everything (including detectors, environments, and humans) is composed of the same fundamental fields, leading to complex renormalization. It is precisely these renormalization that prompts us to question the validity of the derivation of quantum paradoxes and lead us to propose a very natural and relativistically compatible interpretation of quantum mechanics.
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Some Unusual Topics in Quantum Mechanics
- Pankaj Sharan 0
Department of Physics, Jamia Millia Islamia University (emeritus), New Delhi, India
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With a chapter on non-locality and Bell's inequality – Nobel prize 2022
Gives a readable account of the mathematical foundations of quantum mechanics
Addresses topics needed for deeper understanding of the subject
Part of the book series: Lecture Notes in Physics (LNP, volume 1020)
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Table of contents (17 chapters)
Front matter, position operators of non-relativistic quantum mechanics.
A Bundle Picture of Quantum Mechanics
A beam of particles \(=\) a plane wave, star-product formulation of quantum mechanics, can there be a non-linear quantum mechanics, interaction \(=\) exchange of quanta, proof of wigner’s theorem, hilbert space: an introduction, what is an “essentially self-adjoint” operator, is there a time-energy uncertainty relation, fock spaces, second quantization, relativistic configuration states and quantum fields, minimum uncertainty states, path integrals in general coordinates, a brief pre-history of matrix mechanics and o(4) symmetry of the hydrogen atom, non-locality in quantum mechanics and bell’s inequality, back matter.
This second edition of Some Unusual Topics in Quantum Mechanics builds upon the topics covered in the first, with additional chapters that delve deeper into the mathematical foundations of the subject. New topics include Hilbert spaces and unbounded operators, minimum uncertainty states, path integrals in general coordinates, Fock spaces, second quantization, relativistic particle states, and quantum fields. Historical insights are also included, such as a pre-history of matrix mechanics and Pauli's proof of the H-atom spectrum using O(4) symmetry. Finally, readers are introduced to Bell's inequality and the non-locality in quantum mechanics that is revealed through its violation. These topics are rarely covered in introductory textbooks but are crucial to developing a student's interest and deeper understanding of quantum mechanics. This book serves as valuable supporting material for graduate-level core courses on the subject.
- Bundle with Hilbert space fiber
- History of matrix mechanics
- Non-linear quantum mechanics
- Position operators
- Relativistic configuration states
- Rigged Hilbert spaces
- Self-adjoint operators
- Time re-parametrization in path integral
- Time-energy uncertainty relations
- Wigner's theorem
Pankaj Sharan is a professor retired from the Department of Physics at Jamia Millia Islamia, New Delhi. His research interests are chiefly in quantum mechanics, quantum field theory and general relativity. Over a teaching career spanning four decades, he has taught courses on quantum mechanics, quantum field theory, classical mechanics, general relativity and mathematical physics. He is the author of the book Spacetime, Geometry and Gravitation (Birkhäuser, 2009).
Book Title : Some Unusual Topics in Quantum Mechanics
Authors : Pankaj Sharan
Series Title : Lecture Notes in Physics
DOI : https://doi.org/10.1007/978-3-031-35962-0
Publisher : Springer Cham
eBook Packages : Physics and Astronomy , Physics and Astronomy (R0)
Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023
Softcover ISBN : 978-3-031-35961-3 Published: 22 September 2023
eBook ISBN : 978-3-031-35962-0 Published: 21 September 2023
Series ISSN : 0075-8450
Series E-ISSN : 1616-6361
Edition Number : 2
Number of Pages : XXII, 314
Number of Illustrations : 18 b/w illustrations
Topics : Quantum Physics , Elementary Particles, Quantum Field Theory , Mathematical Physics
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100+ Quantum Mechanics Research Topics
Quantum mechanics is a basic physics theory that describes how matter and energy behave at the atomic and subatomic levels. It lays the groundwork for comprehending topics such as quantum entanglement, spin, and tunnelling. Quantum mechanics provides a vast field for theoretical and experimental studies for graduate students and researchers.
This page compiles approximately 100 prospective study topics in quantum mechanics disciplines such as quantum information science, quantum optics, quantum computing, quantum chaos, topological quantum matter, and quantum technologies.
