A unitary transformation is a linear operation on a complex vector space that preserves the inner product, meaning it maintains the length and angle between vectors. This property makes unitary transformations fundamental in quantum mechanics, where they describe the evolution of quantum states in a way that conserves probability amplitudes.

Unitary Transformation

unitary transformation

linear operation

A complex vector space is a vector space where the field of scalars is the complex numbers, allowing for operations such as addition and scalar multiplication to be performed with complex values. This structure is fundamental in quantum mechanics and other areas of physics and engineering, as it allows for the representation and manipulation of complex-valued functions and vectors.

complex vector space

The inner product is a fundamental operation in linear algebra that generalizes the dot product to abstract vector spaces, providing a way to define angles and lengths. It is essential for understanding orthogonality, projections, and the structure of Hilbert spaces, with applications across mathematics and physics.

inner product

length and angle between vectors

fundamental in quantum mechanics

evolution of quantum states

Probability amplitudes are complex numbers used in quantum mechanics to describe the likelihood of a quantum system transitioning from one state to another. The square of the absolute value of a probability amplitude gives the probability of the corresponding transition, embodying the fundamental principle of superposition in quantum states.

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