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Register a new userUsing Mathematica for Quantum Mechanics: A Students Manual
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Authors
Roman Schmied
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This book is an attempt to help students transform all of the concepts of quantum mechanics into concrete computer representations, which can be constructed, evaluated, analyzed, and hopefully understood at a deeper level than what is possible with more abstract representations. It was written for a Masters and PhD lecture given yearly at the University of Basel, Switzerland. The goal is to give a language to the student in which to speak about quantum physics in more detail, and to start the student on a path of fluency in this language. On our journey we approach questions such as: -- You already know how to calculate the energy eigenstates of a single particle in a simple one-dimensional potential. How can such calculations be generalized to non-trivial potentials, higher dimensions, and interacting particles? -- You have heard that quantum mechanics describes our everyday world just as well as classical mechanics does, but have you ever seen an example where such behavior is calculated in detail and where the transition from classical to quantum physics is evident? -- How can we describe the internal spin structure of particles? How does this internal structure couple to the particles motion? -- What are qubits and quantum circuits, and how can they be assembled to simulate a future quantum computer?
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What are cosmic particles and where do they come from? These are questions which are not only fascinating for scientists in astrophysics. With the CosMO experiment (Cosmic Muon Observer) students can autonomously study these particles. They can perform their own hands-on experiments to become familiar with modern scientific working methods and to obtain a direct insight into astroparticle physics. In this contribution we present the experimental setup and possible measurements. The detector consists of three scintillator boxes. Events are triggered and read out by a data acquisition board developed for the QuarkNet Project. With a Python program running on a netbook under Linux, the trigger and data taking conditions can be defined. The program displays the particle rates in real-time and stores the data for offline analysis. Possible student experiments are the measurement of cosmic particle rates dependent on the zenith angle, the distribution of geometrical size of particle showers, and the lifetime of muons. Twenty CosMO detectors have been built at DESY. They are used within the German outreach network Netzwerk Teilchenwelt at 15 astroparticle-research institutes and universities for project work with students.
This Mathematica 7.0/8.0 package upgrades and extends the quantum computer simulation code called QDENSITY. Use of the density matrix was emphasized in QDENSITY, although that code was also applicable to a quantum state description. In the present version, the quantum state version is stressed and made amenable to future extensions to parallel computer simulations. The add-on QCWAVE extends QDENSITY in several ways. The first way is to describe the action of one, two and three- qubit quantum gates as a set of small ($2 times 2, 4times 4$ or $8times 8$) matrices acting on the $2^{n_q}$ amplitudes for a system of $n_q$ qubits. This procedure was described in our parallel computer simulation QCMPI and is reviewed here. The advantage is that smaller storage demands are made, without loss of speed, and that the procedure can take advantage of message passing interface (MPI) techniques, which will hopefully be generally available in future Mathemati
We consider the hypothesis that quantum mechanics is not fundamental, but instead emerges from a theory with less computational power, such as classical mechanics. This hypothesis makes the prediction that quantum computers will not be capable of sufficiently complex quantum computations. Utilizing this prediction, we outline a proposal to test for such a breakdown of quantum mechanics using near-term noisy intermediate-scale quantum (NISQ) computers. Our procedure involves simulating a non-Clifford random circuit, followed by its inverse, and then checking that the resulting state is the same as the initial state. We show that quantum mechanics predicts that the fidelity of this procedure decays exponentially with circuit depth (due to noise in NISQ computers). However, if quantum mechanics emerges from a theory with significantly less computational power, then we expect the fidelity to decay significantly more rapidly than the quantum mechanics prediction for sufficiently deep circuits, which is the experimental signature that we propose to search for. Useful experiments can be performed with 80 qubits and gate infidelity $10^{-3}$, while highly informative experiments should require only 1000 qubits and gate infidelity $10^{-5}$.
Computer-aided design (CAD) programs are essential to engineering as they allow for better designs through low-cost iterations. While CAD programs are typically taught to undergraduate students as a job skill, such software can also help students learn engineering concepts. A current limitation of CAD programs (even those that are specifically designed for educational purposes) is that they are not capable of providing automated real-time help to students. To encourage CAD programs to build in assistance to students, we used data generated from students using a free, open source CAD software called Aladdin to demonstrate how student data combined with machine learning techniques can predict how well a particular student will perform in a design task. We challenged students to design a house that consumed zero net energy as part of an introductory engineering technology undergraduate course. Using data from 128 students, along with the scikit-learn Python machine learning library, we tested our models using both total counts of design actions and sequences of design actions as inputs. We found that our models using early design sequence actions are particularly valuable for prediction. Our logistic regression model achieved a >60% chance of predicting if a student would succeed in designing a zero net energy house. Our results suggest that it would be feasible for Aladdin to provide useful feedback to students when they are approximately halfway through their design. Further improvements to these models could lead to earlier predictions and thus provide students feedback sooner to enhance their learning.
We demonstrate that certain class of infinite sums can be calculated analytically starting from a specific quantum mechanical problem and using principles of quantum mechanics. For simplicity we illustrate the method by exploring the problem of a particle in a box. Twofold calculation of the mean value of energy for the polynomial wave function inside the well yields even argument $p$ ($p>2$) of Riemann zeta and related functions. This method can be applied to a wide class of exactly solvable quantum mechanical problems which may lead to different infinite sums. Besides, the analysis performed here provides deeper understanding of superposition principle and presents useful exercise for physics students.
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