Related to the work: 3, Quantum Mechanics

Quantum mechanics was developed after the establishment of old quantum theory. The old quantum theory made certain artificial modifications or additional conditions to classical physical theories in order to explain some phenomena in the microscopic realm. Due to the unsatisfactory nature of old quantum theory, people established quantum mechanics through two different paths while searching for the laws governing the microscopic domain

The difference between quantum mechanics and classical mechanics is primarily reflected in the description of the state of particles and the laws governing mechanical quantities and their changes. In quantum mechanics, the state of a particle is described by a wave function, which is a complex function of coordinates and time. To depict the laws governing the time evolution of microscopic particle states, it is necessary to identify the motion equation that the wave function satisfies. This equation was first discovered by Schrödinger in 1926 and is known as the Schrödinger equation. To depict the laws governing the time evolution of microscopic particle states, it is necessary to identify the motion equation that the wave function satisfies. This equation was first discovered by Schrödinger in 1926 and is known as the Schrödinger equation

Due to the wave-particle duality of microscopic particles, the laws of motion that govern these particles differ from those that govern macroscopic objects. Consequently, quantum mechanics, which describes the motion of microscopic particles, is distinct from classical mechanics, which describes the motion of macroscopic objects. As the size of a particle transitions from the microscopic to the macroscopic, the governing laws also transition from quantum mechanics to classical mechanics

m > , E_n is the energy eigenvalue, H is the Hamiltonian energy operator

After separating the variables, one can obtain the evolution equation H in a time-independent state

The fundamental principles of quantum mechanics include the concept of quantum states, the equations of motion, the correspondence rules between theoretical concepts and observable physical quantities, as well as the underlying physical principles

n >= δm, where n is the Dirac delta function, satisfying the orthonormality property

The state function satisfies the Schrödinger wave equation, i ( d / dt )

In 1900, Planck proposed the quantum hypothesis of radiation, assuming that the exchange of energy between the electromagnetic field and matter occurs in a discrete form (quanta of energy). The magnitude of the energy quanta is proportional to the radiation frequency, with the proportionality constant known as Planck's constant. This led to the derivation of the energy distribution formula for black body radiation, successfully explaining the phenomenon of black body radiation

Quantum mechanics characterizes the states of microscopic systems through the concept of quantum states, deepening our understanding of physical reality. The properties of microscopic systems are always manifested in their interactions with other systems, particularly with observational instruments

The hypothesis of wave-particle duality proposed by de Broglie: E = ω, p = h / λ, where h = h / 2π, can be derived from E = p² / 2m to obtain λ = √(h² / 2mE)

The interpretation of quantum mechanics involves numerous philosophical questions, with the core issues being causality and the nature of physical reality. According to the dynamical sense of causality, the equations of motion in quantum mechanics are also equations of causality. When the state of a system at a certain moment is known, it is possible to predict its states at any future or past moment based on the equations of motion.

In 1913, Bohr established the quantum theory of the atom based on Rutherford's nuclear atomic model. According to this theory, electrons in an atom can only move in discrete orbits, and the atom possesses a definite energy. This state is referred to as a "stationary state," and an atom can only absorb or emit energy when transitioning from one stationary state to another. Although this theory has many successful aspects, it still faces numerous challenges in further explaining experimental phenomena.

According to the Dirac notation, the state function is represented as < Ψ

In quantum mechanics, the state of a system undergoes two types of changes: one is the evolution of the system's state according to the equations of motion, which is a reversible change; the other is the irreversible change caused by measurement that alters the system's state. Therefore, quantum mechanics cannot provide definite predictions and diagrams for the physical quantities that determine the state, but can only offer probabilities for the values of these physical quantities. In this sense, the principle of causality in classical physics fails in the microscopic realm. Thus, quantum mechanics cannot provide definite predictions and diagrams for the physical quantities that determine the state, but can only offer probabilities for the values of these physical quantities. In this sense, the principle of causality in classical physics fails in the microscopic realm

Some have cited the randomness in quantum mechanics to support the notion of free will. However, first, there remains an insurmountable gap between this randomness at the microscopic scale and the macro-level free will in the conventional sense. Second, it is still difficult to prove whether this randomness is irreducible, as our observational capabilities at the microscopic level are still limited. Whether true randomness exists in nature remains an open question. Many examples of random events in statistics are, strictly speaking, deterministic. Quantum mechanics developed on the foundation of old quantum theory, which includes Planck's quantum hypothesis, Einstein's photon theory, and Bohr's atomic theory.

State functions can be expressed as expansions in the orthogonal space set of state vectors

m ≥ H

and

Accordingly, some physicists and philosophers assert that quantum mechanics abandons causality, while others believe that the causal laws of quantum mechanics reflect a new type of causality—probabilistic causality. In quantum mechanics, the wave function representing a quantum state is defined throughout the entire space, and any change in the state occurs simultaneously across the entire space.

In 1925, Heisenberg, based on the understanding that physical theories only deal with observable quantities, discarded the concept of unobservable orbits and, starting from observable radiation frequencies and their intensities, established matrix mechanics together with Born and Jordan. In 1926, Schrödinger, based on the recognition that quantum nature reflects the wave-like behavior of microscopic systems, found the motion equations for these systems, thereby establishing wave mechanics. Shortly thereafter, he also proved the mathematical equivalence of wave mechanics and matrix mechanics. Dirac and Jordan independently developed a universal transformation theory, providing a concise and complete mathematical expression for quantum mechanics.

