Additions:
In words, ""U"" is a ""linear operator"".
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability Bayesian]] ideas, [[http://en.wikipedia.org/wiki/Probability probabilities]] describe our ideas concerning the likelihood of events. They are not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by a ""continuous"" operator valued function of time, ""U"". Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).
So ""U"" is ""unitary"".
Deletions:
In words, ""U"" is a linear operator.
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability Bayesian]] ideas, [[http://en.wikipedia.org/wiki/Probability probabilities]] describe our ideas concerning the likelihood of events. They are not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, ""U"". Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).
So ""U"" is unitary.
Additions:
Deletions:
<<""Theorem: The wave function satisfies the Schrödinger Equation,""
Additions:
====""""Linearity of Time Evolution====
====""""Continuity of Time Evolution====
====""""Unitarity of Time Evolution====
====""""Stone’s Theorem====
====""""The Heisenberg Picture====
====""""The Wave Function====
====""""Newton’s First Law====
====""""Wave Mechanics====
Deletions:
====""""Linearity of Time Evolution====
====""""Continuity of Time Evolution====
====""""Unitarity of Time Evolution====
====""""Stone’s Theorem====
====""""The Heisenberg Picture====
====""""The Wave Function====
====""""Newton’s First Law====
====""""Wave Mechanics====
Additions:
====""""Linearity of Time Evolution====
====""""Continuity of Time Evolution====
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability Bayesian]] ideas, [[http://en.wikipedia.org/wiki/Probability probabilities]] describe our ideas concerning the likelihood of events. They are not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, ""U"". Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).
====""""Unitarity of Time Evolution====
====""""Stone’s Theorem====
====""""The Heisenberg Picture====
====""""The Wave Function====
====""""Newton’s First Law====
====""""Wave Mechanics====
Deletions:
====""""Linearity of Time Evolution====
====""""Continuity of Time Evolution====
Irrespective of whether a model of discrete particle interactions might appear continuous on the large scale, the evolution of kets is expected to be continuous because kets are not physical states of matter, but are rather probabilistic statements about what might happen in measurement, given current information. According to modern [[http://en.wikipedia.org/wiki/Bayesian_probability Bayesian]] ideas, [[http://en.wikipedia.org/wiki/Probability probabilities]] describe our ideas concerning the likelihood of events. They not a direct description of physical reality. Whether or not reality is fundamentally discrete, changes in probability can be properly described on a mathematical continuum. A discrete interaction will not lead to a discrete change in probability because we do not have information on when the interaction takes place. This being so, time evolution will be modelled by an continuous operator valued function of time, ""U"". Together with the considerations below, continuity is sufficient to ensure differentiability (formal proof omitted).
====""""Unitarity of Time Evolution====
====""""Stone’s Theorem====
====""""The Heisenberg Picture====
====""""The Wave Function====
====""""Newton’s First Law====
====""""Wave Mechanics====
