# Mixing angle > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Mixing_angle > Markdown URL: https://mediated.wiki/source/Mixing_angle.md > Source: https://en.wikipedia.org/wiki/Mixing_angle > Source revision: 1343945424 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Angle between orthogonal basis vectors in particle physics and quantum mechanics In [particle physics](/source/Particle_physics) and [quantum mechanics](/source/Quantum_mechanics), **mixing angles** are the [angles](/source/Angle) between two sets of ([complex](/source/Complex_number)-valued) [orthogonal basis](/source/Orthogonal_basis) vectors, or [states](/source/Quantum_state), usually the [eigenbases](/source/Eigenbasis) of two [quantum mechanical operators](/source/Operator_(quantum_mechanics)).[1] 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](/source/Unitary_matrix), the analogue of a [rotation matrix](/source/Rotation_matrix) in a [complex](/source/Complex_number) [vector space](/source/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](/source/Euler_angles). ## Probabilistic Interpretation Given a [quantum state](/source/Quantum_state) (vector in a [Hilbert space](/source/Hilbert_space)) | ψ ⟩ {\displaystyle |\psi \rangle } , its [inner product](/source/Inner_product) with another state | ϕ ⟩ {\displaystyle |\phi \rangle } is a probability amplitude. When the square-modulus is taken, | ⟨ ϕ | ψ ⟩ | 2 {\displaystyle \left|\langle \phi |\psi \rangle \right|^{2}} gives the probability that the system will be in state | ϕ ⟩ {\displaystyle |\phi \rangle } .[2] 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 vectors](/source/Basis_vector) denoted | 0 ⟩ , | 1 ⟩ {\displaystyle |0\rangle ,\ |1\rangle } . An arbitrary state in this basis can be parametrized by an angle θ {\displaystyle \theta } : one can write | Ψ ⟩ = cos ⁡ θ | 0 ⟩ + sin ⁡ θ | 1 ⟩ {\displaystyle |\Psi \rangle =\cos \theta |0\rangle +\sin \theta |1\rangle } . Such a parametrization is [normalizable](/source/Normalized_solution_(mathematics)), and allows us to define different states in terms of θ {\displaystyle \theta } . The **mixing angle** between these two states is the difference in exactly the angle θ {\displaystyle \theta } between the states. As was previously stated, this angle is deeply related to the probability of finding state | ψ ⟩ {\displaystyle |\psi \rangle } in state | ϕ ⟩ {\displaystyle |\phi \rangle } , computed by | ⟨ ϕ | ψ ⟩ | 2 = cos 2 ⁡ ϑ mixing {\displaystyle \left|\langle \phi |\psi \rangle \right|^{2}=\cos ^{2}\vartheta _{\text{mixing}}} . **Proof** Let | ψ ⟩ = cos ⁡ θ 1 | 0 ⟩ + sin ⁡ θ 1 | 1 ⟩ and let | ϕ ⟩ = cos ⁡ θ 2 | 0 ⟩ + sin ⁡ θ 2 | 1 ⟩ . Then, | ⟨ ϕ | ψ ⟩ | 2 = ( cos ⁡ θ 1 cos ⁡ θ 2 + sin ⁡ θ 1 sin ⁡ θ 2 ) 2 = cos 2 ⁡ ( θ 1 − θ 2 ) . Defining ϑ mixing ≡ | θ 1 − θ 2 | , | ⟨ ϕ | ψ ⟩ | 2 = cos 2 ⁡ θ mixing {\displaystyle {\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}}} ## Notable mixing angles Some notable mixing angles in particle physics are: - *[Neutrino](/source/Neutrino) mixing angles* ([PMNS matrix](/source/Pontecorvo%E2%80%93Maki%E2%80%93Nakagawa%E2%80%93Sakata_matrix)), describing the mixing between the mass and flavour eigenstates of neutrinos, which explains [neutrino oscillations](/source/Neutrino_oscillation).[1] - *[Quark](/source/Quark) mixing angles* including the *[Cabbibo angle](/source/Cabbibo_angle)* ([CKM matrix](/source/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix)), describing the mixing between the mass and flavour eigenstates of quarks. - The *[Weinberg angle](/source/Weinberg_angle)* or *weak mixing angle*, describing the mixing between and relative strength of the electromagnetic and weak forces.[1] - *[Higgs mixing angle](https://en.wikipedia.org/w/index.php?title=Higgs_mixing_angle&action=edit&redlink=1)* ## References 1. ^ [***a***](#cite_ref-:0_1-0) [***b***](#cite_ref-:0_1-1) [***c***](#cite_ref-:0_1-2) Griffiths, David J. (2007). *Introduction to elementary particles*. Weinheim: Wiley. [ISBN](/source/ISBN_(identifier)) [978-0-471-60386-3](https://en.wikipedia.org/wiki/Special:BookSources/978-0-471-60386-3). 1. **[^](#cite_ref-2)** Born, M. Zur Quantenmechanik der Stoßvorgänge. Z. Physik 37, 863–867 (1926). [https://doi.org/10.1007/BF01397477](https://doi.org/10.1007/BF01397477) This particle physics–related article is a stub. You can help Wikipedia by adding missing information. - [v](https://en.wikipedia.org/wiki/Template:Particle-stub) - [t](/source/Template_talk%3AParticle-stub) - [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Particle-stub) This quantum mechanics–related article is a stub. You can help Wikipedia by adding missing information. - [v](https://en.wikipedia.org/wiki/Template:Quantum-stub) - [t](/source/Template_talk%3AQuantum-stub) - [e](https://en.wikipedia.org/wiki/Special:EditPage/Template:Quantum-stub) --- Adapted from the Wikipedia article [Mixing angle](https://en.wikipedia.org/wiki/Mixing_angle) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Mixing_angle?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.