Editing Superposition principle (section)

===Quantum superposition===
{{main|Quantum superposition}}

In [[quantum mechanics]], a principal task is to compute how a certain type of wave [[wave propagation|propagates]] and behaves. The wave is described by a [[wave function]], and the equation governing its behavior is called the [[Schrödinger equation]]. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "[[quantum superposition]]") of (possibly infinitely many) other wave functions of a certain type—[[stationary state]]s whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.<ref name="QuaMech">Quantum Mechanics, [[Hendrik Anthony Kramers|Kramers, H.A.]] publisher Dover, 1957, p. 62 {{ISBN|978-0-486-66772-0}}</ref>

{{Anchor|projective2016-01-30}}The projective nature of quantum-mechanical-state space causes some confusion, because a quantum mechanical state is a ''ray'' in [[projective Hilbert space]], not a ''vector''.
According to [[Paul Dirac|Dirac]]: "''if the ket vector corresponding to a state is multiplied by any complex number, not zero, the resulting ket vector will correspond to the same state'' [italics in original]."<ref>[[Paul Adrien Maurice Dirac|Dirac, P. A. M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford, UK: Oxford University Press, p.&nbsp;17.</ref>
However, the sum of two rays to compose a superpositioned ray is undefined. As a result, Dirac himself
uses ket vector representations of states to decompose or split,
for example, a ket vector <math>|\psi_i\rangle</math>
into superposition of component ket vectors <math>|\phi_j\rangle</math> as:
<math display="block">|\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle,</math>
where the <math>C_j\in \textbf{C}</math>.
The equivalence class of the <math>|\psi_i\rangle</math> allows a well-defined meaning to be given to the relative phases of the <math>C_j</math>.,<ref>{{cite journal|last1=Solem|first1=J. C.|last2=Biedenharn|first2=L. C.|year=1993|title=Understanding geometrical phases in quantum mechanics: An elementary example|journal=Foundations of Physics|volume=23|issue=2|pages=185–195|bibcode = 1993FoPh...23..185S |doi = 10.1007/BF01883623 |s2cid=121930907}}</ref> but an absolute (same amount
for all the <math>C_j</math>) phase change on the <math>C_j</math>
does not affect the equivalence class of the <math>|\psi_i\rangle</math>.

There are exact correspondences between the superposition presented in the main on this page and the quantum superposition.
For example, the [[Bloch sphere]] to represent [[pure state]] of a [[two-level system|two-level quantum mechanical system]]
([[qubit]]) is also known as the [[Bloch sphere|Poincaré sphere]] representing different types of classical
pure [[Polarization (waves)|polarization]] states.

Nevertheless, on the topic of quantum superposition, [[Hans Kramers|Kramers]] writes: "The principle of [quantum] superposition ... has no analogy in classical physics"{{Citation needed|date=March 2023|reason=hopefully not only conclusion but also reasoning}}.
According to [[Paul Dirac|Dirac]]: "''the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory'' [italics in original]."<ref>[[Paul Adrien Maurice Dirac|Dirac, P. A. M.]] (1958). ''The Principles of Quantum Mechanics'', 4th edition, Oxford, UK: Oxford University Press, p.&nbsp;14.</ref>
Though reasoning by Dirac includes atomicity of observation, which is valid, as for phase,
they actually mean phase translation symmetry derived from [[time translation symmetry]], which is also
applicable to classical states, as shown above with classical polarization states.