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.
I am currently reading Dirac Equation from Peskin-Schroeder. In a particular para it says,
"Now let us find Dirac Matrices \gamma^\mu for four-dimensional Minkowski Space. It turns out that these matrices must be at least 4X4."
What is the proof of the above statement? I think (not sure)...
1) show that if AB = AC and A is nonsingular, then B = C.
2) show that if A is nonsingular and AB = 0 for an n x n matrix B, then B = 0.
3) Consider the homogenous system Ax=0, where A is n x n. If A is nonsingular, show that the only solution is the trivial one, x=0.
4) Prove that if A...
I have a question regarding the slide:
http://theory.physics.helsinki.fi/~kvanttilaskenta/Lecture3.pdf
On page 18-21 it gives the proof of the theorem that | \psi_i^{~} \rangle and |\phi_{i}^{~}\rangle generate the same density matrix iff |\psi_{i}^{~}\rangle = \sum_{j} u_{ij}...
Say I was given a 2x2 matrix made from a certain basis {|x\rangle, |y\rangle} , and I split that matrix into two parts, one being the diagonal part and one being the off-diagonal part.
for example, if I had H = H_0 + W = \left(\begin{array}{cc}a&c\\b&d\end{array}\right) =...
Simultaneous "eigenspace" of non-commuting matrices
Hello!
I have been working on the following "brain teaser" the whole day long without any success. I am not even sure there is a "clean" solution. I would love to hear your opinion. Before presenting the whole problem, here is an easy...
1) Is the matrix [upper row 3 0 and lower row 0 2] a linear combination of the matrices [upper row 1 0 and lower row 0 1] and [upper row 1 0 and lower row 0 0]? Justify your answer.
Is it I just have to add the two matrices to see if they are equal the matrix, [upper row 3 0 and lower row 0...
Not a Homework problem, but I think it belongs here.
Homework Statement
Consider four dirac matrices that obey
M_i M_j + M_j M_i = 2 \delta_{ij} I
knowing the property that Tr ABC = Tr CAB = Tr BCA show that the matrices are traceless. Homework Equations
Tr MN = Tr NM
The Attempt...
[b]1. Homework Statement [/
from the ets general physics practice test (ill take it in april) the state of spin 1/2 particles
using the eigenstates up and down Sz up= 1/2 hbar Sz down= -1/2 hbar
Homework Equations
given sigmax (pauli spin matrix) which of the following list...
In most of the physics textbooks I read they only give one or two representations of gamma matrices, but none gives a proof, so how can I prove it from the Clifford algebra?
Homework Statement
Let A, B, C, D be nxn complex matrices such that AB and CD are Hermitian, i.e., (AB)*=AB and (CD)*=CD.
Show that AD-B*C*=I implies that DA-BC=I
The symbol * indicates the conjugate transpose of a matrix, i.e., M* is the conjugate transpose of M.
I refers to the identity...
Can a symmetric matrix contain complex elements(terms).
If no, how is it that 'eigen values of a symmetric matrix are always real'(from a theorem)
Is a symmetric matrix containing complex terms called a hermitian matrix or is there any difference?
Can we call the following matrix...
Why is it a (for example) 3x3 matrix of linear forms cannot necessarily be written as the sum of at most 3 rank one matrices of linear forms but the statement is true if "linear forms" is replaced with scalars? Does it have something to do with the 2x2 minors being calculated differently when...
Homework Statement
Hi all. I am doing this work and can't seem to find any information on this in any of my notes or textbooks. The question is, "Evaluate (if possible) AB, BA, CD and DC", this is what i need some help with.
I also have further on the question, "Evaluate | u |, | v |, u . v...
Homework Statement
If A = \[ \left( \begin{array}{ccc}
a & b \\
c & d \end{array} \right)\][\tex]
and B=\[ \left( \begin{array}{ccc}
\alpha & \beta \\
\gamma & \delta \end{array} \right)\] [\tex]
in the basis |e1>,|e2>, find
AxB (where "x" is the tensorproduct) in the basis...
The Pauli Spin matrices:
\sigma_1=\left[
\begin{array}{ c c }
0 & 1 \\
1 & 0
\end{array} \right],\sigma_2=\left[
\begin{array}{ c c }
0 & -i \\
i & 0
\end{array} \right],\sigma_3=\left[
\begin{array}{ c c }
1 & 0 \\
0 & -1
\end{array} \right]
are used...
Hello
Just took on a new hobby..being game design know a little c# and decided to try my hand at xna...the problem is I am also trying my hand at vectors and matrices also, as I know this with gaming goes hand in hand.
Kinda picking up the basics to vectors but with matrices not got a great...
why is S^n/S^m homotopic to S^n-m-1. the book just made this remark how do you see this geometrically.
how do you compute fundamental groups of matrices like O(3) and SO(3) or SL(2) and whatnot.
Homework Statement
exp^\prime(0)B=B for all n by n matrices B.
Homework Equations
exp(A)= \sum_{k=0}^\infty A^k/k!
The Attempt at a Solution
Obviously I want to calculate the limit of some series, but I don't know what series to calculate. I wanted to try \lim_{h \to...
Does anybody know a good thread, homepage or book that takes up different interpretations of Pauli and Dirac matrices with the connection to for example quaternions or bivectors?
Maybe someone could comment on this?
I need help about similarity transformation in matrices.
Is there anyone who knows how can I decide whether "the two matrices having the same eigenvalues" are similar or not without using eigenvectors?
For example, following two matrices have the same characteristic polynomial. But they are...
