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
s
u
(
2
)
{\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.
Homework Statement
Show that all ##n \times n## unitary matrices ##U## leave invariant the quadratic form ##|x_{1}|^{2} + |x_{2}|^{2} + \cdots + |x_{n}|^{2}##, that is, that if ##x'=Ux##, then ##|x'|^{2}=|x|^{2}##.
Homework Equations
The Attempt at a Solution
##|x'|^{2} = (x')^{\dagger}(x')...
Homework Statement
Show that the set of all ##n \times n## orthogonal matrices forms a group.
Homework Equations
The Attempt at a Solution
For two orthogonal matrices ##O_{1}## and ##O_{2}##, ##x'^{2} = x'^{T}x' = (O_{1}O_{2}x)^{T}(O_{1}O_{2}x) = x^{T}O_{2}^{T}O_{1}^{T}O_{1}O_{2}x =...
Trace of six gamma matrices
I need to calculate this expression:
$$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$
I know that I can express this as:
$$...
Homework Statement
Show that all ##n \times n## (real) orthogonal matrices ##O## leave invariant the quadratic form ##x_{1}^{2} + x_{2}^{2}+ \cdots + x_{n}^{2}##, that is, that if ##x'=Ox##, then ##x'^{2}=x^{2}##.
Homework Equations
The Attempt at a Solution
##x'^{2} = (x')^{T}(x') =...
Homework Statement
An error matrix is in the form, has a characteristic equation:
## CE: s^2 + 120s + 7200 = 0 ##
A state variable feedback system is described by:
## A_F = \begin{bmatrix}0 & 1 \\-616.8 & -40 \end{bmatrix} ##
## B = \begin{bmatrix}0 \\ 1 \end{bmatrix} ##
## C =...
1)I use Linux Mint 17.2 and wxMaxima 13.04.2. In wxMaxima 13.04.2, the code below, It plays correctly:
plot2d([x,x^3,[discrete,[[0,0],[1,1],[-1,-1]]]],[x,-5,5],[y,-5,5],
[style,[lines,2,1],[lines,2,4],[points,3,2]],[point_type,bullet],
[legend,"x","x^2",""],[xlabel, "x"], [ylabel...
Homework Statement
Show that no matrix A ∈ M3 (ℝ) exists so that A2 = -I3
Homework EquationsThe Attempt at a Solution
This is from a french textbook of first year linear algebra. I'm quite familiar with properties of matrices but I don't have any idea of how to prove this.
Thanks for the help!
Homework Statement
A thin lens is placed 2m after the beam waist. The lens has f = 200mm. Find the appropriate system matrix.
This is a past exam question I want to check I got right.
Homework Equations
For some straight section [[1 , d],[0 , 1]] and for a thin lens [[1 , 0],[-1/f , 1]]...
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...
Homework Statement
Hi!
I have the 3x3 matrix for L below, which I calculated. But now I need to figure out how the equation below actually means! Is it just the inverse of L (L^-1)? I cannot proceed if I don't know this step.
Homework Equations
See image
The Attempt at a Solution
I put in...
Light reflecting off a mirror actually penetrates a short distance into the
mirror surface material. In metals, this distance is very short (much less
than a wavelength) and so can be neglected. But metals tend to also absorb
~10% of the light, which is undesirable. Today’s modern multilayer...
So that's the question in the text.
I having some issues I think with actually just comprehending what the question is asking me for.
The texts answer is: all 3x3 matrices.
My answer and reasoning is:
the basis of the subspace of all rank 1 matrices is made up of the basis elements...
Whenever a problem seems too easy, I assume I'm missing something :-)
This is in a section on Legendre polynomials ...
Given the series $ \alpha_0 + \alpha_2Cos^2\theta +\alpha_4C^4 +\alpha_6C^6 = a_0P_0 + a_2P_2 + a_4P_4 +a_6P_6 $ (abbreviating $Cos^n\theta$ to $C^n$)
Express both...
I thought I had this clear, then I met operators and - at least to me - the new information overlapped with, and potentially changed, that understanding. Research on the web didn't help as there seem to be different uses & opinions ...
So what I am trying to do is NOT make a summary of what...
Create a matrix with 4 rows and 8 columns with every element equal to 0. Create a second, smaller matrix with 2 rows and 4 columns where each row is [1 2 3 4]. Replace the 0s in the upper left-hand corner of the bigger matrix with the values from the smaller matrix. (If you do this correctly...
Is there any chart/graph/website online or in a ebook that has a clear concise list of special matrices used in physics?
I'm just getting into an intro to quantum mechanics class and we are going over all types of matrices, Identity, hermitian, diagonal, transpose, unitary, and so on.
I want...
