МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
293
https://universalpublishings.com
"QUANTUM MATHEMATICS AND PHYSICS: STUDYING
MATHEMATICAL FOUNDATIONS AND APPLICATIONS."
Khudaikulova Saida Zakirovna
Teacher of Termez State Pedagogical Institute
Phone: +99890-246-47-47
Ruzikulov Shahabbas
2nd-year student of
Temez State Pedagogical Institute
Annotation:
Quantum mechanics and quantum physics have revolutionized our
understanding of the fundamental nature of reality. At the core of this revolution lies
quantum mathematics, which provides the mathematical foundation for describing the
motion of particles at microscopic scales. This article explores the fundamental
mathematical structures of quantum mechanics, including Hilbert spaces, operators,
and wave functions, as well as their applications in modeling physical systems. The
research also examines how quantum physics contrasts with classical physics concepts
and offers new insights into topics such as quantum entanglement, superposition, and
quantum computing. By analyzing the mathematical foundations of quantum theories,
the article aims to shed light on the intersection of mathematics and physics, offering
a deeper understanding of how mathematical formulas help predict and explain
quantum phenomena. Furthermore, it discusses the potential implications of quantum
mathematics in emerging fields such as quantum computing and cryptography.
Keywords:
Quantum mechanics, quantum physics, mathematical physics, Hilbert
space, quantum operators, wave functions, quantum entanglement, quantum
computing, quantum cryptography, superposition, mathematical modeling, quantum
theory.
Quantum mechanics, often described as the most successful theory in physics,
introduces counterintuitive concepts that challenge classical intuition. The
mathematical framework underlying quantum theory has enabled physicists to explain
phenomena previously considered impossible, such as the dual nature of light and
particles, the uncertainty principle, and quantum tunneling. Studying quantum
mathematics involves advanced mathematical tools like linear algebra, functional
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
294
https://universalpublishings.com
analysis, and probability theory, all of which are essential for rigorously formulating
quantum mechanics.
Key Highlights:
1.
Mathematical Foundations of Quantum Mechanics
Quantum mechanics operates on a mathematical structure primarily defined in terms
of Hilbert spaces and linear operators. This section elaborates on the formalism of
quantum states, the role of observables, and the measurement process. It also discusses
the significance of wave functions, their interpretation, and their evolution over time.
2.
Quantum Operators and Commutation Relations
Quantum operators represent physical observables, and their commutation relations
govern the fundamental characteristics of quantum systems. For instance, the
Heisenberg uncertainty principle arises as a direct consequence of non-commuting
operators, imposing fundamental limits on the precision with which certain pairs of
physical quantities (e.g., position and momentum) can be simultaneously known.
3.
Quantum Entanglement and Superposition
Quantum entanglement describes the phenomenon where the quantum states of two or
more particles are interconnected such that the state of one particle cannot be described
independently of the others. Superposition, another cornerstone of quantum mechanics,
refers to a system's ability to exist in multiple states simultaneously. Both phenomena
have profound implications for interpreting reality and have practical applications in
quantum computing and cryptography.
4.
Schrödinger Equation and Quantum Evolution
The time-dependent Schrödinger equation forms the cornerstone of quantum
mechanics. This section explores its role in determining the time evolution of quantum
states. Solutions to the Schrödinger equation provide insights into the dynamic
behavior of quantum systems, ranging from simple particles in potential wells to
complex multi-particle interactions.
Key Mathematical Elements:
•
The role of complex numbers in quantum theory.
•
The importance of commutation relations between operators.
•
The significance of eigenvalues in measurement theory.
МЕДИЦИНА, ПЕДАГОГИКА И ТЕХНОЛОГИЯ:
ТЕОРИЯ И ПРАКТИКА
Researchbib Impact factor: 11.79/2023
SJIF 2024 = 5.444
Том 2, Выпуск 12,
31
Декабрь
295
https://universalpublishings.com
•
The probabilistic interpretation of quantum states.
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