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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
||
| (같은 사용자의 중간 판 3개는 보이지 않습니다) | |||
| 39번째 줄: | 39번째 줄: | ||
==여러가지 관계식== | ==여러가지 관계식== | ||
| − | + | :<math> | |
\sigma_{+}^2=\sigma_{-}^2=0 | \sigma_{+}^2=\sigma_{-}^2=0 | ||
| − | + | </math> | |
| − | + | :<math> | |
\{\sigma_{+},\sigma_{-}\}=1 | \{\sigma_{+},\sigma_{-}\}=1 | ||
| − | + | </math> | |
| − | + | :<math> | |
\sigma_{+}\sigma_{-}=(1+\sigma_z)/2 | \sigma_{+}\sigma_{-}=(1+\sigma_z)/2 | ||
| − | + | </math> | |
| − | + | :<math> | |
\exp(i \frac{\pi}{2}\sigma_z)=i\sigma_z | \exp(i \frac{\pi}{2}\sigma_z)=i\sigma_z | ||
| − | + | </math> | |
| 86번째 줄: | 86번째 줄: | ||
[[분류:리군과 리대수]] | [[분류:리군과 리대수]] | ||
[[분류:수리물리학]] | [[분류:수리물리학]] | ||
| + | |||
| + | == 노트 == | ||
| + | |||
| + | ===말뭉치=== | ||
| + | # the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded.<ref name="ref_c22282a2">[https://encyclopediaofmath.org/wiki/Pauli_matrices Encyclopedia of Mathematics]</ref> | ||
| + | # In May 1927 Pauli published "Zur Quantenmechanik des magnetischen Elektrons", in which he introduced "Pauli matrices".<ref name="ref_bba6b967">[https://library.ethz.ch/en/locations-and-media/platforms/virtual-exhibitions/wolfgang-pauli-and-modern-physics/the-old-testament-and-the-pauli-matrices.html The "Old Testament" and the Pauli matrices]</ref> | ||
| + | # , it is often more convenient to generate it from a basis formed by the Pauli matrices augmented by the unit matrix.<ref name="ref_f29c46b1">[https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Book%3A_Applied_Geometric_Algebra_(Tisza)/02%3A_The_Lorentz_Group_and_the_Pauli_Algebra/2.04%3A_The_Pauli_Algebra 2.4: The Pauli Algebra]</ref> | ||
| + | # This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example.<ref name="ref_7a7b4c73">[https://en.wikiversity.org/wiki/Pauli_matrices Pauli matrices]</ref> | ||
| + | # 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.<ref name="ref_7a7b4c73" /> | ||
| + | # In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions.<ref name="ref_7a7b4c73" /> | ||
| + | # 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.<ref name="ref_8692b5e1">[https://www.sciencedirect.com/topics/mathematics/pauli-matrix Pauli Matrix - an overview]</ref> | ||
| + | # Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.<ref name="ref_14ff85e1">[https://en.wikipedia.org/wiki/Pauli_matrices Pauli matrices]</ref> | ||
| + | # 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.<ref name="ref_14ff85e1" /> | ||
| + | # Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices.<ref name="ref_14ff85e1" /> | ||
| + | # The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics.<ref name="ref_c5e26bd8">[https://mathworld.wolfram.com/PauliMatrices.html Pauli Matrices -- from Wolfram MathWorld]</ref> | ||
| + | # These matrices X, Y, and Z are called the Pauli matrices.<ref name="ref_c227cc9b">[https://www.sciencedirect.com/topics/engineering/pauli-matrix Pauli Matrix - an overview]</ref> | ||
| + | # Pauli matrices will be discussed in greater detail in a later chapter, as they play a key role in quantum computing and quantum communication.<ref name="ref_c227cc9b" /> | ||
| + | # 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.<ref name="ref_5bd32b7f">[http://www.scientificlib.com/en/Physics/LX/PauliMatrices.html Pauli matrices]</ref> | ||
| + | # 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.<ref name="ref_5bd32b7f" /> | ||
| + | # It is possible to form generalizations of the Pauli matrices in order to describe higher spin systems in three spatial dimensions.<ref name="ref_5bd32b7f" /> | ||
| + | # For arbitrarily large j, the Pauli matrices can be calculated using the ladder operators.<ref name="ref_5bd32b7f" /> | ||
| + | # Demonstrate that the three Pauli matrices given in below are unitary.<ref name="ref_1bb3667f">[https://scipython.com/book/chapter-6-numpy/questions/the-unitary-property-of-the-pauli-matrices/ The unitary property of the Pauli matrices]</ref> | ||
| + | # The rotation performed by a Pauli matrix occurs along the X, Y, or Z axis, repectively, of our visualization.<ref name="ref_e7712ed4">[https://1ijk.dev/personal/hobbies/mathematics/eigenvalue-decomposition-of-pauli-matrices/ eigenvalue decomposition of Pauli matrices]</ref> | ||