- 100+ IoT Research Topics for Final Year Projects
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Foundations of Quantum Mechanics
- Investigations of quantum contextuality and nonlocality using graph theory approaches
- Information-theoretic reconstructions of quantum theory foundations and dynamics
- Operational probabilistic theories as generalizations of quantum mechanics
- Quantum measurement theory and modeling of continuous measurement processes
- Ontological models and hidden variable theories for quantum paradoxes
- Categorical semantics approaches for quantum processes and computations
- Compositionally dynamic system theory as a framework for quantum mechanics
- Quantum generalizations of classical stochastic processes and Markov chains
- Quantum logics as alternative propositional structures with nonclassical features
- Lattice theoretic methods for quantum logic and structure of Hilbert spaces
Quantum Information Science
- Capacities of quantum channels and optimal encoding/decoding methods
- Quantum error correction codes tailored for specific noise models
- Quantum machine learning algorithms using Hamiltonian based models
- Quantum advantage in search algorithms using Grover's algorithm
- Quantum key distribution systems for secure communication applications
- Quantum algorithms for linear algebra and machine learning problems
- Quantum cryptographic protocols for trusted communication networks
- High-dimensional quantum entanglement characterization using entropy
- Quantum memory materials and devices for superconducting circuits
- Quantum simulations of chemical reactions and collision dynamics
- Nonclassical states of light generation using optical nonlinearities
- Photon blockade and single photon sources using cavity quantum electrodynamics
- Quantum metrology techniques using optical interferometers
- Quantum optics experiments testing foundations of quantum mechanics
- Photonic quantum information processing using integrated optical circuits
- Quantum optical implementations of quantum walks algorithms
- Exploration of quantum limits in optical metrology and sensing
- Quantum optics with optomechanical systems and levitated nanospheres
- Quantum optics experiments using Rydberg atoms as giant artificial atoms
- Phase space methods for nonclassical light and ultrafast pulses
- Design of logical qubits and error correction codes for fault tolerance
- Benchmarking quantum advantage in algorithms like quantum Fourier transforms
- Quantum machine learning models for near-term noisy devices
- Quantum compilers and software stacks optimized for NISQ devices
- Materials research for quantum bits with better coherence properties
- Topological quantum computing proposal implementations
- Quantum neural networks algorithms for pattern recognition
- Analysis of quantum algorithm speedups over classical counterparts
- Characterizing quantum supremacy in sampling problems using randomness tests
- Quantum simulations of many-body dynamics in condensed matter systems
Quantum Chaos and Mesoscopics
- Energy level statistics in quantum chaotic systems using random matrix theory
- Experimental signatures of quantum chaos in microwave cavities and billiards
- Quantum chaotic scattering models and cross-over regimes from integrability
- Field theoretical methods for quantum chaotic systems and non-Hermitian models
- Decoherence models and quantum-to-classical correspondence in chaotic systems
- Semiclassical analysis of chaos assisted tunneling and transport in mixed systems
- Non-Gaussian fluctuations and higher-order statistics of spectral form factor
- Superconducting and normal metal mesoscopic devices realization and measurements
- Bosonization methods for quantum impurity problems in mesoscopic fermion systems
- Persistent currents, quantum Hall effect and AB oscillations in quantum rings
Topological Quantum Systems
- Majorana bound states detection in semiconductor-superconductor structures
- Model realizations and precise quantization of anomalous Hall effects
- Hofstadter butterfly spectrum and Chern numbers in lattice systems
- Topological quantum computing using Majorana zero modes
- Quantum spin Hall insulators and emerging quasi-particles like skyrmions
- Synthetic topological matter using ultracold atoms in optical lattices
- Braiding non-Abelian anyons for fault tolerant topological quantum computation
- Ultracold atom experiments for observation of edge states in Chern insulators
- PhD topological materials characterization via transport and spectroscopy probes
- Quantum spin liquids physics and fractionalized excitations
- Nitrogen vacancy centers in diamond for quantum sensing applications
- Development of quantum standards for metrology using entanglement
- Testing quantum mechanics foundations using quantum optomechanics
- Quantum enhanced microscopy techniques beating classical limits
- Building blocks for modular quantum computers and simulations
- Materials research for quantum transducers and interconnects
- High coherence quantum dots systems for quantum photonics
- Quantum algorithms implementations on noisy intermediate-scale devices
- Ultracold molecule production using photoassociation and magnetoassociation
- Experimental tests of hidden variable theories and noncontextuality
- Quantum biological phenomena evidenced in photosynthesis, avian magnetoreception
- Quantum thermodynamics laws and emergence of classicality
- Quantitative finance applications using quantum probability theory
- Quantum tetris toy model investigations for many-body localization
- Quantum optics with hybrid light-matter systems based on cold atoms, ions, solid state etc.
- Quantum transport models and nonequilibrium Green functions methods
- Quantum cellular automata as discrete models between classical and quantum systems
- Potential connections between quantum foundations and consciousness
- Quantum mechanics interpretations and philosophical implications
This compilation summarizes over 100 potential quantum mechanics research topics spanning fundamental theoretical studies, experimental quantum physics, and interdisciplinary quantum information science domains. For graduate students and researchers, it aims to provide inspiration for identifying meaningful and interesting problems for investigation. Quantum mechanics continues to be an active research frontier with many open questions to explore.
Q1. How do I select a good research topic in quantum mechanics?
Tips for choosing a good quantum mechanics research topic:
- Select an area of personal interest within quantum physics
- Identify open theoretical or experimental problems in the field
- Choose a focused topic with well-defined goals and objectives
- Ensure necessary resources and tools are accessible for investigation
- Align topic with advisor's expertise to receive good mentorship
- Expanding on existing work and models is better than reinventing the wheel
- Topic should be novel and provide scope for original contributions
- Evaluate ongoing progress worldwide to identify gaps and new approaches
Q2. What are good sources to find quantum research topics?
Some good sources to find interesting quantum research topics include:
- Latest quantum physics journals like PRX Quantum, Quantum Science and Technology, npj Quantum Information etc.
- Recent quantum conferences - program tracks and invited talks
- Preprint archives - arXiv quantum physics and quantum information theory categories
- PhD dissertations in university repositories
- Research topics from leading quantum groups and labs
- Quantum science roadmaps published by government organizations
- Discussions with professors working in quantum domains
- Funding opportunities, grants and fellowships focused on quantum
Q3. How should a quantum mechanics research proposal be structured?
A good research proposal in quantum mechanics should include:
- Overview - summary of key objectives and significance
- Background - foundational concepts and literature review
- Problem statement - current challenges and gaps being addressed
- Proposed approach and methodology - theoretical modeling, experiments, computations etc.
- Preliminary studies performed and initial results
- Work plan - tasks, milestones and timeline
- Required resources - equipment, tools, materials etc.
- Expected outcomes and deliverables
- Future directions for investigation
Duration, budgeting and other constraints should be considered. Follow advisor guidance on specifics of formatting and presentation.
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A maverick physicist is building a case for scrapping quantum gravity.
Gravity might be classical, not quantum, physicist Jonathan Oppenheim suggests
Scientists are developing a theory of gravity that describes spacetime in a classical, or non-quantum, way, yet preserves known quantum effects, such as interference patterns (represented by the striped circle at right) observed in certain experiments.
By Emily Conover
December 8, 2023 at 8:00 am
A rift runs deep through the heart of physics. The general theory of relativity, which describes gravity, clashes with quantum physics. In an effort to seal that physics fissure, untold numbers of physicists have spent their careers working to build a theory of quantum gravity.