Homework Statement
This is a homework problem for my Honors Calculus I class. The problem I'm having is that though I can solve a traditional function composition problem, I'm stumped as to how to do this for multivariate functions. I read that it requires an extension of the notion of...
The exercise is: (b) describe all the subspaces of D, the space of all 2x2 diagonal matrices.
I just would have said I and Z initially, since you can't do much more to simplify a diagonal matrix.
The answer given is here, relevant answer is (b):
Imgur link: http://i.imgur.com/DKwt8cN.png...
I'm taking a Differential Equations class and we're dealing with matrices and determinants. I've dealt with them before but I was always annoyed by the fact that I don't know what the heck is going on. So I know that matrices are a way to organise linear equations and make transformations...
Homework Statement
Find an example of two unitary matrices that when summed together are not unitary.
Homework EquationsThe Attempt at a Solution
A = \begin{pmatrix}
0 & -i\\
i & 0\\
\end{pmatrix}
B = \begin{pmatrix}
0 & 1\\
1 & 0\\
\end{pmatrix}
A+B =
A = \begin{pmatrix}
0 & 1-i\\
1+i &...
Homework Statement
Find all 2x2 Matrices which are both hermitian and unitary.
Homework Equations
Conditions for Matrix A:
A=A^†
A^†A=I
I = the identity matrix
† = hermitian conjugateThe Attempt at a Solution
1. We see by the conditions that A^† = A and by the second condition, we see that...
Homework Statement
Let K and L be symmetric PSD matrices of size N*N, with all entries in [0,1]. Let i be any number in 1...N and K’, L’ be two new symmetric PSD matrices, each with only row i and column i different from K and L. I would like to obtain an upper bound of the equation below...
Homework Statement
Show that the product of two nxn unitary matrices is unitary. Is the same true of the sum of two nxn unitary matrices?
Homework Equations
Unitary if A†A=I
Where † = hermitian conjugate
I = identity matrix.
The Attempt at a Solution
[/B]
We have the condition: (AB)†(AB)=I
I...
Homework Statement
Show that the sum of two nxn Hermitian matrices is Hermitian.Homework Equations
Hermitian conjugate means that you take the complex conjugate of the elements and transpose the matrix. I will denote it with a †.
I will denote the complex conjugate with a *.
The Attempt at a...
Homework Statement
Find all diagonal unitary matrices.
Homework Equations
The Attempt at a Solution
I think I am starting to get the hang of this type of material.
I hope I am right in my thinking.
So if we have a diagonal matrix, let's say a 2x2 for a simple example:
\begin{pmatrix}
a &...
Homework Statement
Show that |A_ij| ≤ 1 for every entry A_ij of a Unitary Matrix A.
Homework Equations
A matrix is unitary when A^†*A=I
Where † is the hermitian operator, meaning you Transpose and take the complex conjugate
and I = the identity matrix
The Attempt at a Solution
I'm having a...
Homework Statement
Show that (A+B)*=A*+B*
Homework Equations
I think I am missing a property to prove this.
The Attempt at a Solution
This should be easier then I am making it out to be. But I seem to be missing one key property to do this.
A*+B* is just A(ij)*+B(ij)* = Right hand side...
Homework Statement
Let ##G=GL_n(F)## for ##F## a field, and define an equivalance relation by ##A\sim B## iff ##A## and ##B## are conjugate, that is, iff ##A=PBP^{-1}## for some ##P\in GL_n(F)##. Does ##\sim## respect multiplication?Homework Equations
The equivalency respects multiplication...
Good afternoon all,
I'm taking a linear algebra course this semester, and upon entering the topic of 'Applications of Matrix Operations', my professor has given our class the opportunity to earn some extra credit points by writing a paragraph or two on the application of stochastic matrices in...
A very small country town has a population that can be grouped according to three categories: adults teenagers and children.
Each year statistics show that:
Children are born at the rate of 4% of the adult population 12% of children become teenagers 15% of teenagers become adults 0.5% of...
Homework Statement
Assuming I understand the problem correctly, I need to define the set of all orthogonal matrices.
Homework Equations
The Attempt at a Solution
Per the definition of orthogonal matrix: Matrix ##A\in Mat_n(\mathbb{R})## is orthogonal if ##A^tA = I##
If ##O## is the set of all...
Homework Statement
Find all 2 x 2 and 3 x 3 orthogonal matrices which are diagonal. Construct an example of a 3 x 3 orthogonal matrix which is not diagonal.
Homework Equations
Diagonal Matrix = All components are 0 except for the diagonal, for a 2x2 matrix, this would mean components a and d...
$$A$$ is a hermitian matrix with eigenvalues +1 and -1. Let $$\left|+\right>$$ and $$\left|-\right>$$ be the eigenvector of $$A$$ with respect to eigenvalue +1 and eigenvalue -1 respectively.
