Summary

Quantum mechanics is one of the most prolific theories of all time. Despite the many controversies it has aroused since its conception, its predictions have been confirmed experimentally with incredible accuracy. The quantum mechanics has been used in a variety of fields. Like, quantum economics—a very promising novel field of its application; quantum computation; quantum information science; quantum electronics; quantum cosmology; quantum chemistry.

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Editors
Marek Szopa

Institution: Department of Operations Research, College of Informatics and Communication, University of Economics in Katowice, Ul. Bogucicka 3, 40-287 Katowice, Poland

Interests: game theory; quantum games; matching markets; fair share; quantum communication; application of game theory to negotiation and decision making

Sandro Sozzo

Institution: Department of Humanities and Cultural Heritage (DIUM), Centre for Quantum Social and Cognitive Science (CQSCS), University of Udine, Vicolo Florio 2/b, 33100 Udine, Italy

Interests: foundations of science; epistemology; quantum mechanics; entanglement; cognitive modelling

Entries
Topic Review Peer Reviewed
Grover Quantum Algorithm: Applications and Limits
The Grover algorithm is a fundamental quantum algorithm that achieves a quadratic speedup for unstructured search problems, requiring O(√N) queries instead of O(N) classically. It works by repeatedly applying an oracle and a diffusion operator to amplify the probability of marked states. This advantage makes it relevant to cryptography, optimization, and constraint satisfaction and as a general primitive via amplitude amplification in areas like quantum machine learning and simulation. However, practical implementations are severely constrained by current noisy intermediate-scale quantum (NISQ) machines with limited coherence, deep oracle circuits, and lack of scalable Quantum RAM, restricting demonstrations to small-scale experiments with reproducibility challenges.
  • 14
  • 14 Apr 2026
Topic Review Peer Reviewed
Derivation of the Schrödinger Equation from Fundamental Principles
Schrödinger’s path to the quantum mechanical wave equation was heuristic and guided more by physical intuition than formal deduction. Here we derive the Schrödinger equation for the particle’s wave function Ψ, assuming that the complex function Ψ(t,𝑟⃗ ) has a meaning of the probability amplitude to find the particle at time 𝑡 at point 𝑟⃗ and the relations 𝐸=ℏ𝜔, 𝑝⃗ =ℏ𝑘⃗ expressing particle energy and momentum in terms of the frequency and wave vector of the associated probability wave.
  • 137
  • 14 Apr 2026
Topic Review Peer Reviewed
Wavefunction Collapse Broadens Molecular Spectrum
Spectral lines in the optical spectra of atoms, molecules, and other quantum systems are characterized by a range of frequencies ω or a range of wavelengths λ=2πc/ω, where c is the speed of light. Such a frequency or wavelength range is called the width of the spectral lines (linewidth). It is influenced by many specific factors. Thermal motion of the molecules results in broadening of the lines as a result of the Doppler effect (thermal broadening) and by their collisions (pressure broadening). The electric fields of neighboring molecules lead to Stark broadening. The linewidth to be considered here is the so-called parametric broadening (PB) of spectral lines in the optical spectrum. PB can be considered the fundamental type of broadening of the electronic vibrational–rotational (rovibronic) transitions in a molecule, which is the direct manifestation of the basic concept of the collapse of a wavefunction that is postulated by the Copenhagen interpretation of quantum mechanics. Thus, that concept appears to be not only valid but is also useful for predicting physically observable phenomena.
  • 1.9K
  • 11 Apr 2023
Topic Review Peer Reviewed
Undecidability and Quantum Mechanics
Recently, great attention has been devoted to the problem of the undecidability of specific questions in quantum mechanics. In this context, it has been shown that the problem of the existence of a spectral gap, i.e., energy difference between the ground state and the first excited state, is algorithmically undecidable. Using this result herein proves that the existence of a quantum phase transition, as inferred from specific microscopic approaches, is an undecidable problem, too. Indeed, some methods, usually adopted to study quantum phase transitions, rely on the existence of a spectral gap. Since there exists no algorithm to determine whether an arbitrary quantum model is gapped or gapless, and there exist models for which the presence or absence of a spectral gap is independent of the axioms of mathematics, it infers that the existence of quantum phase transitions is an undecidable problem. 
  • 2.1K
  • 18 Oct 2022
Topic Review Peer Reviewed
Foundations of Quantum Mechanics
Quantum mechanics is a mathematical formalism that models the dynamics of physical objects. It deals with the elementary constituents of matter (atoms, subatomic and elementary particles) and of radiation. It is very accurate in predicting observable physical phenomena, but has many puzzling properties. The foundations of quantum mechanics are a domain in which physics and philosophy concur in attempting to find a fundamental physical theory that explains the puzzling features of quantum mechanics, while remaining consistent with its mathematical formalism. Several theories have been proposed for different interpretations of quantum mechanics. However, there is no consensus regarding any of these theories.
  • 3.3K
  • 07 Jun 2022

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