But one physicist is championing a radically different path. Jonathan Oppenheim thinks that gravity might be fundamentally classical, meaning it isn’t quantum at all. It’s an unconventional idea, to say the least.
“When we started, maybe 99 percent of our colleagues thought we were crackpots and that’s now down to maybe 70 percent,” quips Oppenheim, of University College London.
All known forces except gravity are formulated in terms of quantum physics. The prevailing view is that gravity will need to assimilate with its quantum colleagues. But gravity is different, Oppenheim argues. While other forces evolve within a landscape of spacetime, gravity is the warping of spacetime itself. So, Oppenheim says, “it is pretty unclear that it should have a quantum nature, in my view.”
Physicists have devised several “no-go” theorems that seemingly forbid a classical theory of gravity. Such theorems highlight inconsistencies, apparently fatal to the idea, that arise when classical gravity is applied to quantum particles. But it’s possible to get around those prohibitions by adding some randomness to the way that spacetime bends in response to quantum particles, Oppenheim reports December 4 in Physical Review X .
Consider the famous double-slit experiment of quantum physics ( SN: 5/3/19 ). Particles are sent toward a detector, separated by a barrier with two slits in it. When those particles arrive at the detector, they create a stripy pattern called an interference pattern. That pattern arises because, in quantum physics, the particle isn’t constrained to pass through one slit or the other. Instead, it can exist in a superposition, taking a quantum combination of both possible routes. If a scientist makes a measurement to determine which slit the particle passed through, that pattern disappears.
If a standard classical picture of gravity were correct, it would be possible to measure the gravitational field of that particle so precisely that you could determine which slit the particle went through. This possibility would destroy the interference pattern, even without actually doing the measurement. Because scientists do observe interference patterns in the lab, that’s a big blow for a standard classical theory of gravity.
But the randomness baked into Oppenheim’s theory means that, instead of a particle having a determined gravitational field, the field fluctuates. That means, unlike for the standard version of classical gravity, it’s not possible to determine which slit a particle went through by precisely measuring its gravitational field. Particles can pass through the slits in a superposition, and the interference pattern is saved, restoring the possibility gravity could be classical.
Experiments can test this theory by searching for evidence of those random gravitational fluctuations , Oppenheim and colleagues report December 4 in Nature Communications . “Essentially, you very precisely measure the response of a mass to a gravitational field,” says study coauthor Zach Weller-Davies, who completed the work at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.
This is not the first time scientists have proposed a way to make classical gravity comport with quantum physics. But Oppenheim has been “leading a renaissance,” says physicist Vivishek Sudhir of MIT. Sudhir hopes to test the theory with another type of experiment, measuring the correlations between the motions of two masses that interact gravitationally, he and a colleague report September 16 at arXiv.org.
However, the theory has features some physicists might find unsatisfying. For example, the randomness involved means that the theory is not reversible: Unlike other theories, there’s no way to start from the endpoint of an interaction and trace its steps backward.
Still, even some quantum gravity believers think that the work has merit.
“The reason why this work is interesting for me is not really because I would believe that gravity is classical,” says Flaminia Giacomini of ETH Zurich. The result, she says, is interesting regardless of whether gravity is found to be classical or quantum. That’s because, in order for an experiment to confidently proclaim that gravity is quantum, scientists need to understand the possibilities for classical gravity. “Only in that way will we be able to prove in a strong way that gravity is not compatible with a classical description.”
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MIT researchers observe a hallmark quantum behavior in bouncing droplets
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In our everyday classical world, what you see is what you get. A ball is just a ball, and when lobbed through the air, its trajectory is straightforward and clear. But if that ball were shrunk to the size of an atom or smaller, its behavior would shift into a quantum, fuzzy reality. The ball would exist as not just a physical particle but also a wave of possible particle states. And this wave-particle duality can give rise to some weird and sneaky phenomena.
One of the stranger prospects comes from a thought experiment known as the “quantum bomb tester.” The experiment proposes that a quantum particle, such as a photon, could act as a sort of telekinetic bomb detector. Through its properties as both a particle and a wave, the photon could, in theory, sense the presence of a bomb without physically interacting with it.
The concept checks out mathematically and is in line with what the equations governing quantum mechanics allow. But when it comes to spelling out exactly how a particle would accomplish such a bomb-sniffing feat, physicists are stumped. The conundrum lies in a quantum particle’s inherently shifty, in-between, undefinable state. In other words, scientists just have to trust that it works.
But mathematicians at MIT are hoping to dispel some of the mystery and ultimately establish a more concrete picture of quantum mechanics. They have now shown that they can recreate an analog of the quantum bomb tester and generate the behavior that the experiment predicts. They’ve done so not in an exotic, microscopic, quantum setting, but in a seemingly mundane, classical, tabletop setup.
In a paper appearing today in Physical Review A , the team reports recreating the quantum bomb tester in an experiment with a study of bouncing droplets. The team found that the interaction of the droplet with its own waves is similar to a photon’s quantum wave-particle behavior: When dropped into a configuration similar to what is proposed in the quantum bomb test, the droplet behaves in exactly the same statistical manner that is predicted for the photon. If there were actually a bomb in the setup 50 percent of the time, the droplet, like the photon, would detect it, without physically interacting with it, 25 percent of the time.
The fact that the statistics in both experiments match up suggests that something in the droplet’s classical dynamics may be at the heart of a photon’s otherwise mysterious quantum behavior. The researchers see the study as another bridge between two realities: the observable, classical world and the fuzzier quantum realm.
“Here we have a classical system that gives the same statistics as arises in the quantum bomb test, which is considered one of the wonders of the quantum world,” says study author John Bush, professor of applied mathematics at MIT. “In fact, we find that the phenomenon is not so wonderful after all. And this is another example of quantum behavior that can be understood from a local realist perspective.”