Since the 1970s, experiments regarding distant particle correlations have indicated the existence of quantum mechanical predictions of correlations in spacelike-separated events. This correlation contradicts the viewpoint of special relativity, which states that physical interactions between objects can only be transmitted at speeds not exceeding that of light. Consequently, some physicists and philosophers have proposed the existence of a global or holistic causality in the quantum realm to explain the presence of such correlations. This differs from the local causality established on the basis of special relativity, as it can simultaneously determine the behavior of correlated systems as a whole.

Quantum mechanics is a branch of physics that studies the laws of motion of microscopic particles. It primarily investigates the fundamental theories regarding the structure and properties of atoms, molecules, condensed matter, atomic nuclei, and elementary particles. Together with relativity, it forms the theoretical foundation of modern physics. Quantum mechanics is not only one of the foundational theories of contemporary physics but has also found extensive applications in related disciplines such as chemistry and many modern technologies

Quantum mechanics indicates that the reality of microscopic physics is neither a wave nor a particle; the true reality is the quantum state. The real state is decomposed into hidden states and observable states, which is caused by measurement, where only the observable states conform to the classical meaning of reality. The reality of microscopic systems is also reflected in their inseparability. Quantum mechanics views the research object and its environment as a whole, and it does not allow the world to be seen as composed of separate, independent parts. The conclusions from experiments on the correlation of distant particles also quantitatively support and illustrate the inseparability of quantum states. Quantum mechanics views the research object and its environment as a whole, and it does not allow the world to be seen as composed of separate, independent parts. The conclusions from experiments on the correlation of distant particles also quantitatively support and illustrate the inseparability of quantum states

Heisenberg also proposed the uncertainty principle, which is expressed in the following formula: Δx Δp ≥ ℏ / 2

ρi > , where.

The square of the wave function represents the probability of the physical quantity appearing as its variable. Based on these fundamental principles and supplemented by other necessary assumptions, quantum mechanics can explain various phenomena at the atomic and subatomic levels

The combination of quantum mechanics and special relativity led to the development of relativistic quantum mechanics. This was further advanced by the work of Dirac, Heisenberg, and Pauli, resulting in quantum electrodynamics. After the 1930s, a quantization theory describing various particle fields was established—quantum field theory—which forms the theoretical foundation for describing the phenomena of fundamental particles

When people describe observational results using the language of classical physics, they find that under different conditions, a microscopic system may primarily exhibit wave-like patterns or primarily exhibit particle behavior. The concept of quantum state expresses the possibilities of a microscopic system's behavior as either a wave or a particle, resulting from the interaction between the system and the measuring instrument

m >= En

In 1905, Einstein introduced the concept of the light quantum (photon) and provided the relationship between the energy and momentum of photons and the frequency and wavelength of radiation, successfully explaining the photoelectric effect. Subsequently, he proposed that the vibrational energy of solids is also quantized, thereby explaining the specific heat problem of solids at low temperatures

The fundamental content of quantum mechanics

When a microscopic particle is in a certain state, its mechanical quantities (such as position, momentum, angular momentum, energy, etc.) generally do not have definite values, but rather a range of possible values, each appearing with a certain probability. When the state of the particle is determined, the probability of the mechanical quantity having a certain possible value is also completely determined. This is the uncertainty principle derived by Heisenberg in 1927, while Bohr proposed the complementarity principle, which provided further elucidation of quantum mechanics. When the state of the particle is determined, the probability of the mechanical quantity having a certain possible value is also completely determined. This is the uncertainty principle derived by Heisenberg in 1927, while Bohr proposed the complementarity principle, which provided further elucidation of quantum mechanics.

After people recognized that light possesses the dual nature of both waves and particles, in order to explain phenomena that classical theories could not account for, the French physicist Louis de Broglie proposed in 1923 the hypothesis that microscopic particles exhibit wave-particle duality. De Broglie posited that, just as light has wave-particle duality, material particles (such as electrons, atoms, etc.) also possess this property, meaning they exhibit both particle-like and wave-like characteristics. This hypothesis was soon confirmed by experiments. De Broglie believed that, just as light has wave-particle duality, material particles (such as electrons, atoms, etc.) also possess this property, meaning they exhibit both particle-like and wave-like characteristics. This hypothesis was soon confirmed by experiments

However, the predictions of quantum mechanics differ in nature from those of classical physics, specifically the equations of motion (the equations of motion for particles and wave equations). In classical physics, measuring a system does not alter its state; it undergoes only one type of change and evolves according to the equations of motion. Therefore, the equations of motion can make definite predictions about the mechanical quantities that determine the state of the system.

Ψ represents the probability, and its probability current density is expressed as ( / 2mi ) ( Ψ * ▽ Ψ - Ψ ▽ Ψ * ). The probability is the spatial integral of the probability density

Ψ ( x ) ≥ ∑

In quantum mechanics, the state of a physical system is represented by a wave function, and any linear superposition of wave functions still represents a possible state of the system. The evolution of the state over time follows a linear differential equation, which predicts the behavior of the system. Physical quantities are represented by operators that satisfy certain conditions and represent specific operations. The operation of measuring a physical quantity of a physical system in a certain state corresponds to the action of the operator representing that quantity on its wave function. The possible measurement values are determined by the eigenvalue equation of that operator, and the expected value of the measurement is calculated using an integral equation that includes that operator

The probability density of the wave function is represented as \( \rho = < \Psi \)

Thus, the quantization problem of classical physical quantities reduces to the problem of solving the Schrödinger wave equation

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