Homework Statement
[PLAIN]http://img530.imageshack.us/img530/6672/linn.jpg
The Attempt at a Solution
For parts (a) and (b) I've found the eigenvalues to be -\frac{1}{3} and -1 with corresponding eigenvectors \begin{bmatrix} -1 \\ 3 \end{bmatrix} and \begin{bmatrix} -1 \\ 1...
Hello. I am writing an encryption algorithm for a program and have decided to use a hill cipher. My problem is that for the hill cipher, I have to have matrices with very specific properties. How might I go about finding 3 matrices of size 4x4 such that all of the matrices are integer-only and...
[input_pattern, input_class]=loading(file_name,no_sample,no_vector);
input_pattern and input_class are 60 X 60 matrices, how can I display these matrices ?
Thank you.
Hello, I attached a copy of the problem and my attempted solution. The three Pauli spin matrices are given above the problem. I'm having trouble getting the right side to equal the left side, so I'm assuming I'm doing something wrong. When I got towards the end it just wasn't looking right...
So since I learning QFT a while ago, I've always struggled to understand fermions. I can do computations, but I feel at some level, something fundamental is missing in my understanding. The spinors encountered in QFT develop a lot from "objects that transform under the fundamental representation...
hi
how to calculate the traces of product of Dirac matrices in QED.
i want caculate crossection of process scattering in QED. a program to calculate it
I was surprised that I have never had to do this in so long and forgot the basic way to factor out a scalar multiple when a matrix is raised to a certain power (for example -1 for inverse matrices).
Basically, I just want some confirmation:
(λT)^n= λ^n (T^n ) ∶ for λ ϵ F and Tϵ L(V)...
Homework Statement
I'm supposed to write a proof for the fact that det(A)=det(B) if A and B are similar matrices.
Homework Equations
Similar matrices have an invertible matrix P which satisfies the following formula:
A=PBP^{-1}
det(AB) = det(A)det(B)
The Attempt at a Solution...
I try to understand how to calculate derivatives of functions, which contain matrices.
For a start I am looking at derivatives by a single variable.
I have x=f(t) and I want to calculate \frac{dx}{dt}. The caveat is that f contains matrices, that depend on t. Can I use the ordinary chain rule...
Homework Statement
Show that A = 3 4 3
-1 0 -1
1 2 3
is not diagonalizable but is triangulable and carry out triangulation (A has rational entries)
I found that the only eigenvalue is 2, and that the characteristic equation is (x-2)3, but I'm...
Homework Statement
Show that if A is nilpotent then det(I-A)=det(I+A)=1.
Homework Equations
I know that det(A)=0 if A is nilpotent and det(I)=1, so this seems like it follows logically. I also know that the tr(A)=0 and that tr(I-A)=tr(I+A)=n, and that the characteristic polynomial of...
Homework Statement
The actual problem is
Find a basis for the null-space of the matrix
(1 0 1 2 1)
(0 1 2 0 1)
(0 1 -1 3 1)
Homework Equations
there are no relevant equations.
The Attempt at a Solution
I attempted to get the matrix into RREF i got
(1 0 3 0 1)...
Hi,
I have a question about linear transformation. So given a matrix A in the basis u (denoted as A_u). Now in another basis that I don't know, A_u becomes A_v.
How can I find v? (I know u, A_u and A_v).
Thank you very much,
Homework Statement
A biconconvex (n_l=1.5) lens have radii worth 20 and 10 cm and an axial width of 5 cm. Describe the image of an object whose height is 2.5 cm and situated at 8 cm from the first vertex.Homework Equations
Transfer matrices.
The Attempt at a Solution
I used the ray transfer...
I have stumbled upon a problem which I have so far been unable to solve.
I we consider a general set of linear equations:
Ax=b,
I know the the system is inconsistent which makes least square method the logical choice.
So the mission is to minimize ||Ax-b||
And the usual way I do...
Homework Statement
Let V=Rⁿ and let AεMnxn(R) Prove that <x,Ay> = <A^T x, y> for all x,yεV
Homework Equations
The Attempt at a Solution
Can someone tell me if I'm on the right track?
<x,Ay> = x^T Ay = (x^TA)y = <A^T x,Y>
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...
Dear All,
I have inherited a few rotation matrices through some old computer code I am updating. The code is used to construct some geometry.
The matrices I have inherited are left handed rotation matrices and they are being applied to a right handed coordinate system, but they give the...
A,B are nxn. If AB is invertible. Show that A and B are invertible.
I know how to prove it by determinant, using linear transformations and contradictions.
I am looking for a direct way using a proof by matrices. Can anyone think of one?
Thank you.
Show the following:
If A and B are n x n matrices such that A - B is singular then A2 - AB + BA - B2 is also singular.
I really have no clue how to solve this, but I am guessing that AB does not equal BA, I don't know how that can help or be relevant but just in case
Thanks alot...
"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...
Homework Statement
By taking the trace of both sides prove that there are no finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p,x] = -ih/2pi
Why does this argument fail if the matrices are infinite dimensional?
Homework...
Homework Statement
Ok so
I have to construct a real symmetric matrix R whose eigenvalues are 2,1,-2 and who corresponding normalized eigenvectors are bla bla bla..
So let the matrix of eigenvalues down diagonal be E and matrix of eigen vectors be V
Is R = VEV^T or R = V^TEV??
How...
If I have a matrix for which all eigenvalues are zero, what can be said about its properties?
If I multiply two such matrices, will the product also have all zero eigenvalues?
Thanks
Hi, I have a bunch of images saved as .fig and I need to obtain the data in these files. I tried something as simple as
image1 = imread('figure1.fig');
but it does not recognize the format. If anyone could help me out that would be greatly appreciated, thanks!