Bush’s co-author is former MIT postdoc Valeri Frumkin.
To some physicists, quantum mechanics leaves too much to the imagination and doesn’t say enough about the actual dynamics from which such weird phenomena supposedly arise. In 1927, in an attempt to crystallize quantum mechanics, physicist Louis de Broglie presented the pilot wave theory — a still-controversial idea that poses a particle’s quantum behavior is determined not by an intangible, statistical wave of possible states but by a physical “pilot” wave of its own making, that guides the particle through space.
The concept was mostly discounted until 2005, when physicist Yves Couder discovered that de Broglie’s quantum waves could be replicated and studied in a classical, fluid-based experiment. The setup involves a bath of fluid that is made to subtly vibrate up and down, though not quite enough to generate waves on its own. A millimeter-sized droplet of the same fluid is then dispensed over the bath, and as it bounces off the surface, the droplet resonates with the bath’s vibrations, creating what physicists know as a standing wave field that “pilots,” or pushes the droplet along. The effect is of a droplet that appears to walk along a rippled surface in patterns that turn out to be in line with de Broglie’s pilot wave theory.
For the last 13 years, Bush has worked to refine and extend Couder’s hydrodynamic pilot wave experiments and has successfully used the setup to observe droplets exhibiting emergent, quantum-like behavior, including quantum tunneling, single-particle diffraction, and surreal trajectories.
“It turns out that this hydrodynamic pilot-wave experiment exhibits many features of quantum systems which were previously thought to be impossible to understand from a classical perspective,” Bush says.
In their new study, he and Frumkin took on the quantum bomb tester. The thought experiment begins with a conceptual interferometer — essentially, two corridors of the same length that branch out from the same starting point, then turn and converge, forming a rhombus-like configuration as the corridors continue on, each ending in a respective detector.
According to quantum mechanics, if a photon is fired from the interferometer’s starting point, through a beamsplitter, the particle should travel down one of the two corridors with equal probability. Meanwhile, the photon’s mysterious “wave function,” or the sum of all its possible states, travels down both corridors simultaneously. The wave function interferes in such a way to ensure that the particle only appears at one detector (let’s call this D1) and never the other (D2). Hence, the photon should be detected at D1 100 percent of the time, regardless of which corridor it traveled through.
If there is a bomb in one of the two corridors, and a photon heads down this corridor, it predictably triggers the bomb and the setup is blown to bits, and no photon is detected at either detector. But if the photon travels down the corridor without the bomb, something weird happens: Its wave function, in traveling down both corridors, is cut short in one by the bomb. As it’s not quite a particle, the wave does not set off the bomb. But the wave interference is altered in such a way that the particle will be detected with equal probability at D1 and D2. Any signal at D2 therefore would mean that a photon has detected the presence of the bomb, without physically interacting with it. If the bomb is present 50 percent of the time, then this weird quantum bomb detection should occur 25 percent of the time.
In their new study, Bush and Frumkin set up an analogous experiment to see if this quantum behavior could emerge in classical droplets. Into a bath of silicon oil, they submerged a structure similar to the rhombus-like corridors in the thought experiment. They then carefully dispensed tiny oil droplets into the bath and tracked their paths. They added a structure to one side of the rhombus to mimic a bomb-like object and observed how the droplet and its wave patterns changed in response.
In the end, they found that 25 percent of the time a droplet bounced through the corridor without the “bomb,” while its pilot waves interacted with the bomb structure in a way that pushed the droplet away from the bomb. From this perspective, the droplet was able to “sense” the bomb-like object without physically coming into contact with it. While the droplet exhibited quantum-like behavior, the team could plainly see that this behavior emerged from the droplet’s waves, which physically helped to keep the droplet away from the bomb. These dynamics, the team says, may also help to explain the mysterious behavior in quantum particles.
“Not only are the statistics the same, but we also know the dynamics, which was a mystery,” Frumkin says. “And the inference is that an analogous dynamics may underly the quantum behavior.”
"This system is the only example we know which is not quantum but shares some strong wave-particles properties," says theoretical physicist Matthieu Labousse, of ESPCI Paris, who was not involved in the study. "It is very surprising that many examples thought to be peculiar to the quantum world can be reproduced by such a classical system. It enables to understand the barrier between what it is specific to a quantum system and what is not. The latest results of the group at MIT pushes the barrier very far."
This research is supported, in part, by the National Science Foundation.
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- Published: 10 January 2022
40 years of quantum computing
Nature Reviews Physics volume 4 , page 1 ( 2022 ) Cite this article
This year we celebrate four decades of quantum computing by looking back at the milestones of the field and forward to the challenges and opportunities that lie ahead.
In science there are few true eureka moments experienced by lone geniuses, but rather a continuous exchange and development of ideas that drive the collective human curiosity in new directions. Before a new field of research is born, there is usually a time when many similar ideas are in the air and scientists start to see something new forming, but cannot quite put their finger on it. Then someone manages to articulate a new concept, opening a new direction. From there, it may take years or even decades until the full implications are grasped. Such is the case of quantum computing.
In the early 1980s a deep connection between physics and computation was becoming evident. Twenty years earlier, Rolf Landauer had linked thermodynamics and information . In 1980, mathematician Yuri Manin mentioned in the introduction of his book Computable and Uncomputable (in Russian) the idea of a quantum automaton that used superposition and entanglement (see the English translation in ref. 1 ) and Paul Benioff discussed 2 a microscopic quantum mechanical Hamiltonian as a model of Turing machines. Then, in May 1981, a conference on the ‘ Physics of Computation ’ organized by MIT and IBM brought together physicists and computer scientists. Among the participants were some well-known scientists like Freeman Dyson, John Archibald Wheeler or Richard Feynman, Landauer and Benioff, and others whose names resonate with anyone having worked in quantum computing: Charles Bennett, Tommaso Toffoli, Edward Fredkin. The talks were published the next year in the International Journal of Theoretical Physics .
