In linear algebra, linear transformations can be represented by matrices. If
T
{\displaystyle T}
is a linear transformation mapping
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
and
x
{\displaystyle \mathbf {x} }
is a column vector with
n
{\displaystyle n}
entries, then
T
(
x
)
=
A
x
{\displaystyle T(\mathbf {x} )=A\mathbf {x} }
for some
m
×
n
{\displaystyle m\times n}
matrix
A
{\displaystyle A}
, called the transformation matrix of
T
{\displaystyle T}
. Note that
A
{\displaystyle A}
has
m
{\displaystyle m}
rows and
n
{\displaystyle n}
columns, whereas the transformation
T
{\displaystyle T}
is from
R
n
{\displaystyle \mathbb {R} ^{n}}
to
R
m
{\displaystyle \mathbb {R} ^{m}}
. There are alternative expressions of transformation matrices involving row vectors that are preferred by some authors.
I just want to make sure I'm doing this right. I know how to do the rotation, but reflection isn't demonstrated in the text. From what I'm seeing in the book, it seems like you take the matrix and simply multiply it by ##\begin{bmatrix} x \\ y \end{bmatrix} ##
Assuming that's the case, I get...
When one does the phase transformation ##\psi(\vec{x},t)=R(\vec{x},t)\exp^{iS(\vec{x},t)/\bar{h}}##
For this transformation to be valid doesn't it need to have the same asymptotic behaviour ##x \to \pm \infty##, and also at ##x=0 ## as the original wave-function ##\psi## does? How come this is...
Proof:
Consider the transformation ## x=\frac{1}{\sqrt{1+e^{-2q}}} ## and ## y=\frac{1}{\sqrt{1+e^{-2p}}} ## with the Hamiltonian function ## H(q, p)=ap-b\cdot ln(e^{p}+\sqrt{1+e^{2p}})+cq-d\cdot ln(e^{q}+\sqrt{1+e^{2q}}) ##.
Let ## \dot{x}=\frac{dx}{dt}=(a-by)x(1-x^2)=(a-by)(x-x^3) ## and ##...
So, I linked an image to Problem 1.26 below. As far as that problem, to save you the trouble, the answers (at least from what I have) are:
a) ##\gamma = 1.7 ##
b) ## t = 3x10^{-8} ##
c) ## Pions = 30,555 ##
d) ## Pions = 29,628 ##
Fairly confident those are correct, but now this question...
In p. 84, Zee says “In the new coordinates, M is replaced by M’ = R[-1]MR.” However, I figure out M is replaced by M’ = RMR[-1]. Why is M replaced by M’ = R[-1]MR?
I'm so confused here. If we make the transformation of the coordinates x -> x', are we not suppose to consider the transformation of the coordinates only
$$ \phi(x) \rightarrow \phi(x') $$ ? Then why are they writing $$ \phi(x) \rightarrow \phi'(x') $$ ? If $$ \phi(x) $$ is a scalar function...
My interest is on how they arrived at ##r^2 \sin θ##
My approach using the third line, is as follows
##\cos θ[r^2 \cos θ \sin θ \cos^2 ∅ + r^2 \sin θ \cos θ cos^2∅ ] + r\sin θ [r\sin^2 θ \cos^2∅ + r \sin^2 θ \sin^2 ∅]=##
##\cos θ[r^2 \cos θ \sin θ[\cos^2 ∅+ \sin^2 ∅]] + r\sin θ [r\sin^2 θ...
Hello, I doing my thesis, I'm beginner in MCNPX. I want to ask about transformation in MCNPX, I want to transform this head phantom in inside and outside field ( radiation using LINAC from face to ear) in angle 90° and 270 °. Can anyone help me to solve this problem with transformation code?
I was watching this video by minutephysics on the No-cloning theorem.
Henry very plainly shows why the no-cloning theorem holds, given the setup.
However, I am no quantum physicist and lack the necessary background to truly understand what's going on there.
What are the origins of the 3...
Namaste & G'day!
Imagine a helicopter view of a Polo ground. It's length & breadth are known.
Now you are seated where the blue dot is. Your view is such:
How do mathematicians calculate the distance travelled by a ball from the second perspective?
From the top view, this would be...
This is the statement in question:
But if they were scalar fields, they would not transform at all. How could they contribute differently if they didn't change?
What type of chemical and physical transformation occurs in the blue sphere silica gel when it is subjected to very high temperatures above the temperature supported by it for several minutes?
does it turn into a compound that will absorb and store moisture and water inside and can leak if it...
