Mathematical spaces {complex vector space} {Hilbert space, quantum mechanics} can have complex-number vectors that originate at origin.
dimensions
Mathematical spaces can have from zero to infinite number of dimensions (coordinates), all of same type. Mathematical-space points have values for all coordinates.
vectors
Complex vectors are not lines, like real vectors, but are planes because they have two components, real and imaginary. Complex vectors can vary over time and so are waves with phase and amplitude. Phase goes from 0 to 2 * pi. Vector length is wave amplitude.
Hilbert-space vectors represent same state no matter what length, because only space direction is a physical property.
vectors: normalization
Because only direction matters, normalized vectors can all have amplitude one (unit vector), making square equal one.
vectors: scalar product
Vectors have scalar products with themselves {Hermitean scalar product}, to make squared length. Scalar products commute, so relations are symmetrical. If two coordinate vectors have scalar product zero, they are orthogonal and independent. Two vectors typically are not orthogonal, but spin states of spin-1/2 particles are orthogonal, as are integer multiples of spin 1/2.
transformations
If coordinate relations are linear, coordinate systems can transform, using translation, rotation, and reflection.
Physical Sciences>Physics>Quantum Mechanics>Theory>Spaces
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Date Modified: 2022.0224