It is difficult to tell to what extent these papers were influenced by the discussions at the meeting or whether the ideas presented had been articulated by individual scientists beforehand. Most contributors referenced the other papers, except Feynman who did not cite anyone (although he did credit Fredkin for inspiration) and just transcribed his keynote speech with its colloquialisms (“Nature isn’t classical, dammit.”). His paper 3 has become a landmark in quantum computation and simulation, and has been credited for the birth of these fields. Feynman took the ideas that were in the air — computation is a physical process, perhaps even a quantum mechanical one — then turned them around by asking how to compute (simulate) physics. He showed that “quantum mechanics can’t seem to be imitable by a local classical computer”, but could be tacked by “quantum computers — universal quantum simulators”. Manin had had a similar intuition 1 (“the quantum behaviour of the system might be much more complex than its classical simulation”), but he did not develop it further.
Although it is hard to assign a single moment in time as the starting point of quantum computing, as a journal, we like to take the 1982 issue of the International Journal of Theoretical Physics as the crystallization of the idea of a quantum computer. We would also like to credit all the pioneers whose ideas connected quantum mechanics with computing.
From 1982 to today quantum computing has been on a journey with many ups and downs and unexpected encounters. It saw great excitement after Shor’s quantum algorithm for factorization in 1994, followed by the first proposals for building a quantum computer. Hopes were high, but then came the realization of how difficult it would be in practice. No other algorithms to rival the potential of Shor’s were found. Despite disappointment, momentum was not lost and the field branched into different directions. Unexpected connections to fundamental physics and insight into the foundations of quantum mechanics were uncovered and numerous advances were made both in theory and experiment. Things started to pick up again for quantum computing and the past five years have witnessed a renewed commercial interest and the first demonstrations of quantum computers performing tasks that are hard for classical computers, a quantum advantage.
To celebrate four decades of quantum computing we put together a Collection of relevant content from our pages. As we have done in the past, we will revisit milestone papers and their legacy in ‘then and now’-type retrospective pieces. We will also look ahead with a Roadmap article and other upcoming content. Watch this space.
Mathematics as Metaphor. Selected Essays of Yuri I. Manin 77–78 (American Mathematical Society, 2007).
Benioff, P. The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines. J. Stat. Phys. 22 , 563–591 (1980).
Article ADS MathSciNet Google Scholar
Feynman, R. P. Simulating physics with computers. Int. J. Theor. Phys. 21 , 467–488 (1982).
Article MathSciNet Google Scholar
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40 years of quantum computing. Nat Rev Phys 4 , 1 (2022). https://doi.org/10.1038/s42254-021-00410-6
Published : 10 January 2022
Issue Date : January 2022
DOI : https://doi.org/10.1038/s42254-021-00410-6
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Browse Course Material
- Prof. Krishna Rajagopal
As taught in.
- Quantum Mechanics
Learning Resource Types
Quantum physics iii, 1) project summary.
Everyone in 8.06 will be expected to research, write and “publish” a short paper on a topic related to the content of 8.05 or 8.06. The paper can explain a physical effect or further explicate ideas or problems covered in the courses. It can be based on the student’s own calculations and/or library research. The paper should be written in the style and format of a brief journal article and should aim at an audience of 8.06 students.
Writing, editing, revising and “publishing” skills are an integral part of the project. Each of you will ask another student to edit your draft and will then prepare a final draft on the basis of the suggestions of your “peer editor”. We will supply templates for the Revtex version of LaTeX (used by the Physical Review ) so that you can prepare your paper in a finished, publishable form. We will also arrange a LaTeX tutorial, likely in place of sections one day in April.
You will submit your first draft marked up with editorial comments by your peer editor. This first draft will then be critiqued by a “writing assistant” (see below) and returned to you. Two weeks after the first draft is due, you will submit your final draft. Your papers will be graded on the intellectual quality of your work, the effectiveness of your presentation and the success of your prose style. A part of your grade will also be determined by how carefully and constructively you edited the draft of the paper for which you were the peer editor. The grade you earn for your paper will count 20% towards your final grade in 8.06.
Because 8.06 is a CI-M ( Communication Intensive in the Major ) Subject, in order to pass 8.06 you must obtain a grade of C or better on your term paper. If you do not succeed in this, you will get a grade of Incomplete until you revise your term paper sufficiently to earn at least a C, and only at that time you will be assigned a final grade, with your term paper grade counting 20%.
When a practicing physicist writes a research paper, he or she often asks a few colleagues to comment on a first draft. The final draft is then reviewed anonymously by one or several peers before it is accepted by a journal like the Physical Review . The goal of this informal and formal peer review process is to push authors to write papers which successfully communicate ideas among a community of peers. Your goal is to write a paper which presents a phenomenon or problem in quantum physics in a way which communicates your ideas clearly and effectively to your fellow 8.06 students, namely to your peers. Do not seek to teach Profs. Liu and Rajagopal, although they are always happy to learn. Do seek to teach your peers. If your peers cannot understand what you write, you have not succeeded. Note that writing for your peers is a much higher standard than writing for the faculty. Presenting a topic sufficiently clearly and logically that one of your peers new to this topic can learn about it requires clarity of thought and depth of understanding. These are the prerequisites for an effective written (or, for that matter, verbal) presentation.
We have obtained resources to support four “writing assistants” who can help you with writing, editing and preparing the paper. Each of you will be contacted by email by one of the writing assistants on March 29. (See the schedule below.) You should arrange to meet soon thereafter, and should seek their assistance from then on as you need it. They will critique the proposal and outline for your paper, and will also critique the first draft which you submit after it has been peer edited. In between, you may also ask them to help you with parts of your paper as you write them. Think of your writing assistant as a coach. They are there to help you, and are good at it. If you wish to get their help earlier than March 29, please submit your paper proposal and the name of your peer editor earlier, and one of the writing assistants will be assigned to help you.