I am a physics enthusiast reading Covariant Physics by Moataz H. Emam. In his chapter about Point Particle mechanics there is a transformation equation for a displacement vector. I don't see how he arrived at the final equation 3.6. Is it a chain rule or product rule? Can't seem to figure it...
I have encountered a problem related to the Galilean Transformation. Let's consider two observers who will be referred to as ##O## and ##O^{'}##, with their corresponding coordinates ##(t,x,y,z)## and ##(t^{′},x^{′},y^{′},z^{'})## respectively. They are initially at the same location, at time...
I need to prove that under an infinitesimal coordinate transformation ##x^{'\mu}=x^\mu-\xi^\mu(x)##, the variation of a vector ##U^\mu(x)## is $$\delta U^\mu(x)=U^{'\mu}(x)-U^\mu(x)=\mathcal{L}_\xi U^\mu$$ where ##\mathcal{L}_\xi U^\mu## is the Lie derivative of ##U^\mu## wrt the vector...
Question:
Eq. 12.109:
My solution:
We’ll first use the configuration from figure 12.35 in the book Griffiths. Where the only difference is
that v_0 is in the z-direction. The electric field in the y-direction will be the same.
$$E_y = \frac{\sigma}{\epsilon _0}$$
Now we're going to derive the...
I started by expanding ##dx## and ##dt## using chain rule:
$$dt = \frac{dt}{dX}dX+\frac{dt}{dT}dT$$
$$dx = \frac{dx}{dX}dX+\frac{dx}{dT}dT$$
and then expressing ##ds^2## as such:
$$ds^2 =...
I am given an initial vector potential let's say:
\begin{equation}
\vec{A} = \begin{pmatrix}
g(t,x)\\
0\\
0\\
g(t,x)\\
\end{pmatrix}
\end{equation}
And I would like to know how it will transform under the Lorenz Gauge transformation. I know that the Lorenz Gauge satisfy...
So, I've been watching eigenchris's video series "Tensors for Beginners" on YouTube. I am currently on video 14. I am a complete beginner and just want some clarification on if I'm truly understanding the material.
Basically, is everything below this correct?
In summary of the derivation of the...
Hi,
Unfortunately I am not getting anywhere with task three, I don't know exactly what to show
Shall I now show that from ##S(T,V,N)## using Legendre I then get ##S(E,V,N)## and thus obtain the Sackur-Tetrode equation?
Hi,
unfortunately, I'm not that fit concerning the Lagrangian formalism, so I'm not sure if I solved the problem 1a correctly.
I have now proceeded as follows
the Lagrangian is
$$L=T-U$$
Since there are no constraining or other forces acting on the point mass, I assume that the...
T(α1), T(α2), T(α3) written in terms of β1, β2:
Tα1 =(1,−3)
Tα2 =(2,1)
Tα3 =(1,0).
Then there is row reduction:
Therefore, the matrix of T relative to the pair B, B' is
I don't understand why the row reduction takes place? Also, how do these steps relate to ## B = S^{-1}AS ##? Thank you.
This is a cyclic transformation. Is it safe to say thay it's irreversible because if you reverse it, it means I could extract an amount of heat from a cold reservoir and move it into a hotter reservoir with no other effect?
My Progress:
I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change:
$$
\frac{\partial}{\partial\omega} f =...
I'm trying to investigate the possibility of martensitic transformation in a non-iron alloy, described as a single-phase alpha-solid-solution (Nickel-Silver CuNi12Zn25Pb1, CW404J). I know that Cu-Ni-Zn alloys with higher zinc amounts show even shape memory effects. And that CuNi12Zn25Pb1 is no...
"There is a linear transformation T from R3 to R3 such that T (1, 0, 0) = (1,0,−1), T(0,1,0) = (1,0,−1) and T(0,0,1) = (1,2,2)" - why is this the case?
Thank you.
In《Introducing Einstein's Relativity Ed 2》on page 106"lowering the first index with the metric,then it is easy to establish,for example by using geodesic coordinates..."
In 《A First Course in General Relativity - 2nd Edition》on page 159 "If we lower the index a,we get(in the locally flat...
So,
##\hat{p}(\omega)=\int_{-\infty}^{\infty} p(t)e^{-i\omega t}\mathrm{d}t=A\int_{0}^{\infty}e^{-t(\gamma+i(\omega+\omega_0))}=A\left[-\frac{e^{-t(\gamma+i(\omega+\omega_0))}}{\gamma+i(\omega+\omega_0)}\right]_0^\infty,##
provided ##\gamma+ i(\omega+\omega_0)\neq 0## for the last equality. Now...
