What is Pauli matrices: Definition and 59 Discussions

In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter sigma (σ), they are occasionally denoted by tau (τ) when used in connection with isospin symmetries.

These matrices are named after the physicist Wolfgang Pauli. In quantum mechanics, they occur in the Pauli equation which takes into account the interaction of the spin of a particle with an external electromagnetic field.
Each Pauli matrix is Hermitian, and together with the identity matrix I (sometimes considered as the zeroth Pauli matrix σ0), the Pauli matrices form a basis for the real vector space of 2 × 2 Hermitian matrices.
This means that any 2 × 2 Hermitian matrix can be written in a unique way as a linear combination of Pauli matrices, with all coefficients being real numbers.
Hermitian operators represent observables in quantum mechanics, so the Pauli matrices span the space of observables of the 2-dimensional complex Hilbert space. In the context of Pauli's work, σk represents the observable corresponding to spin along the kth coordinate axis in three-dimensional Euclidean space R3.
The Pauli matrices (after multiplication by i to make them anti-Hermitian) also generate transformations in the sense of Lie algebras: the matrices iσ1, iσ2, iσ3 form a basis for the real Lie algebra



{\displaystyle {\mathfrak {su}}(2)}
, which exponentiates to the special unitary group SU(2). The algebra generated by the three matrices σ1, σ2, σ3 is isomorphic to the Clifford algebra of R3, and the (unital associative) algebra generated by iσ1, iσ2, iσ3 is isomorphic to that of quaternions.

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  1. G

    A Question about commutator involving fermions and Pauli matrices

    Suppose ##\lambda_A## and ##\bar{\lambda}_A## are fermions (A goes from 1 to N) and ##\{ \lambda_{A \alpha}, \bar{\lambda}_B^{\beta}\} = \delta_{AB}\delta_{\alpha}^{\beta}##. Let ##\sigma^i## denote the Pauli matrices. Does it follow that ##[\bar{\lambda}_A \sigma^i \lambda_A, \bar{\lambda_B}...
  2. Frigorifico9

    I In what chapter do Mehra and Rechenberg discuss Pauli matrices?

    I am very interested in how Pauli found the Pauli matrices, so I read his original paper, but it didn't give me the perspective I wanted, so I went to Mehra and Rechenberg, but here's the thing, after reading Volumes 1, 2 and most of volume 3, I can't find any mention of Pauli matrices anywhere...
  3. ARoyC

    I Confusion Regarding a Spectral Decomposition

    Hi. I am not being able to understand how we are getting the following spectral decomposition. It would be great if someone can explain it to me. Thank you in advance.
  4. P

    Spin matrix representation in any arbitrary direction

    I've tried to use the 1st equation as a matrix to determine, but it clearly isn't a diagonal matrix. My guess is that I need to find the spin matrix along the direction ##\hat{n}##, but do I need to find the eigenstates of ##\sigma \cdot \hat{n}## first and check if they form a diagonal matrix...
  5. docnet

    I Where do these time derivatives of Pauli matrices come from?

    Wolfgang Pauli's matrices are $$\sigma_x=\begin{bmatrix}0& 1\\1 & 0\end{bmatrix},\quad \sigma_y=\begin{bmatrix}0& -i\\i & 0\end{bmatrix},\quad \sigma_z=\begin{bmatrix}1& 0\\0 & -1\end{bmatrix}$$ He introduces these equations as "the equations of motion" of the spin in a magnetic field. $$...
  6. M

    A Diagonalization of 2x2 Hermitian matrices using Wigner D-Matrix

    Motivation: Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...
  7. hamad12a

    I How Peskin & Schroeder simplified this horrible product of bilinears?

    P&S had calculated this expression almost explicitly, except that I didn't find a way to exchange the $$\nu \lambda$$ indices, but I'm sure the below identity is used, $$ \begin{aligned}\left(\overline{u}_{1 L} \overline{\sigma}^{\mu} \sigma^{\nu} \overline{\sigma}^{\lambda} u_{2...
  8. Sunny Singh

    I Pauli matrices and shared eigenvectors

    We know that S2 commutes with Sz and so they share their eigenspace. Now since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx. But since the paulic matrices σx and σy are not diagonlized in the eigenbasis of S2, it is clear...
  9. QuasarBoy543298

    I Exploring the Pauli Matrices: Questions Answered

    Hi :) I have several questions about the Pauli matrices, I have seen them when the lecturer showed us Stern-Gerlach experiment , and we did some really weird assumptions on what we think they should be. 1- why did we assume that all of those matrices should satisfy σ2 = I (the identity...
  10. M

    How Do Lorentz Group Commutation Relations Apply to Spin Matrices?

