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Register a new userUsing quantum mechanics for calculation of different infinite sums
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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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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?
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 $10^{-5}$.
Unconditionally secure quantum bit commitment (QBC) was widely believed to be impossible for more than two decades. But recently, basing on an anomalous behavior found in quantum steering, we proposed a QBC protocol which can be unconditionally secure in principle. The protocol requires the use of infinite-dimensional systems, thus it may seem less feasible at first glance. Here we show that such infinite-dimensional systems can be implemented with quantum optical methods, and propose an experimental scheme using Mach-Zehnder interferometer.
In this paper we use the framework of generalized probabilistic theories to present two sets of basic assumptions, called axioms, for which we show that they lead to the Hilbert space formulation of quantum mechanics. The key results in this derivation are the co-ordinatization of generalized geometries and a theorem of Soler which characterizes Hilbert spaces among the orthomodular spaces. A generalized Wigner theorem is applied to reduce some of the assumptions of the theorem of Soler to the theory of symmetry in quantum mechanics. Since this reduction is only partial we also point out the remaining open questions.
In 1956 Dyson analyzed the low-energy excitations of a ferromagnet using a Hamiltonian that was non-Hermitian with respect to the standard inner product. This allowed for a facile rendering of these excitations (known as spin waves) as weakly interacting bosonic quasi-particles. More than 50 years later, we have the full denouement of non-Hermitian quantum mechanics formalism at our disposal when considering Dysons work, both technically and contextually. Here we recast Dysons work on ferromagnets explicitly in terms of two inner products, with respect to which the Hamiltonian is always self-adjoint, if not manifestly Hermitian. Then we extend his scheme to doped antiferromagnets described by the t-J model, in hopes of shedding light on the physics of high-temperature superconductivity.
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