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
Find 2 more orthonormal polynomials on the interval [-2,1] up to degree 2 given that the first polynomial p(x) = 1/√3. ( Note: Take the highest coefficient to be positive and enter your answer as a decimal.)
Homework Equations
This is a web assign equation so the answer...
Homework Statement
Divide
A(x)=
[x3+2x2+3 -4x3-x2-5]
[3x2-2 x3-2x2+x+4]
by
B(x) =
[x+4 -3]
[-x+6 x+2]
on both the right side and the left side.
Homework Equations
The Attempt at a Solution
I am...
This is not a homework, only something embarrasing..
[T_8, T_4 + i T_5] = (3^(1/2) / 2) T_4 + i T_5
from http://phys.columbia.edu/~cyr/notes/QFT_3/lecture3.pdf"
I can't see how to get the structure constant (3^(1/2) / 2).
T_4 + i T_5 is a 3x3 matrix with a one at (2,3), the rest zeroes. I...
Given matrices in a vectorspace, how do you go about determining if they are independent or not?
Since elements in a given vectorspace (like matrices) are vector elements of the space, I think we'd solve this the same way as we've solved for vectors in R1 -- c1u1 + c2u2 + c3u3 = 0. But I'm...
Homework Statement
Given that \gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}*1 where 1 is the identity matrix and the \gamma are the gamma matrices from the Dirac equation, prove that:
\gamma_{\mu}\gamma_{\nu}+\gamma_{\nu}\gamma_{\mu}=2g_{\mu\nu}*1 Homework Equations...
Hey all,
I was hoping someone could explain to me how to calculate the angle between matrices, ie. two square matrices
[ 2 0
0 -1]
and
[0 1
1 3^(1/2)]
under the inner product <A|B> = trace (A^TB)
Also, how would you go about determining an angle between...
Hi,
I already posted this in solid state physics forum, but no one answered, so I guess this topic might belong to Mathematics.
I read a text about crystallography where matrices were designated in the form:
(S2 O S1)
where S1 is input coordinate system, S2 is output coordinate...
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...
Homework Statement
Is it possible to solve this system of equations using matrices?
x^2 + y^2 = 42
x+3y+2y^2=6
Homework Equations
The Attempt at a Solution
I solved the system of equations using the following MATLAB code. I'm kind of confused by the results. Are there two x values...
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}...
Can someone direct me to a good deep exposition of Jacobians and Hessians? I am especially looking for stuff that pertains to their being generalizations of derivatives of vector and scalar functions as well as div, grad, curl. Book sources or web links are appreciated.
I start to learn about matrices and their algebra, but I am wondering what physical application they have. I know that matrices have application in optics, which is called “Matrix Optics”, but do they have other applications? Can you give different and real physical examples with matrix algebra...
Hi :smile:
I don't really know how to post a matrix here so I will try and make it as clear as possible.
Matrix A: 4 -1
2 3
Matrix B: 6 4
-5 -3
Matrix C: 1 2
3 4
We are given that APB = C and we are asked for P
What I did was I...
Homework Statement
Prove the following theorem by induction:
Let P be the transition matrix of a Markov chain. The ijth entry p(n)ij of the matrix Pn gives the probability that the Markov chain, starting in state si, will be in state sj after n steps.
Homework Equations
p(2)ij =...
Homework Statement
Find all 2x2 square matrices A which are their own inverses.
Homework Equations
A2=I
A=A-1
The Attempt at a Solution
I know that the diagonal is comprised of 1s and or -1s and the other entries are zero but I can't seem to show it algebraically.
I went the...
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
Homework Statement
From Principles of Quantum Mechanics, 2nd edition by R Shankar, problem
1.8.10:
By considering the commutator, show that the following Hermitian matrices may be
simultaneously diagonalized. Find the eigenvectors common to both and verify
that under a unitary transformation...
Hi,
I read a text about crystallography where matrices were designated in the form:
(S2 O S1)
Where S1 is input coordinate system, S2 is output coordinate system and O is the operator
corresponding to the matrix. I found this designation is often more useful than the
usual...
Homework Statement
What is the result of operating on the state |+> with the operator Sx?
here, |+> denotes the eigenstate of Sz with eigenvalue 1/2. I am working in units where h-bar is 1 (for simplicity, and because I don't know how to type it)
Homework Equations
S_i = \frac{1}{2} σ_i The...
Homework Statement
I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix.
For example, studying the lattice gas model produces the transfer...
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
Let A = [a b; c d] a 2x2 matrix with complex entries. Suppose that A is row-reduced and also that a+b+c+d =0 . Prove that there are exactly three such matrices...
so i realize that there are seven possible 2x2 matrices that are row-reduced.
[1 0; 0 1], [0 1; 1 0], [0 0; 1 0], [0 0;0 1]...
If v is in Rn and is an eigenvector of matrix A, and P is an invertible matrix, how would you go about finding an eigenvector w of PAP-1?
I'm thinking you have to use a fact about similarity?
I want to see if the matrix w = (1,0;0,1) is a linear combination of the matrices
v1 = (1,2;-2,1) and v2 = (3,2;-1,1) where ; denotes a new line in the matrix.
I know for example if w and v were 1xn matrices i.e vectors such as w = [1,1,1]
v1= [2,-1,3] v2=[1,1,2] then i setup a matrix with...
I probably can remember the matrices by just trying to, but I hate having to "remember" things without actually understanding them.