    Homework Statement Prove that the sets ##(S_{\mu\nu})_L## and ##(S_{kl})_R##, where $$ \left( S _ { k \ell } \right) _ { L } = \frac { 1 } { 2 } \varepsilon _ { j k \ell } \sigma _ { j } = \left( S _ { k \ell } \right) _ { R } \quad\text{and}\quad \left( S _ { 0 k } \right) _ { L } = \frac {...
  11. P

    I Why choose traceless matrices as basis?

    While writing down the basis for SU(2), physicists often choose traceless hermitian matrices as such, often the Pauli matrices. Why is this? In particular why traceless, and why hermitian?
  12. AwesomeTrains

    Density matrix for a mixed neutron beam

    Homework Statement A beam of neutrons (moving along the z-direction) consists of an incoherent superposition of two beams that were initially all polarized along the x- and y-direction, respectively. Using the Pauli spin matrices: \sigma_x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \\...
  13. W

    A How spin projector got included in inverse of Matrix?

    The following matrix A is, \begin{equation} A= \begin{bmatrix} a+b-\sigma\cdot p & -x_1 \\ x_2 & a-b-\sigma\cdot p \end{bmatrix} \end{equation} The inversion of matrix A is, \begin{equation} A^{-1}= \frac{\begin{bmatrix} a-b-\sigma\cdot p & x_1 \\ -x_2 & a+b-\sigma\cdot p...
  14. Destroxia

    Pauli Matrices in the Basis of Y?

    Homework Statement [/B] I know the pauli matrices in terms of the z-basis, but can't find them in terms of the other bases. I would like to know what they are. Homework Equations The book says they are cyclic, via the relations XY=iZ, but this doesn't seem to apply when I use this to find the...
  15. T

    I How is Graphene's Hamiltonian rotationally invariant?

    Graphene's Hamiltonian contains first order derivatives (from the momentum operators) which aren't invariant under simple spatial rotations. So it initially appears to me that it isn't invariant under rotation. From reading around I see that we also have to perform a rotation on the Pauli...
  16. fresh_42

    Insights Representations and Why Precision is Important - Comments

    fresh_42 submitted a new PF Insights post Representations and Why Precision is Important Continue reading the Original PF Insights Post.
  17. F

    Calculation with Pauli matrices

    Homework Statement Hey :-) I just need some help for a short calculation. I have to show, that (\sigma \cdot a)(\sigma \cdot b) = (a \cdot b) + i \sigma \cdot (a \times b) The Attempt at a Solution I am quiet sure, that my mistake is on the right side, so I will show you my...
  18. M

    A How Does the Dirac Spin Exchange Operator Work in Quantum Mechanics?

    The spin exchange operator would have the property $$\begin{align*}P\mid \chi_{\uparrow\downarrow} \rangle = \mid\chi_{\downarrow\uparrow} \rangle & &P\mid \chi_{\downarrow\uparrow} \rangle =\mid \chi_{\uparrow\downarrow} \rangle \end{align*}$$ This also implies ##P\mid \chi_{\text{sym.}}...
  19. C

    Transformation of a 2x2 matrix with Pauli matrices

    Homework Statement Suppose the vector ##\phi## transforms under SU(2) as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where ## (t_j)_{kl} = −i \epsilon_{jkl}## and ##j, k, l \in \left\{1, 2, 3\right\}.## Based on ##\phi,## we define the ##2 \times 2## matrix ##\sigma = \tau \cdot...
  20. polyChron

    Rotations in Bloch Sphere about an arbitrary axis

    Hey, (I have already asked the question at http://physics.stackexchange.com/questions/244586/bloch-sphere-interpretation-of-rotations, I am not sure this forum's etiquette allows that!) I am trying to understand the following statement. "Suppose a single qubit has a state represented by the...
  21. F

    Can Any Traceless Self-Adjoint 2x2 Matrix Be Expressed Using Pauli Matrices?

    Hi everybody, a teacher of mine has told me that any complex, self adjoint matrix 2*2 which trace is zero can be written as a linear combination of the pauli matrices. I want to prove that, but I haven't been able to. Please, could somebody point me a book where it is proven, or tell me how to...
  22. M

    Eigenvalues and eigenvectors, pauli matrices

    Homework Statement Look at the matrix: A = sin t sin p s_x + sin t sin p s_y +cos t s_z where s_i are the pauli matrices a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)? b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the...
  23. HeavyMetal

    Dirac Equation and Pauli Matrices

    I have been reading through Mark Srednicki's QFT book because it seems to be well regarded here at Physics Forums. He discusses the Dirac Equation very early on, and then demonstrates that squaring the Hamiltonian will, in fact, return momentum eigenstates in the form of the momentum-energy...
  24. L

