파울리 행렬

개요

• 전자의 스핀과 전자기장의 상호작용을 기술하기 위한 파울리 방정식 을 찾는 과정에서 등장
• 파울리 행렬

\[\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} \]\[\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix} \]\[\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}\]

교환자 관계식

• \([\sigma _i,\sigma _j]=2i \epsilon _{i j k}\sigma _k\)

anti-commutator

• \(\left\{\sigma _i,\sigma _j\right\}=2\delta _{i j}\)
• \(\left\{I,\sigma _1,\sigma _2,\sigma _3,iI,i \sigma _1,i \sigma _2,i \sigma _3\right\}\) 를 기저로 갖는 클리포드 대수를 얻는다
• 3차원 유클리드 공간 \(E_{3}\)의 클리포드 대수\(C(E_{3})\)와 동형이다

사원수와의 관게

• 해밀턴의 사원수 참조

sl(2)

• raising and lowering 연산자

\[\sigma_{\pm}=\frac{1}{2}(\sigma_{x}\pm i\sigma_{y})\] \[\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}\] \[\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}\] \[[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}\]

여러가지 관계식

\[ \sigma_{+}^2=\sigma_{-}^2=0 \]

\[ \{\sigma_{+},\sigma_{-}\}=1 \]

\[ \sigma_{+}\sigma_{-}=(1+\sigma_z)/2 \]

\[ \exp(i \frac{\pi}{2}\sigma_z)=i\sigma_z \]

스핀

• 양자역학적 시스템의 간단한 예
• 스핀과 파울리의 배타원리 항목 참조

역사

• 수학사 연표

관련된 항목들

• 클리포드 대수와 스피너
• 파울리 방정식
• 스핀과 파울리의 배타원리
• 요르단-위그너 변환(Jordan-Wigner transformation)

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxUUxmTjdqM1VEamM/edit

노트

말뭉치

1. the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded.
2. In May 1927 Pauli published "Zur Quantenmechanik des magnetischen Elektrons", in which he introduced "Pauli matrices".
3. , it is often more convenient to generate it from a basis formed by the Pauli matrices augmented by the unit matrix.
4. This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example.
5. As the quaternions of unit norm is group-isomorphic to SU(2), this gives yet another way of describing SU(2) via the Pauli matrices.
6. In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions.
7. The mathematical significance of this operator is seen by noticing that, from the properties of the Pauli matrices, all even powers of n˙ σ are equal to 1, and all odd powers are equal to n˙ σ.
8. Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.
9. In quantum mechanics, each Pauli matrix is related to an angular momentum operator that corresponds to an observable describing the spin of a spin ½ particle, in each of the three spatial directions.
10. Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices.
11. The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics.
12. These matrices X, Y, and Z are called the Pauli matrices.
13. Pauli matrices will be discussed in greater detail in a later chapter, as they play a key role in quantum computing and quantum communication.
14. In the language of quantum mechanics, hermitian matrices are observables, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.
15. In quantum mechanics, each Pauli matrix is related to an operator that corresponds to an observable describing the spin of a spin ½ particle, in each of the three spatial directions.
16. It is possible to form generalizations of the Pauli matrices in order to describe higher spin systems in three spatial dimensions.
17. For arbitrarily large j, the Pauli matrices can be calculated using the ladder operators.
18. Demonstrate that the three Pauli matrices given in below are unitary.
19. The rotation performed by a Pauli matrix occurs along the X, Y, or Z axis, repectively, of our visualization.
20. It can be generalized to the arbitrary number of dimensions, if we replace Pauli matrices with generalized Gell-Mann matrices .
21. Convert to a list or array of Pauli matrices.
22. This is a lazy iterator that converts each row into the Pauli matrix representation only as it is used.
23. I have so far misrepresented the term Pauli matrices.
24. The Pauli matrices in this form are not the exact equivalent of quaternions this is because, if we square them, we get +1 and not -1.
25. In this form Pauli matrices have different properties, they don't form a normed division algebra.