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M. next
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Is the transformation of an operator under INFINITESIMAL unitary transformation, U^-1AU or UAU^-1?? I saw that two books defined it differently?
A unitary transformation is a mathematical operation that preserves the length and angles of vectors in a vector space. In other words, it is a transformation that maintains the overall structure and geometry of a space.
A regular transformation can change the length and angles of vectors, while a unitary transformation preserves these properties. Additionally, unitary transformations are always reversible, meaning that they have an inverse transformation that can undo the original transformation.
In quantum mechanics, unitary transformations play a crucial role in representing the evolution of quantum systems. These transformations ensure that the laws of quantum mechanics, such as the conservation of energy and probability, are preserved.
Unitary transformations and Hermitian operators are closely related in quantum mechanics. In fact, every unitary transformation can be represented by a corresponding Hermitian operator. This relationship is known as the spectral theorem.
Yes, unitary transformations have many practical applications in fields such as signal processing, data compression, and image processing. They are also used in quantum computing and machine learning algorithms.