    Prove Pauli Matrices: 65-Character Title

    Homework Statement Prove \exp (\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x)=\cosh \sqrt{\alpha^2+\beta^2}+\frac{\sinh \sqrt{\alpha^2+\beta^2}}{\sqrt{\alpha^2+\beta^2}}(\alpha \hat{\sigma}_z+\beta \hat{\sigma}_x) Homework Equations e^{\hat{A}}=\hat{1}+\hat{A}+\frac{\hat{A}^2}{2!}+...
  25. D

    Pauli matrices and the Levi-Civita tensor : commutation relations

    Homework Statement Whats up guys! I've got this question typed up in Word cos I reckon its faster: http://imageshack.com/a/img5/2286/br30.jpg Homework Equations I don't know of any The Attempt at a Solution I don't know where to start! can u guys help me out please? Thanks!
  26. S

    Calculating the Determinant of a 2x2 Matrix with Pauli Matrices

    1. Consider the 2x2 matrix \sigma^{\mu}=(1,\sigma_{i}) where \sigma^{\mu}=(1,\sigma) where 1 is the identity matrix and \sigma_{i} the pauli matrices. Show with a direct calcuation that detX=x^{\mu}x_{\mu} 3. I'm not sure how to attempt this at all...
  27. S

    Writing a random 2N by 2N matrix in terms of Pauli Matrices

    Hi, Wasn't sure if I should post this to Linear Algebra or here. My question is really simple: Can a 2N by 2N random, and Hermitian Matrix ( Hamiltonian ) be always written as: H = A \otimes I_{2\times 2} + B \otimes \sigma_x + C \otimes \sigma_y + D \otimes \sigma_z where A,B,C,D are all...
  28. U

    Commuting Pauli Matrices: A Tricky Homework Challenge

    Homework Statement Express the product where σy and σz are the other two Pauli matrices defined above. Homework Equations The Attempt at a Solution I'm not sure if this is a trick question, because right away both exponentials combine to give 1, where the result is...
  29. C

    Deriving the pauli matrices from general su(2) matrix

    Hi, I think I need a sanity check, because I've been working on this for a while and I can't see what I'm doing wrong! According to several authors, including Sakurai (Modern QM eq 3.3.21), a general way to write an operator from SU(2) is...
  30. shounakbhatta

    Exploring Properties of Pauli Matrices

    Hello, I am new to this: Taking the first Pauli Matrix: σ1=[0 1 1 0] Doing the transpose it becomes: [0 1 1 0] So is it a unitary matrix? Similarly σ2= [0 -i i 0] Doing a transpose =[0 i [-i 0] Does it mean the complex conjugates are...
  31. H

    Find the inner product of the Pauli matrices and the momentum operator?

    Homework Statement Show that the inner product of the Pauli matrices, σ, and the momentum operator, \vec{p}, is given by: σ \cdot \vec{p} = \frac{1}{r^{2}} (σ \cdot \vec{r} )(\frac{\hbar}{i} r \frac{\partial}{\partial r} + iσ \cdot \vec{L}), where \vec{L} is the angular momentum operator and...
  32. A

    Pauli Matrices: Calculating Expression

    Hey guys, I was wondering how to get the expression for pauli matrices. I know that for one electron: S_i = \frac{\hbar}{2} \sigma_i But I also know that you can get to the above expression by explicitly calculating the matrix elements of the Sz, Sx and Sy operators (in the basis generated...
  33. M

    Pauli matrices forming a basis for 2x2 operators

    Hi, We know that the Pauli matrices along with the identity form a basis of 2x2 matrices. Any 2x2 matrix can be expressed as a linear combination of these four matrices. I know of one proof where I take a_{0}\sigma_{0}+a_{1}\sigma_{1}+a_{2}\sigma_{2}+a_{3}\sigma_{3}=0 Here, \sigma_{0} is...
  34. 0

    Rotation group representation and pauli matrices

    Kindly ignore if some +- signs are placed wrongly in the equations. Thank you. Rotation in three dimensions can be represented using pauli matrices \sigma^{i}, by writing coordinates as X= x_{i}\sigma^{i}, and applying the transform X'= AXA^{-1}. Here A= I + n_{i}\sigma^{i}d\theta/2. The pauli...
  35. 1

    Propagator with Pauli matrices

    Homework Statement Consider a 1/2-spin particle. Its time evolution is ruled by operator U(t)=e^{-i\Omega t} with \Omega=A({\vec{\sigma}}\cdot {\vec{L}})^{2}. A is a constant. If the state at t=0 is described by quantum number of {\vec{L}}^2, L_{z} and S_{z}, l=0, m=0 and s_{z}={1/2}...
  36. M

    Proving the Pauli Matrices Solution for a Given Relation | Help Needed

    Hi, Given the two relations below, is it true and if yes, can anyone help me show that the solution to this must be the Pauli matrices? The alphas are matrices here. \alpha_{i}\alpha_{j}+\alpha_{j}\alpha_{i} = 2\delta_{ij}*1. 1 is the identity matrix \alpha_{i}^{2} = 1 Thank you
  37. E

    Four vector made of Pauli matrices commute?