By the time you turn in your final paper, it will have been edited by one of your peers and you will also have had time to implement the suggestions of one of the writing assistants. Past 8.06 students have found that their papers improve enormously through this process. Based on experience from previous years, by the time you turn in your finished paper, very many of you will have produced an account of a piece of physics written to a very high standard. It would be a shame if these papers were not “published”. We shall have as our goal the “publication” of a journal consisting of all your papers. There are two important caveats: (i) only papers which are submitted electronically, using the LaTeX template provided, will be published; (ii) only papers which earn a grade of B or higher will be published. Subject to these caveats, we hope to produce a compilation of all of your papers. We will circulate this “journal” to all of you, so that you can in the end read the work of all your peers, and not just of the one person whose work you edited.
2) Schedule and Due Dates for the Paper
You should use the first part of the term to consider possible topics and to choose a peer editor. Your peer editor must be an 8.06 student, and must be someone whose own 8.06 paper topic is unrelated to yours. A list of suggested topics is given below, but you are free to choose topics not on this list upon first obtaining Prof. Rajagopal’s approval. By the time Spring Break is upon us, you should have a good idea of what you are going to write about and should be well into the process of reading about your topic and doing the calculations, if any are involved. You should spend Spring Break completing your understanding of the physics that you plan to write about, completing any calculations that you plan, and outlining your paper. You will then be ready to write your proposal:
Your proposal is due on Tuesday March 29, in lecture. This must consist of: a title, a one paragraph description of what you plan to write about, an outline of your proposed paper, a list of several references you plan to use, the name of your peer editor, and your name and email address.
You will then be contacted by one of the writing assistants. They may either accept your proposal, or request that you revise it in response to their suggestions. You should arrange to meet with them as soon as possible (even if they accept your proposal). Anyone who has not met with their writing assistant at least once before submitting their first draft will be penalized.
Your peer edited first draft is due on Tuesday April 12 in lecture. This means that you must give your first draft to your peer editor several days earlier, to give that person sufficient time to critique it substantively by April 12. Each of you should then meet with your writing assistant by Friday April 15 in order to obtain their comments on your first draft. In fact, if A edits for B and B edits for A, I will make sure that A and B have the same writing assistant and would therefore suggest that you both meet him or her together, to obtain comments on both your papers simultaneously. You will get your first drafts back when you meet with your writing assistant.
A hard copy of your final, polished paper is due in lecture on Tuesday April 26. Think of this as submitting your paper to The 8.06 Physical Review . If you get back a positive report (i.e. grade of B or better) from the editor (Prof. Rajagopal) you will then be expected to submit your paper for publication electronically. You will all get a copy of the 2005 Physical Review .
3) Nature of the Paper
The aim of this project is to give a clear and pedagogical presentation of a “problem” or “phenomenon” in quantum mechanics.
- A “problem” could be similar to but more elaborate than the type of problems that appear on problem sets. For example, coherent states were introduced briefly in the context of the harmonic oscillator in 8.05. A student might delve deeper into the coherent state formalism, describe the properties of coherent states, explain the types of problems where they are useful, and give some examples of their applications. Such a paper would resemble a short chapter in some hypothetical text book for 8.05. The principal references for a paper like this could be existing quantum mechanics texts and the references to the original literature to be found in them.
- A paper focused on a “phenomenon” would introduce the phenonomenon and explain its origins in terms of the concepts and language of 8.06. For example, when we treated systems of identical particles at the end of 8.05 we alluded very briefly to the “allotropic forms of hydrogen” known as ortho and para hydrogen. A student might find out what they are, how their properties are understood in terms of Fermi-Dirac statistics, and describe the interesting role they played in the early history of quantum mechanics. Once again the principal references would likely be texts, perhaps modern physics texts in this case, histories of quantum physics, and the original literature.
Papers on “problems” might be based at least in part on your own calculations. Papers on “phenomena” might involve some library research. In either case reference must be given for any material taken from other sources. Do not plagiarize. Anyone who contemplates borrowing material directly from mainstream texts should consider how difficult it is to find a text that presents quantum physics at the level appropriate to 8.06.
We encourage students to write papers which expand upon a problem or phenomenon which was already introduced in either 8.05 or 8.06 lectures. If you do this, you should begin at the level of whatever we have already covered and then go farther. Students may also choose topics which have not appeared at all in class, but whose quantum mechanical explanation can be understood based upon what we have learned in 8.05 and 8.06.
Please do not try to choose subjects which are obscure, difficult or controversial. Misguided attempts like this to gain the respect of the faculty inevitably have the opposite effect. There are plenty of deep, interesting and challenging subjects in the mainstream of quantum mechanics.
Papers can range between 8 - 15 pages (in the LaTeX template provided) in length. These limits are firm.
Students are encouraged to use equations and figures to aid their presentation, much as they are used in articles and sophisticated textbooks.
4) Possible Topics
Students are welcome to suggest topics of their own. You should do this by sending Prof. Rajagopal a brief paragraph by email, summarizing the topic. There is no separate deadline by which you must do this, but note that your complete proposal is due on March 29. At the time you submit your proposal, you should already know that Prof. Rajagopal has approved your choice of topic. (Note that your writing assistant may nevertheless require you to revise your proposal.)
Here is a list of possible topics. In some cases, either Prof. Liu or Prof. Rajagopal will have ideas for where to begin reading about these topics. Not in all cases, however.
- Coherent states.
- The allotropic forms of hydrogen.
- Nuclear Magnetic Resonance. For example, you might take off from where we stopped in 8.05 and explain how NMR is applied in a particular experimental context.
