The time development operator in quantum theory is unitary, because the Hamiltonian is hermitian. Consequently the transition probability matrix is doubly stochastic, which implies the Second Law of Thermodynamics. This derivation is quite general, based on the Shannon entropy, and does not require any assumptions beyond unitarity, which is universally accepted. It is a consequence of the irreversibility or singular nature of the general transition matrix. I do not fully understand this, but it seems to imply that all possible choices in the quantum state must add up to 100%. The quantum state collapse gives up information through the entropy laws. The original concepts of decoherence proposed by David Bohm and John Bell were held in little regard for many years, but the advances in quantum computing have caused a resurgence in the study of dephasing. While Bohm felt that a carrier wave was responsible for the duality of particles, dephasing holds that the particle is real in many states, but releases information (irreversibility) when interaction occurs.
Vector phasing, may be functions or frequencies; instead of matrix multiplication, linear transformations may be operators such as the derivative from calculus. These are only a few of countless examples where eigenvectors and eigenvalues are important.
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In this shear graph, the yellow arrow
undergoes a phase shift but the red arrow does not.
Therefore the red arrow is an eigenvector,
with eigenvalue 1, as its length is unchanged. |
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The Bloch sphere is a representation of a qubit,
the fundamental building block of quantum computers. |
The Protium atom is an example where both types of spectra appear. The eigenfunctions of the hydrogen atom Hamiltonian are called eigenstates and are grouped into two categories. The bound states of the hydrogen atom correspond to the discrete part of the spectrum (they have a discrete set of eigenvalues that can be computed by Rydberg formula) while the ionization processes are described by the continuous part (the energy of the collision/ionization is not quantized). In mathematics, a shear mapping or transvection is a particular kind of linear mapping. Its effect leaves fixed all points on one axis and other points are shifted parallel to the axis by a distance proportional to their perpendicular distance from the axis. It is notable that shear mappings carry areas into equal areas.
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Folding Space. A simple cube being folded and multiplied |
The concept of multiple spacial eigenvector values coexisting in the wave function is not new. Some of the quantum information computing work being done is also focusing on time eigenvalue vector analysis.
Superposition, Quantum Algorithms (a fixed sequence of quantum logic gates), Particle Entanglement, and the
Quantum Decoherence are all important aspects being studied. The goal in quantum computing is to get the quantum state to release information. The theory of decoherence has made a huge comeback in main-stream thinking. Now decoherence is seen less as a complete collapse of the probability wave (or the disappearance of a carrier wave), and more as the release of information to our timeline through a process of entropic information leak. The probability wave never fully collapses, but must make decisions based on interactions to remain in concert with the laws of physics (a partial collapse).
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| The Protium atom orbital shells. |