| + | # It can be generalized to the arbitrary number of dimensions, if we replace Pauli matrices with generalized Gell-Mann matrices .<ref name="ref_821a06ef">[https://quantumsim.gitlab.io/architecture/pauli.html Pauli Bases — Quantumsim documentation]</ref> | ||
| + | # Convert to a list or array of Pauli matrices.<ref name="ref_a92d1c9c">[https://qiskit.org/documentation/stubs/qiskit.quantum_info.PauliTable.html info.PauliTable — Qiskit 0.23.2 documentation]</ref> | ||
| + | # This is a lazy iterator that converts each row into the Pauli matrix representation only as it is used.<ref name="ref_a92d1c9c" /> | ||
| + | # I have so far misrepresented the term Pauli matrices.<ref name="ref_e540a1e0">[https://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/notations/pauli/index.htm Pauli Matricies]</ref> | ||
| + | # 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.<ref name="ref_e540a1e0" /> | ||
| + | # In this form Pauli matrices have different properties, they don't form a normed division algebra.<ref name="ref_e540a1e0" /> | ||
| + | ===소스=== | ||
| + | <references /> | ||
| + | |||
| + | == 메타데이터 == | ||
| + | |||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q336233 Q336233] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'pauli'}, {'LEMMA': 'matrix'}] | ||
2020년 12월 27일 (일) 18:15 기준 최신판
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
관련된 항목들
매스매티카 파일 및 계산 리소스
노트
말뭉치
- the Pauli matrices form a complete system of second-order matrices by which an arbitrary linear operator (matrix) of dimension 2 can be expanded.[1]
- In May 1927 Pauli published "Zur Quantenmechanik des magnetischen Elektrons", in which he introduced "Pauli matrices".[2]
- , it is often more convenient to generate it from a basis formed by the Pauli matrices augmented by the unit matrix.[3]
- This relationship between the Pauli matrices and SU(2) can be explored further, as can be seen from the following simple example.[4]
- 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.[4]
- In quantum mechanics, each Pauli matrix represents an observable describing the spin of a spin ½ particle in the three spatial directions.[4]
- 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.[5]
- Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space.[6]
- 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.[6]
- Hence the Pauli matrices or the Sigma matrices operating on these spinors have to be 4 × 4 matrices.[6]
- The Pauli matrices, also called the Pauli spin matrices, are complex matrices that arise in Pauli's treatment of spin in quantum mechanics.[7]
- These matrices X, Y, and Z are called the Pauli matrices.[8]
- Pauli matrices will be discussed in greater detail in a later chapter, as they play a key role in quantum computing and quantum communication.[8]
- 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.[9]
- 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.[9]
- It is possible to form generalizations of the Pauli matrices in order to describe higher spin systems in three spatial dimensions.[9]
- For arbitrarily large j, the Pauli matrices can be calculated using the ladder operators.[9]
- Demonstrate that the three Pauli matrices given in below are unitary.[10]
- The rotation performed by a Pauli matrix occurs along the X, Y, or Z axis, repectively, of our visualization.[11]
- It can be generalized to the arbitrary number of dimensions, if we replace Pauli matrices with generalized Gell-Mann matrices .[12]
- Convert to a list or array of Pauli matrices.[13]
- This is a lazy iterator that converts each row into the Pauli matrix representation only as it is used.[13]
- I have so far misrepresented the term Pauli matrices.[14]
- 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.[14]
- In this form Pauli matrices have different properties, they don't form a normed division algebra.[14]
소스
- ↑ Encyclopedia of Mathematics
- ↑ The "Old Testament" and the Pauli matrices
- ↑ 2.4: The Pauli Algebra
- ↑ 4.0 4.1 4.2 Pauli matrices
- ↑ Pauli Matrix - an overview
- ↑ 6.0 6.1 6.2 Pauli matrices
- ↑ Pauli Matrices -- from Wolfram MathWorld
- ↑ 8.0 8.1 Pauli Matrix - an overview
- ↑ 9.0 9.1 9.2 9.3 Pauli matrices
- ↑ The unitary property of the Pauli matrices
- ↑ eigenvalue decomposition of Pauli matrices
- ↑ Pauli Bases — Quantumsim documentation
- ↑ 13.0 13.1 info.PauliTable — Qiskit 0.23.2 documentation
- ↑ 14.0 14.1 14.2 Pauli Matricies
메타데이터
위키데이터
- ID : Q336233
Spacy 패턴 목록
- [{'LOWER': 'pauli'}, {'LEMMA': 'matrix'}]