파울리 행렬 - 편집 역사

Pythagoras0: /* Spacy 패턴 목록 */

2020-12-28T02:15:25Z

Spacy 패턴 목록

Pythagoras0: /* 메타데이터 */ 새 문단

2020-12-27T13:08:04Z

메타데이터: 새 문단

Pythagoras0: /* 노트 */ 새 문단

2020-12-27T13:08:01Z

노트: 새 문단

2020년 11월 16일 (월) 12:08에 Pythagoras0님의 편집

2020-11-16T12:08:25Z

2013년 12월 16일 (월) 03:45에 Pythagoras0님의 편집

2013-12-16T03:45:48Z

2013년 3월 24일 (일) 23:00에 Pythagoras0님의 편집

2013-03-24T23:00:39Z

2013년 3월 13일 (수) 08:57에 Pythagoras0님의 편집

2013-03-13T08:57:27Z

Pythagoras0: /* 관련된 항목들 */

2013-02-08T10:44:48Z

Pythagoras0: /* sl(2) */

2013-02-08T10:42:43Z

sl(2)

2013년 2월 8일 (금) 10:37에 Pythagoras0님의 편집

2013-02-08T10:37:31Z

@@ 123번째 줄: / 123번째 줄: @@
 * ID :  [https://www.wikidata.org/wiki/Q336233 Q336233]
 ===Spacy 패턴 목록===
-# [{'LOWER': 'pauli'}, {'LEMMA': 'matrix'}]
+* [{'LOWER': 'pauli'}, {'LEMMA': 'matrix'}]

← 이전 판		2020년 12월 27일 (일) 13:08 판
117번째 줄:		117번째 줄:
	===소스===		===소스===
	<references />		<references />
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q336233 Q336233]
		+	===Spacy 패턴 목록===
		+	# [{'LOWER': 'pauli'}, {'LEMMA': 'matrix'}]

← 이전 판		2020년 12월 27일 (일) 13:08 판
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 />

@@ 39번째 줄: / 39번째 줄: @@
 ==여러가지 관계식==
-$$
+:<math>
 \sigma_{+}^2=\sigma_{-}^2=0
-$$
+</math>
-$$
+:<math>
 \{\sigma_{+},\sigma_{-}\}=1
-$$
+</math>
-$$
+:<math>
 \sigma_{+}\sigma_{-}=(1+\sigma_z)/2
-$$
+</math>
-$$
+:<math>
 \exp(i \frac{\pi}{2}\sigma_z)=i\sigma_z
-$$
+</math>

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
-*  전자의 스핀과 전자기장의 상호작용을 기술하기 위한 [[파울리 방정식]] 을 찾는 과정에서 등장<br>
+* 전자의 스핀과 전자기장의 상호작용을 기술하기 위한 [[파울리 방정식]] 을 찾는 과정에서 등장
-*  파울리 행렬:<math>\sigma_1 = \sigma_x = \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix} </math>:<math>\sigma_2 = \sigma_y = \begin{pmatrix} 0&-i\\ i&0 \end{pmatrix}  </math>:<math>\sigma_3 = \sigma_z = \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}</math><br>
+* 파울리 행렬
-−
-−
-==commutator==
+==교환자 관계식==
 ==anti-commutator==
-* <math>\left\{\sigma _i,\sigma _j\right\}=2\delta _{i j}</math><br>
+* <math>\left\{\sigma _i,\sigma _j\right\}=2\delta _{i j}</math>
-* <math>\left\{I,\sigma _1,\sigma _2,\sigma _3,iI,i \sigma _1,i \sigma _2,i \sigma _3\right\}</math> 를 기저로 갖는  [[클리포드 대수와 스피너|클리포드 대수]]를 얻는다<br>
+* <math>\left\{I,\sigma _1,\sigma _2,\sigma _3,iI,i \sigma _1,i \sigma _2,i \sigma _3\right\}</math> 를 기저로 갖는  [[클리포드 대수와 스피너|클리포드 대수]]를 얻는다
-*  3차원 유클리드 공간 <math>E_{3}</math>의 [[클리포드 대수와 스피너|클리포드 대수]]<math>C(E_{3})</math>와 동형이다<br>
+*  3차원 유클리드 공간 <math>E_{3}</math>의 [[클리포드 대수와 스피너|클리포드 대수]]<math>C(E_{3})</math>와 동형이다
 ==사원수와의 관게==
-* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]] 참조<br>
+* [[해밀턴의 사원수(quarternions)|해밀턴의 사원수]] 참조
-−
-−
 ==sl(2)==
@@ 39번째 줄: / 35번째 줄: @@
 :<math>\sigma_{+}=\frac{1}{2}(\sigma_{x}+ i\sigma_{y})=\begin{pmatrix} 0&1\\ 0&0 \end{pmatrix}</math>
 :<math>\sigma_{-}=\frac{1}{2}(\sigma_{x}- i\sigma_{y})=\begin{pmatrix} 0&0\\ 1&0 \end{pmatrix}</math>
-:<math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math><br>
+:<math>[\sigma_{z},\sigma_{\pm}]=\pm 2\sigma_{\pm}</math>
@@ 57번째 줄: / 53번째 줄: @@
 $$
 \exp(i \frac{\pi}{2}\sigma_z)=i\sigma_z
 $$
 ==스핀==
-−
-−
 ==역사==
@@ 74번째 줄: / 68번째 줄: @@
 * [[수학사 연표]]
-−
 ==관련된 항목들==
@@ 94번째 줄: / 78번째 줄: @@
 * [[스핀과 파울리의 배타원리]]
 * [[요르단-위그너 변환(Jordan-Wigner transformation)]]
 ==매스매티카 파일 및 계산 리소스==
 * https://docs.google.com/file/d/0B8XXo8Tve1cxUUxmTjdqM1VEamM/edit
 [[분류:리군과 리대수]]
 [[분류:수리물리학]]

← 이전 판		2013년 3월 24일 (일) 23:00 판
114번째 줄:		114번째 줄:
	==리뷰논문, 에세이, 강의노트==		==리뷰논문, 에세이, 강의노트==
	[[분류:리군과 리대수]]		[[분류:리군과 리대수]]
		+	[[분류:수리물리학]]

← 이전 판		2013년 3월 13일 (수) 08:57 판
113번째 줄:		113번째 줄:

	==리뷰논문, 에세이, 강의노트==		==리뷰논문, 에세이, 강의노트==
		+	[[분류:리군과 리대수]]