    Hey guys There are those vectors made of Pauli matrices like \bar{\sigma}^\mu and {\sigma}^\mu. So if I have the product \bar{\sigma}^\mu {\sigma}^\nu I wonder if it is commutative? And if not, what is the commutator? Cheers, earth2
  38. G

    Trouble understanding spin-1 Pauli matrices

    If we consider the spin-1/2 pauli matrices it makes sense that [S_x,S^2] = [S_y,S^2] = [S_z,S^2] = 0 since S^2 = I... and this is supposed to be true in general, right? Well, if I attempt to commute the spin-1 pauli matrices given on http://en.wikipedia.org/wiki/Pauli_matrices, with...
  39. J

    Pauli matrices with two spacetime indices

    "Pauli matrices with two spacetime indices" Hi all. This is my first post so forgive me if my latex doesn't show up correctly. I am familiar with defining a zeroth Pauli matrix as the 2x2 identity matrix to construct a four-vector of 2x2 matrices, $\sigma^\mu$. I'm trying to read a paper...
  40. O

    Gamma matrices out of pauli matrices - symmetry/group theory

    I'm reading an article (http://prb.aps.org/abstract/PRB/v82/i4/e045122) but I have some problems understanding certain definitions. The authors have introduced a basis of certain bands (four) and then continue to give the transformation matrices of the symmetry operators. One (rotation) of them...
  41. E

    The average of the three Pauli Matrices

    Homework Statement By using the general density matrix rho find the average of the three Pauli matrices. You can then tell how many independent experiments you must make in order to determine rho. Homework Equations The Attempt at a Solution I know the Pauli matrices and their...
  42. J

    What is the significance of tau and sigma matrices in quantum theory?

    In Zee's quantum theory text, introducing the Dirac equation, he states the gamma matrices as direct products of Pauli matrices. The statements involve the identity matrix, sigma matrices, and tau matrices. It took me a bit to realize that the latter were identical. I hadn't seen the tau...
  43. V

    Is the given Pauli matrix in SU(2)?

    Can I check with someone - is the following pauli matrix in SU(2): 0 -i i 0 Matrices in SU(2) take this form, I think: a b -b* a* (where * represents complex conjugation) It seems to me that the matrix at the top isn't in SU(2) - if b=-i, (-b*) should be -i...
  44. O

    Pauli Matrices as generators of SU(2)

    Why is it that the Pauli spin matrices ( the operators of quantum spin in x,y,z) are the generators of a representation of SU(2)? I understand that we use the 2X2 representation as it is the simplest, but why is it that spin obeys this SU(2) symmetry and how is it that we come up with the Pauli...
  45. D

    Finding the Trace of a Product of 4 SL(2,C) Matrices | Helpful Guide

    Dear All I'd be very grateful if someone could help me out with finding the trace of a product of 4 SL(2,C) matrices, namely: \mathrm{Tr} \left[ \sigma^{\alpha} \sigma^{\beta} \sigma^{\gamma} \sigma^{\delta} \right] where: \sigma^{\alpha} = (\sigma^0, \sigma^1, \sigma^2, \sigma^3)...
  46. E

    Solving Tricky Pauli Matrices with Einstein Notation

    Hello, I am trying to recover the following calculation (where K,A are 4x4 matrices in SL(2,C)): --(start)-- "We expand K'=AKA^{\dagger} in terms of k^a and k'^{a}=(\delta_a^{b} + \lambda_a^{b} d\tau)k^b. Multiplying by a general Pauli matrix and using the relation...
  47. K

    Dot product of two pauli matrices

    In some text, I read something like this \vec{S}_i\cdot\vec{S}_j where \vec{S}_i and \vec{S}_j are "vectors" with each components be the pauli matrices S_x, S_y, S_z individularly. My question is: if all components of this kind of vector are a 3x3 matrix, so how do you carry out the dot...
  48. O

    Tensor Product of Pauli Matrices

    Homework Statement Suppose that [\sigma_a]_{ij} and [\eta_a]_{xy} are Pauli matrices in two different two dimensional spaces. In the four dimensional tensor product space, define the basis: |1\rangle=|i=1\rangle|x=1\rangle |2\rangle=|i=1\rangle|x=2\rangle |3\rangle=|i=2\rangle|x=1\rangle...
  49. R

    Understanding Spin Matrix: Pauli Matrices and 6x6 Matrices

    Hi everyone, I now able to understand spin matrix (if i am correct in other words Pauli matrix). For e.g., for S=5/2 systems the spin matrix (say for SX) is given by: Sx= 1/2[a 6X6 matrix] I hope members will know what is this 6X6 matrix! Since i don't know how to type matrix in this...