Therefore, $$P_{+} = \left|+\right>\left<+\right|$$ is the projection matrix with respect to eigenvalue...
Homework Statement
Let A =
\begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}
Find all 2 x 2 matrices B such that AB = BA.
Homework Equations
http://euclid.colorado.edu/~roymd/m3130/Exam2sol.pdf
The Attempt at a Solution
I let B =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix} and set AB=BA...
MIT OCW 18.06 using Intro to Linear Algebra by Strang
So I was working through some stuff about Cyclic Matrices, and the text was talking about how the column vectors that make up this cyclic matrix, shown here,
are coplanar, and that is the reason that Ax = b will have either infinite...
Homework Statement
I'm a bit at a loss - I thought the last row with '1's would be useful, but it just gave me:
(b2c - bc2) - (a2c - ac2) + (a2b - ab2)
and
bc(b - c) - ac(a - c) + ab(a - b)
But then it is a dead end. I am probably doing something stupid again ...
Any help appreciated.
Homework Statement
Let B1={([u][/1]),([u][/2]),([u][/3])}={(1,1,1),(0,2,-1),(1,0,2)} and
B2={([v][/1]),([v][/2]),([v][/3])}={(1,0,1),(1,-1,2),(0,2,1)}
a) Show that B1 is a basis for [R][/3]
b) Find the coordinates of w=(2,3,1) relative to B1
c)Given that B2 is a basis for [R[/3], find...
Hey all. I am currently reading an article and there is a paragraph that I am having a hard time understand. This is what the paragraph says:
"Since Ar = Arτ and Ai = -Aiτ, we know that only the lower triangular (including the diagonal) elements of Ar are independent and only the strictly lower...
I have just started to study quantum mechanics, so I have some doubts.
1) if I consider the base given by the eigenstates of s_z s_z | \pm >=\pm \frac{\hbar}{2} |\pm> the spin operators are represented by the matrices
s_x= \frac {\hbar}{2} (|+><-|+|-><+|)
s_y= i \frac...
Hi all,
Firstly, I am not sure whether this is the area of the forum to ask this.
I have been learning and researching a completely different topic, and from this I have come across a completely new concept of the Kronecker function. I have done a google search on this to get the intro and...
I have two questions, but the second is only worth asking if the answer to the first is yes:
Are the spin matrices for three particles, with the same spin,
σ ⊗ I ⊗ I,
I ⊗ σ ⊗ I and
I ⊗ I ⊗ σ
for particles 1, 2 and 3 respectively, where σ is the spin matrix for a single one of the particles?
I...
Asked to determine the eigenvalues and eigenvectors common to both of these matrices of
\Omega=\begin{bmatrix}1 &0 &1 \\ 0& 0 &0 \\ 1& 0 & 1\end{bmatrix} and \Lambda=\begin{bmatrix}2 &1 &1 \\ 1& 0 &-1 \\ 1& -1 & 2\end{bmatrix}
and then to verify under a unitary transformation that both can...
Folks,
What is the idea or physical significance of simultaneous diagonalisation? I cannot think of anything other than playing a role in efficient computation algorithms?
Thanks
Homework Statement
[/B]
What are the ##n\times n## matrices over a field ##K## such that ##M^2 = 0 ## ?
Homework Equations
The Attempt at a Solution
Please can you tell me if this is correct, it looks ok to me but I have some doubts. I have reused the ideas that I found in a proof about...
Homework Statement
Show that ##n\times n ## complex matrices such that ##\forall 1\le i \le n,\quad \sum_{k\neq i} |a_{ik}| < |a_{ii}|##, are invertible
Homework EquationsThe Attempt at a Solution
If I show that the column vectors are linearly independent, then the matrix has rank ##n## and...
Hi all,
This isn't actually part of my assigned homework, I was just trying it out as the topic confuses me. I think I might understand what's going on a little more if someone could walk me through this. Any advice on the intuition behind it would be great. Thanks so much.
1. Homework...
Consider the rotation group ##SO(3)##.
I know that ##R_{x}(\phi) R_{z}(\theta) - R_{z}(\theta) R_{x} (\phi)## is a commutator?
But can this be called a commutator ##R_{z}(\delta \theta) R_{x}(\delta \phi) R_{z}^{-1}(\delta \theta) R_{x}^{-1} (\delta \phi)##?
Homework Statement
Are the following matrices hermitian, anti-hermitian or neither
a) x^2
b) x p = x (hbar/i) (d/dx)
Homework EquationsThe Attempt at a Solution
For a) I assume it is hermitian because it is just x^2 and you can just move it to get from <f|x^2 g> to <f x^2|g> but I am not...