- Magnetic monopoles, gauge invariance, and the Dirac quantization condition for the magnetic charge of a magnetic monopole.
- Scattering off a magnetic flux tube.
- Bell’s theorem - can classical mechanics imitate quantum mechanics?
- Neutrino oscillations in vacuum, beyond what we covered in 8.05.
- Oscillation phenomena involving kaons and/or B mesons, beyond what we covered in 8.05.
- The solar neutrino problem.
- Levinson’s theorem - how the scattering phase shift is related to the number of bound states in a potential.
- The shell model of nuclear structure.
- The properties of the deuteron.
- The α-decay of 238 U.
- The rotational and vibrational spectrum of diatomic molecules.
- Dynamical SO(3) × SO(3) symmetry of the hydrogen atom.
- Dynamical SU(n) symmetry of the harmonic oscillator in n-dimensions.
- Supersymmetric quantum mechanics, beyond what we did in 8.05.
- The Zeeman effect in weak, intermediate and strong magnetic fields.
- The Lamb shift in hydrogen - evidence that relativistic quantum mechanics must be replaced by quantum field theory. (This is an example of a topic where you will not be able to give a complete derivation of the effect, but where those of you interested in the history of physics could write a paper which explains the quantum physics more qualitatively while at the same time describing the experiments and the history in full.)
- The non-relativistic quark model of the proton, neutron and related particles.
- Isospin - a quantum symmetry of elementary particles.
- The 21 cm. line of hydrogen and its role in astrophysics.
- The Casimir effect.
- Feynman’s path integral approach to quantum mechanics, and its application to several problems of your choice which we have previously analyzed using other methods (If you choose a formal topic like this, about a method rather than a phenomenon or problem, you must take it far enough to show how the method is applied to a phenomenon or problem.)
- The van der Waals force between hydrogen atoms in excited states.
- Quantum computing? (You may not write a paper that purports to be about “Quantum computing”. You may only choose a topic within this area if you have a focussed idea, perhaps involving presentation of one of the ideas for implementation of a quantum computer, the quantum mechanics of the implementation, the difficulties, etc. Note also that you may not write a paper whose sole purpose is the presentation of Grover’s and/or Shor’s algorithms, since you will see those in lecture at the end of the semester.)
- Quantum teleportation.
- Quantum cryptography.
- Bose-Einstein condensation.
- Integer Quantum Hall Effect (There are a number of ways you could go beyond what we do in lecture.)
- Landauer conductivity in two dimensional systems.
- Photonic Crystals.
- Quantum Dots.
- The deHaas van Alphen effect as a tool for measuring the shapes of fermi surfaces in metals.
- Periodic potentials and band structure.
- An introduction to the quantum statistical mechanics of photons and the spectrum of black body radiation. (You could also include an account of how Planck was led to discover quantum mechanics in the first place, or of how the spectrum of black body radiation appears in the cosmic three degree background radiation.)
- The density matrix formalism in quantum mechanics, and quantum statistical mechanics.
- Optical pumping, masers, lasers.
- Masers in astrophysics.
- Interesting applications of the semiclassical approximation.
- The Ramsauer-Townsend effect.
- The Josephson effect.
- The Wigner-Eckart theorem.
- Fractional statistics in two dimensions.
- Squeezed states and applications.
- Wigner functions and applications.
- Tunnelling, beyond the discussion in class. The Euclidean approach; effects of nonzero temperature.
- The microscopic origin and effects of quantum dissipation, for example on tunnelling.
- Inverse scattering method and its application to solitons.
5) Writing Tips
Here are some tips that you may find useful.
- Identify a well-defined topic area as early as possible. Changing your focus is fine, but you may find that it requires substantial rewriting to keep things clear.
- Work through and understand the physics before writing. You should do this over Spring Break. This will ensure that you have a well-defined topic before you start writing. You will find that this will make structuring the paper infinitely easier.
- Make sure the main points of your paper are clearly indicated. This is especially important for scientific writing, since the reader can easily get bogged down in details. Your main points should be highlighted by the structure of the paper as well as mentioned in the introduction and/or abstract.
- Write the abstract and, possibly, the introduction last.
- After you have your outline ready, don’t be afraid to draft later sections before earlier sections. If you understand the last half of your argument better than the first, start by writing the last half. Doing so will help you think through how to understand and explain the first half.
- In thinking about both style and structure, remember that you are writing a scientific paper and not a work of literature. The writing in great works of literature typically has multiple meanings, and can be understood in many ways, at different levels. It can be read differently by readers at different times or with different backgrounds. It often makes veiled allusions to other great literature. Over the years, great literature takes on meanings that go beyond those intended consciously by its author. In contrast, the central purpose of a scientific paper is the clear communication of your ideas to your readers, with no ambiguity, multiple meanings or veiled allusions. Your goal is to ensure that every one of your readers, who may indeed have varying backgrounds, understands your ideas in precisely the way that you intend. This means that clarity and precision are your paramount goals. You should seek to ensure that no reader can misunderstand what you intend to communicate in any sentence that you write, even should they willfully try to misunderstand you. To this end, write in simple, declarative sentences, avoid contorted constructions and always aim for clarity.
- Feel free to use whichever voice you are most comfortable with. “I will show,” “we will show” or “it will be shown” are all fine. For unknown reasons, some students seem to think that personal pronouns are banned and the passive voice is required. Nothing could be further from the truth. Good scientific writing should be animated and compelling. Your paper should “tell a physics story”. I find the overuse of the passive voice to be deadening. Don’t be dull. Clarity and precision come first, but don’t fall into the trap of thinking that this can only be accomplished via boring your reader to tears. Not true.
- Try to lead your reader along, motivating their interest, building up the physics ground work you need them to understand, drawing them into the story you are telling, and working up to a compelling conclusion.
- All the advice I’ve given you about style is just as important when, later in life, you find yourself preparing a lecture or a seminar.
