Expectation values in quantum mechanics provide the average outcome of a measurement over many identical experiments, reflecting the probabilistic nature of quantum states. They are calculated using the wave function or state vector, offering insights into observable quantities like position, momentum, and energy.

Expectation Values

Expectation values

Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, uncertainty principle, and quantum entanglement, which challenge classical intuitions about the behavior of matter and energy.

quantum mechanics

average outcome of a measurement

identical experiments

probabilistic nature of quantum states

The wave function is a fundamental concept in quantum mechanics that describes the quantum state of a system, encoding information about the probability amplitudes of a particle's position, momentum, and other physical properties. It is typically represented as a complex-valued function, and its squared magnitude gives the probability density of finding a particle in a particular state or location.

wave function

A state vector is a mathematical representation of a physical system's state in a given space, encapsulating all the information necessary to describe the system at a particular time. It is fundamental in quantum mechanics and control theory, where it is used to predict future states and analyze system behavior.

state vector

observable quantities

Position refers to the specific location or arrangement of an object or entity in space or within a particular context. It is a fundamental concept in various fields, influencing how entities interact, relate, and are perceived in both physical and abstract spaces.

position

Momentum is a vector quantity that describes the quantity of motion an object has, calculated as the product of its mass and velocity. It is a conserved quantity in isolated systems, meaning the total momentum remains constant unless acted upon by external forces.

momentum

Energy is a fundamental property of the universe that can neither be created nor destroyed, only transformed from one form to another. It is the driving force behind all physical processes and is essential for the functioning of both natural and engineered systems.

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