Is there no intuition behind these matrices so that I can remember it (the intuition) and then from it produce the wanted matrix?
To me the matrices look like...
Hi all,
Please help me to solve my problem in Mathematica that involves simultaneous equations with large matrices. The code is provided in the attached file, Exercise1.nb
the errors say like this " General::unfl: Underflow occurred in computation. >>
General::unfl: Underflow occurred...
Basically there are 2 equations ;
x+2y+3z = 1 2x+4y+6z=2
I put them into a matrix and row reduce to get
1 2 3 | 1
0 0 0 | 0
so we can say x = 1 - 2y -3z and let y and z = 0 to get a solution is (1,0,0)
Now i need to find the nullspace to find the whole solution set;
so x +...
I also posted this in the homework help for introductory physics, but it wasn't getting any responses, so I guess it's slightly more advanced.
Homework Statement
Let L_b(a) denote the 4x4 matrix that gives a pure boost in the direction that makes an angle a with the x-axis in the xy plane...
Homework Statement
Let L_b(a) denote the 4x4 matrix that gives a pure boost in the direction that makes an angle a with the x-axis in the xy plane. Explain why this can be found as L_b(a) = L_r(-a)*L_b(0)*L_r(a), where L_r(a) denotes the matrix that rotates the xy plane through the angle a and...
Homework Statement
If:
A = l 1 -3 l , B = l -1 2 l C = l -12 -11 l
l 2 1 l l 3 1 l l -10 -1 l
Then, find X if AXB=C
Homework Equations
The Attempt at a Solution
I don't know how I would start this problem??
Homework Statement
If
l -1 2 l l x l = l 2 l
l 1 -2 l l y l l 1 l
Show that the two lines are parallel and so never cross.
Homework Equations
The Attempt at a Solution
I have attempted it, and so far all i have done is find the determentant. When i do this, i get...
Homework Statement
Phil bought 4 meat pizzas, 4 veg pizzas and one loaf of garlic bread, and it costs him $92. By Abyy, who is partial to garlic bread, bough 2 meat pizzas, 2 veg pizzas and 10 loaves of garlic bread, and it cost $84.
a) Use matrices to find Pmv, the most of 1 meat pizza...
Homework Statement
Use Matrices to solve for x and y if:
2y - 4x - 5 = 0 and y = 3x + 1
Homework Equations
The Attempt at a Solution
I have done it, but i get a determinant of zero. So is this right?
My working is the following:
i rewrote both equations in the form of...
Matrix HELP!
Ok this question is in a cd quiz from a textbook, i have no idea what the question even means or how to do it. Please help!
Let M =
7 −3
15 −7 .
Find all 2 × 2 matrices
A =
[ a b
b d ]
such that AM = −MA.
(Note that the solutions wanted should have equal...
Homework Statement
A = {{2,1,0}{0,-2,1}{0,0,1}} and B = {{3,2,-5}{1,2,-1}{2,2,-4}}
Show that A and B are similar by showing that they are similar to the same diagonal matrix. Then find an invertible matrix P such that P-1AP=B
Homework Equations
P-1AP=B, or AP=PB
The Attempt at a...
I am trying to put a lot of data into a matrix to call upon later in plotting. From a masterfile, I need to call on the (:,1) values on x and y, and the (:,2) values in x and y. It is easy to construct a 12x3 matrix to contain all of this information, however when I am plotting I want to call...
Hi all,
I have difficulties about applying the Do-loop command as it takes very long time to run (more than 24 hours and it keeps running).
However, if i do it manually, without Do command, i.e. putting the values of the variables, Mathematica gives me a pretty quick output.
Please see...
A Hermitian matrix is a square matrix that is equal to it's conjugate transpose.
Now let's say I have a Hermitian operator and a function f:
[ H.f ]
The stuff in the square is the complex conjugate as the functions are in general complex. If I do not consider the matrix representation of...
Homework Statement
Matrices A and B are simultaneously unitarily diagonalizeable. Prove that they commute.
Homework Equations
As A and B are simultaneously unitarily diagonalizeable, there exists a unitary matrix P such that
P^{-1}AP = D_{1} and P^{-1}BP = D_{2}, where D_{1} and D_{2}...
Let A and B be nxn matrices which generate a group under matrix multiplication. Assume A and B are not orthogonal. How can I determine an nxn matrix X such that X-1AX and X-1BX are both orthogonal matrices? Is it possible to do this without any special knowledge of the group in question?
Hi
I've just read the statement that a matrix that commutes with all four gamma matrices / Dirac matrices has to be a multiple of the identity. I don't see that; can anyone tell me why this is true?
Thanks in advance
Typically I understand that projection operators are defined as
P_-=\frac{1}{2}(1-\gamma^5)
P_+=\frac{1}{2}(1+\gamma^5)
where typically also the fifth gamma matrices are defined as
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
and.. as we choose different representations the projection...
Typically I understand that projection operators are defined as
P_-=\frac{1}{2}(1-\gamma^5)
P_+=\frac{1}{2}(1+\gamma^5)
where typically also the fifth gamma matrices are defined as
\gamma^5=i\gamma^0\gamma^1\gamma^2\gamma^3
and.. as we choose different representations the projection...
I have a problem regarding concatination of multiple matrices in MATLAB. For finite number of matrices there exists a command called cat or we may even put the matrices directly in a matrix representation format to get the desired concatenation.
Like, for A and B to form a matrix C in...