5.3) Some Details
- Be rigorously consistent in your notation, even at the risk of being repetitive.
- Clearly define every quantity that you introduce.
- Avoid ambiguous references, such as “this shows”. Instead, use references like “Eq. 4.1 shows.” The LaTeX commands \label and \ref are useful here.
6) More on Peer Editing
As described in the project summary, each of you will act as an editor for one of your peers. (Note: if you cannot find someone to act as your editor, ask Prof. Rajagopal. He will pair people up as he gets such requests. You must list the name of your peer editor as part of your proposal, due on March 29.) When you finish your first draft, give it to your editor for editing. You must give your editor time to complete their work in time for you to submit your peer-edited first draft on Tuesday April 12.
As you are editing the work of one of your peers, you should start by praising what the document does well. If the author has made specific requests (i.e. “please see if my argument in this section makes sense to you”) then spend much of your time responding to these specific requests. Do not focus on spelling and the mechanics of writing, unless asked by the author to do so. (Of course, note problems of this sort which you happen to spot, but this is not your main goal and the author should in general not rely on you for this sort of editorial review.) Instead, focus on helping the author to revise content, organization and logic. Do not just criticize. Make suggestions on how to solve the problems you notice in the paper.
As you edit the work of your peer, here are some of the questions which you should be thinking about:
- What is the paper’s main argument?
- How interesting is it? Is the importance of the topic explained?
- How specific is the argument? Would it benefit from being made more general or complete? Would it, in contrast, benefit from being made more focussed?
- Is the paper divided into sections and subsections in a way which makes following its logic easy? Does each section flow logically from the preceding one? Do ideas flow smoothly from one paragraph to the next?
- Early in the text, is there a clear road map of the entire document?
- Are all outside sources documented? If, as will be the case for almost all 8.06 papers, the paper contains ideas which are not the results of calculations done by the author and are not ideas we have all seen in lecture, can you see from which source the author learned each such idea?
- Are all technical terms which are new to you defined clearly, and used consistently?
- If the paper presents the solution to a problem, what are the arguments on which the solution rests? Do you understand each argument and the solution as a whole? Is each part of each argument substantiated? (Either by calculation presented in the paper, or by reference to 8.05 and 8.06 material which you can see substantiates the argument.) Is there anything missing, which would help complete an argument?
- If the paper describes a phenomenon, do you understand the description? Is the nature of the phenomenon clearly described? Are the reasons why the phenomenon is of interest clear? Do you understand the quantum mechanical explanation of the phenomenon presented by the author? What do you wish the author had included that would have given you a better understanding of the phenomenon?
7) The LaTeX Templates
LaTeX (and its ancestor TeX) are widely used in academic and technical publishing. They are “mark-up” languages, like HTML®, that tell a processor how to construct mathematical expressions that look like typeset text. One of the objectives of this assignment is to give you an experience preparing a physics paper for “publication”. When practicing physicists submit papers to the Physical Review , they do so by emailing a LaTeX file, and perhaps some postscript figures, to the editorial office. If you wish to have your paper published, you will do the same.
Many 8.06 students have had previous exposure to LaTeX; some have not. Both to level the playing field and to make possible the publication of your finished papers, we will put a template on the web, for you to download. LaTeX itself is already available as standard MIT Server software.
The most computer-illiterate among you - nevertheless more literate than Professors Rajagopal and Liu by far - need only download the templates, open them in your favorite editor (such as emacs ), and notice the way the LaTeX template deals with title pages, footnotes, references, equations, mathematical symbols in text and set off from text, equation labels, tabs, and so forth. You can construct your paper by cutting the text out of the template text and inserting your own.
In order for students to have access to all necessary macros, already installed on MIT Server, it may first be necessary to type: add newtex. [Note: this was necessary three years ago, but Prof. Rajagopal thinks it is now not necessary.]
You should begin by downloading the template, and making sure that you can LaTeX it successfully, to produce output which looks like the hard copy of the template paper which I will post on the server.
In order to do this, you will need the commands:
- latex filename.tex will run the LaTeX typesetting program to produce typeset output from your input file. If there are errors in your LaTeX file, the file filename.log will contain error messages that are usually helpful. When LaTeX runs successfully, its output is filename.dvi, where dvi means “device independent”. (Note that you will need to run LaTeX twice on the file, in order for all the references to bibliographic items and equation numbers to come out right.)
- xdvi filename.dvi will display your output in its finished form.
- dvips filename.dvi will convert the dvi file to a postscript file, send it to the printer, and then delete the postscript file. If, instead, you want to save a postscript file instead of printing it, use dvips -o filename.ps filename.dvi. This creates a postscript file named filename.ps. One reason to do this is that you can then view your output using ghostview (gv) instead of xdvi. gv is a more sophisticated viewer than xdvi. (A final note here: gv by default does not antialias to save time. It can be turned on and off from within gv or you can use the -antialias flag when calling gv to do it automatically.)
The template provided will contain postscript figures. If you know how to produce illustrations in postscript, the template will illustrate how to incorporate them into your paper. If you don’t or don’t want to bother, you are welcome to draw figures by hand or with your favorite graphics package, and simply staple them onto the end of your paper. Note, however, that if you wish to submit your final paper for publication, you must prepare it using the LaTeX template and must include any figures as encapsulated postscript files, as done in the template.
The template uses a macro called BoxedEPS in order to incorporate encapsulated postscript figures. This macro may be available on MIT Server, but to be safe we will make it available for you to download at the same time that you download the template itself.
We strongly urge people who are new at LaTeX to communicate with classmates. Likewise we strongly encourage LaTeX wizards to help the less experienced with the nuances of the language.
8.06 Sample Term Paper ( PDF )
BoxedEPS ( TEX )
Sample Paper ( TEX )
Energy Levels ( PDF )
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