Mixing angle

{{Short description|Angle between orthogonal basis vectors in particle physics and quantum mechanics}} In [[particle physics]] and [[quantum mechanics]], '''mixing angles''' are the [[angle]]s between two sets of ([[Complex number|complex]]-valued) [[orthogonal basis]] vectors, or [[Quantum state|states]], usually the [[Eigenbasis|eigenbases]] of two [[Operator (quantum mechanics)|quantum mechanical operators]].<ref name=":0">{{Cite book |last=Griffiths |first=David J. |title=Introduction to elementary particles |date=2007 |publisher=Wiley |isbn=978-0-471-60386-3 |location=Weinheim}}</ref> The choice of angles (parameterization) is not unique but based on convention.

== Mathematics == The relation between two eigenbases is described completely by a [[unitary matrix]], the analogue of a [[rotation matrix]] in a [[Complex number|complex]] [[vector space]]. The number of degrees of freedom in this matrix is usually reduced by removing any excess complex phase from the transformation, since in most cases that is not a measurable quantity.

For two-dimensional vector space this reduces the matrix to a rotation matrix, which can be described completely by one mixing angle. In a three dimensional space there are three mixing angles and one additional '''complex phase parameter'''. Different conventions exist for how the three angles are defined, such as [[Euler angles]].

== Probabilistic Interpretation == Given a [[quantum state]] (vector in a [[Hilbert space]]) <math>|\psi\rangle</math>, its [[inner product]] with another state <math>|\phi\rangle</math> is a probability amplitude. When the square-modulus is taken, <math>\left|\langle\phi|\psi\rangle\right|^2</math> gives the probability that the system will be in state <math>|\phi\rangle</math>.<ref>Born, M. Zur Quantenmechanik der Stoßvorgänge. Z. Physik 37, 863–867 (1926). https://doi.org/10.1007/BF01397477</ref>

For a two-state system, where most will first encounter the mixing angle, the basis of this Hilbert space will be two-dimensional, often with [[basis vector]]s denoted <math>|0\rangle,\ |1\rangle</math>. An arbitrary state in this basis can be parametrized by an angle <math>\theta</math>: one can write <math>|\Psi\rangle = \cos\theta|0\rangle + \sin\theta|1\rangle</math>. Such a parametrization is [[Normalized solution (mathematics)|normalizable]], and allows us to define different states in terms of <math>\theta</math>.

The '''mixing angle''' between these two states is the difference in exactly the angle <math>\theta</math> between the states. As was previously stated, this angle is deeply related to the probability of finding state <math>|\psi\rangle</math> in state <math>|\phi\rangle</math>, computed by <math>\left|\langle\phi|\psi\rangle\right|^2= \cos^2\vartheta_\text{mixing}</math>.

{{math proof|<math display="block">\begin{aligned} &\text{Let }|\psi\rangle = \cos\theta_1|0\rangle + \sin\theta_1|1\rangle\\ &\text{and let } |\phi\rangle = \cos\theta_2|0\rangle + \sin\theta_2|1\rangle.\\ &\text{Then, } \left|\langle \phi|\psi\rangle\right|^2 = \left(\cos\theta_1\cos\theta_2 + \sin\theta_1\sin\theta_2\right)^2 = \cos^2(\theta_1 - \theta_2).\\ &\text{Defining }\vartheta_\text{mixing} \equiv |\theta_1 - \theta_2| \text{, } \left|\langle \phi|\psi\rangle\right|^2 = \cos^2\theta_\text{mixing} \end{aligned} </math>}}

== Notable mixing angles == Some notable mixing angles in particle physics are:

* ''[[Neutrino]] mixing angles'' ([[Pontecorvo–Maki–Nakagawa–Sakata matrix|PMNS matrix]]), describing the mixing between the mass and flavour eigenstates of neutrinos, which explains [[neutrino oscillation]]s.<ref name=":0" /> * ''[[Quark]] mixing angles'' including the ''[[Cabbibo angle]]'' ([[Cabibbo–Kobayashi–Maskawa matrix|CKM matrix]]), describing the mixing between the mass and flavour eigenstates of quarks. *The ''[[Weinberg angle]]'' or ''weak mixing angle'', describing the mixing between and relative strength of the electromagnetic and weak forces.<ref name=":0" /> * ''[[Higgs mixing angle]]''

==References== {{Reflist}}

[[Category:Particle physics]] [[Category:Quantum mechanics]]