From the contents: The origin of the quantum concepts.- Problems.- Diffraction, duality and the Schroedinger equations.- Illustrative problems.- Problems.- Properties of the stationary Schroedinger equation.- Illustrative problems.- Problems.- Tunneling effect and other quantum phenomena.- Problems.- Quantum coherence and energy bands.- Problems.- The WKB approximation.- Problems.- Operators and dynamical variables.- Problems.- The harmonic oscillator.- Problems.- Angular momentum and central potentials.From the contents: The origin of the quantum concepts.- Problems.- Diffraction, duality and the Schroedinger equations.- Illustrative problems.- Problems.- Properties of the stationary Schroedinger equation.- Illustrative problems.- Problems.- Tunneling effect and other quantum phenomena.- Problems.- Quantum coherence and energy bands.- Problems.- The WKB approximation.- Problems.- Operators and dynamical variables.- Problems.- The harmonic oscillator.- Problems.- Angular momentum and central potentials.

Nov 22, 20120 citations

Problems in Quantum Mechanics

Emilio d’Emilio, Luigi E. Picasso

UnitextJan 1, 201140 citations

Four Related Combinatorial Problems

Stanley Gudder

This pedagogical article solves an interesting problem in quantum measure theory. Although a quantum measure $μ$ is a generalization of an ordinary probability measure, $μ$ need not satisfy the usual additivity condition. Instead, $μ$ satisfies a grade-2 additivity condition. Besides the quantum measure problem, we present three additional combinatorial problems. These are (1)\enspace A sum of binomial coefficients problem;\enspace (2)\enspace A recurrence relation problem; and\enspace (3)\enspace An interated vector problem. We show that these three problems are equivalent in that they have a common solution. We then show that this solves the original quantum measure problem.This pedagogical article solves an interesting problem in quantum measure theory. Although a quantum measure $μ$ is a generalization of an ordinary probability measure, $μ$ need not satisfy the usual additivity condition. Instead, $μ$ satisfies a grade-2 additivity condition. Besides the quantum measure problem, we present three additional combinatorial problems. These are (1)\enspace A sum of binomial coefficients problem;\enspace (2)\enspace A recurrence relation problem; and\enspace (3)\enspace An interated vector problem. We show that these three problems are equivalent in that they have a common solution. We then show that this solves the original quantum measure problem.

arXiv:2311.07716Nov 13, 20230 citations

Problems in Quantum Mechanics

Emilio d’Emilio, Luigi E. Picasso

UNITEXT for physicsJan 1, 201714 citations

Fundamental Problems in Quantum Theory

Sandu Popescu, Yakir Aharonov, Sidney Coleman +5 more

Jan 1, 1995124 citations

The Quantum Decoding Problem

André Chailloux, Jean–Pierre Tillich

One of the founding results of lattice based cryptography is a quantum reduction from the Short Integer Solution problem to the Learning with Errors problem introduced by Regev. It has recently been pointed out by Chen, Liu and Zhandry that this reduction can be made more powerful by replacing the learning with errors problem with a quantum equivalent, where the errors are given in quantum superposition. In the context of codes, this can be adapted to a reduction from finding short codewords to a quantum decoding problem for random linear codes. We therefore consider in this paper the quantum decoding problem, where we are given a superposition of noisy versions of a codeword and we want to recover the corresponding codeword. When we measure the superposition, we get back the usual classical decoding problem for which the best known algorithms are in the constant rate and error-rate regime exponential in the codelength. However, we will show here that when the noise rate is small enough, then the quantum decoding problem can be solved in quantum polynomial time. Moreover, we also show that the problem can in principle be solved quantumly (albeit not efficiently) for noise rates for which the associated classical decoding problem cannot be solved at all for information theoretic reasons. We then revisit Regev's reduction in the context of codes. We show that using our algorithms for the quantum decoding problem in Regev's reduction matches the best known quantum algorithms for the short codeword problem. This shows in some sense the tightness of Regev's reduction when considering the quantum decoding problem and also paves the way for new quantum algorithms for the short codeword problem.One of the founding results of lattice based cryptography is a quantum reduction from the Short Integer Solution problem to the Learning with Errors problem introduced by Regev. It has recently been pointed out by Chen, Liu and Zhandry that this reduction can be made more powerful by replacing the learning with errors problem with a quantum equivalent, where the errors are given in quantum superposition. In the context of codes, this can be adapted to a reduction from finding short codewords to a quantum decoding problem for random linear codes. We therefore consider in this paper the quantum decoding problem, where we are given a superposition of noisy versions of a codeword and we want to recover the corresponding codeword. When we measure the superposition, we get back the usual classical decoding problem for which the best known algorithms are in the constant rate and error-rate regime exponential in the codelength. However, we will show here that when the noise rate is small enough, then the quantum decoding problem can be solved in quantum polynomial time. Moreover, we also show that the problem can in principle be solved quantumly (albeit not efficiently) for noise rates for which the associated classical decoding problem cannot be solved at all for information theoretic reasons. We then revisit Regev's reduction in the context of codes. We show that using our algorithms for the quantum decoding problem in Regev's reduction matches the best known quantum algorithms for the short codeword problem. This shows in some sense the tightness of Regev's reduction when considering the quantum decoding problem and also paves the way for new quantum algorithms for the short codeword problem.

arXiv:2310.20651Leibniz international proceedings in informaticsOct 31, 20231 citation

This pedagogical article solves an interesting problem in quantum measure theory. Although a quantum measure

μ

is a generalization of an ordinary probability measure,

μ

need not satisfy the usual additivity condition. Instead,

μ

satisfies a grade-2 additivity condition. Besides the quantum measure problem, we present three additional combinatorial problems. These are (1)\enspace A sum of binomial coefficients problem;\enspace (2)\enspace A recurrence relation problem; and\enspace (3)\enspace An interated vector problem. We show that these three problems are equivalent in that they have a common solution. We then show that this solves the original quantum measure problem.