I made a tool for calculating and visualizing how the electric and magnetic fields transform under a Lorentz boost. Thought I'd share it here, in case anyone finds it interesting.
https://em-transforms.vercel.app/
Hey I have a question about the relation between Legendre transformation and Hamilton-Jacobi formalism. Is there some relation? Cause Hamilton-Jacobi is the expression of Hamiltonian with a transformation.
Hello,
I would like to reproduce the following equation, but I don't quite understand how to do the transformation:
$$ \sum_{i=1}^k \left( \frac{\langle y , x_i^* \rangle}{\sqrt{\langle x_i^*, x_i^* \rangle}} \right)^2 = \langle y, y \rangle$$
Where ##x_1^*,...,x_k^*##, are orthogonalized...
Could one derive a set of coordinate transformations that transforms events between different reference frames in the de Sitter metric using the invariant line element, similar to how the Lorentz Transformations leave the line element of the Minkowski metric invariant? Would these coordinate...
You see in the literature that the vector potentials in a gauge covariant derivative transform like:
A_\mu \rightarrow T A_\mu T^{-1} + i(\partial_\mu T) T^{-1}
Where T is not necessarily unitary. (In the case that it is ##T^{-1} = T^\dagger##)
My question is then if T is not unitary, how is...
My notes say that the Resolution of the Aperture(in the Electric field of the wave) is the Fourier transformation of the aperture.
Then gives us the equation of the aperture:
and says that for the circular aperture in particular also:
My attempt at solving this:
We know that the Fourier...
1. The 2nd line on the 3rd page of your notes, you have x=ct and
x'=ct', thus ux=dx/dt and ux'= dx'/dt' =c according to Einstein's
assumptiuon.
2. But near the end of the last page, you wrote dx'/dt' = (ux
-v)/(1-vux/c2) . Compare with 1. This equation can be valid only for ux=c
and...
The Lorentz transform for velocities is as follows:
$$u=\frac{v+w}{1+\frac{vw}{c^{2}}}$$
But which assumption exactly underlies this so that you get exactly this formula and not any other formula with approximately the same properties?
THe question is pretty simple. I was doing an exercise, in which $$p = \lambda P, Q = \lambda q$$ is a canonical transformation.
We can check it by $$\{Q,P \} = 1$$
But, if we add $$t' = \lambda ^2 t$$, the question says that the transformation is not canonical anymore.
I am a little...
I don't understand why the highlighted term is there.
This image was taken from Sean Carroll's notes available here: preposterousuniverse.com/wp-content/uploads/grnotes-three.pdf
There is no possible measurement, no matter how clever, that can measure the one way speed of light. It is a synchronization convention. In this topic I would like to apply this idea on a specific case.
I have a microwave oven with width L. In this oven I have a standing wave.
$$E(t,x)=E...
How do the energy and generalized momenta change under the following coordinate
transformation $$q= f(Q,t)$$The generalized momenta: $$P = \partial L / \partial \dot Q = \partial L / \partial \dot q\times \partial \dot q / \partial \dot Q = p \partial \dot q / \partial \dot Q = p \partial q /...
Hi,
I have to verify the sifting property of ##\frac{1}{2\pi i} \int_{-i\infty}^{i\infty} e^{-sa}e^{st} ds## which is the inverse Mellin transformation of the Dirac delta function ##f(t) = \delta(t-a) ##.
let ##s = iw## and ##ds = idw##
##\frac{1}{2\pi} \int_{-\infty}^{\infty} e^{-iwa}e^{iwt}...
While deriving Lorentz transformation equations, my professor assumes the following:
As ##\beta \rightarrow 1,##
$$-c^2t^2 + x^2 = k$$
approaches 0. That is, ##-c^2t^2 + x^2 = 0.## But the equation of the hyperbola is preserved in all inertial frames of reference. Why would ##-c^2t^2 + x^2##...
Consider a set of ##n## position operators and ##n## momentum operator such that
$$\left[q_{i},p_{j}\right]=i\delta_{ij}.$$
Lets now perform a linear symplectic transformation
$$q'_{i} =A_{ij}q_{j}+B_{ij}p_{j},$$
$$p'_{i} =C_{ij}q_{j}+D_{ij}p_{j}.$$
such that the canonical commutation...
I don't know where to start. I understand that the constraint ##ad-bc=1## gives us one less parameter since ##d=1+bc/a##. So we can rewrite our original function. I know how to compute the generators of matrix groups but in this case the generators will be functions. I also know there should be...
I'm trying to understand the so-called polaron transformation as frequently encountered in quantum optics. Take the following paper as example: "Quantum dot cavity-QED in the presence of strong electron-phonon interactions" by I. Wilson-Rae and A. Imamoğlu. We have the spin-phonon model with...