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The term transformation theory refers to a procedure and a "picture" used by Paul Dirac in his early formulation of quantum theory, from around 1927. [ 1 ] This "transformation" idea refers to the changes a quantum state undergoes in the course of time, whereby its vector "moves" between "positions" or "orientations" in its Hilbert space .
Therefore, physicists have developed mathematical techniques to simplify these problems and clarify what is happening physically. One such technique is to apply a unitary transformation to the Hamiltonian. Doing so can result in a simplified version of the Schrödinger equation which nonetheless has the same solution as the original.
In mathematics, the Stieltjes transformation S ρ (z) of a measure of density ρ on a real interval I is the function of the complex variable z defined outside I by the formula S ρ ( z ) = ∫ I ρ ( t ) d t t − z , z ∈ C ∖ I . {\displaystyle S_{\rho }(z)=\int _{I}{\frac {\rho (t)\,dt}{t-z}},\qquad z\in \mathbb {C} \setminus I.}
Binomial transform; Discrete Fourier transform, DFT . Fast Fourier transform, a popular implementation of the DFT; Discrete cosine transform. Modified discrete cosine transform ...
More precisely, a unitary transformation is an isometric isomorphism between two inner product spaces (such as Hilbert spaces). In other words, a unitary transformation is a bijective function U : H 1 → H 2 {\displaystyle U:H_{1}\to H_{2}}
In theoretical physics, the Weyl transformation, named after German mathematician Hermann Weyl, is a local rescaling of the metric tensor: ...
A diffusionless transformation, commonly known as displacive transformation, denotes solid-state alterations in crystal structures that do not hinge on the diffusion of atoms across extensive distances. Rather, these transformations manifest as a result of synchronized shifts in atomic positions, wherein atoms undergo displacements of distances ...
The Mathieu transformations make up a subgroup of canonical transformations preserving the differential form ∑ i p i δ q i = ∑ i P i δ Q i {\displaystyle \sum _{i}p_{i}\delta q_{i}=\sum _{i}P_{i}\delta Q_{i}\,}