Q stringlengths 70 13.7k | A stringlengths 28 13.2k | meta dict |
|---|---|---|
Composition of Lorentz Transformations If a particle is moving in the $x$-direction with velocity $c/2$, then the Lorentz transformation $\Lambda = \begin{pmatrix}\gamma & -\beta \gamma & 0 & 0 \\ -\beta \gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix}\cosh\ \phi & -\sinh\ \phi... | If you are doing what I think you are doing then, you are trying to get the Boost matrix for an arbitrary direction. Then the way to go about that will be to use the generalized boost matrix (see J D Jackson page547 or http://en.wikipedia.org/wiki/Lorentz_transformation). The the matrix against which you can compare to... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/174624",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Using the uncertainty principle to estimate the ground state energy of hydrogen I have been reading through this estimate of the ground state energy of hydrogen and others like it. In this one it says it is using the uncertainty principle but then proceeded to use the following:
$$pr=\hbar$$
But why is it not using:
$... | Note that $\Delta p_x \Delta r$ does not satisfy the uncertainty principle in the strict sense since $r$ is not conjugate to $p_x$ (or $p_y$ and $p_z$). Instead you can consider $\Delta p_x \Delta x$. The ground state of the hydrogen atom is
\begin{equation}
\psi_0(r) = \frac{1}{\sqrt{\pi a^3}} e^{-r/a},
\end{equation}... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/183960",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 3,
"answer_id": 0
} |
Hindered rotation model for flexible polymers: deriving the Flory characteristic ratio In the hindered rotation model we assumes constant bond angles $\theta$ and lengths $\ell$, with torsion angles between adjacent monomers being hindered by a potential $U(\phi_i)$. In Rubinstein's book problem 2.9 asks us to derive t... | The vector $\mathrm{r}_i$ is between the beads $i$ and $i+1$. Define a local coordinate system so that $x$ is along $\mathrm{r}_i$. The coordinate $y$ is defined so that $\mathrm{r}_{i-1}$, $\mathrm{r}_{i}$ are both on the same plane, and $z$ is normal to this plane. Thus we can write
\begin{align}
\mathrm{r}_{i-1} &= ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/256207",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Generalized Euler substitution doesn't seem to work when the integration variable has a dimension I came across Euler substitutions while trying to evaluate the integral $\int \frac{y^2}{x^2+y^2+z^2 + x\sqrt{x^2+y^2+z^2}} dy$, where $x, y, z$ are length quantities. The generalized substitution at the bottom of the page... | You can always make a substitution that makes the integration variable dimensionless, and then this isn't an issue. Rewrite your integral as
$$
\int \frac{y^2}{ x^2+z^2 + y^2 + x \sqrt{x^2 + z^2} \sqrt{\frac{y^2}{x^2+z^2} + 1} } dy.
$$
Substitute the dimensionless variable $\eta = y/\sqrt{x^2 + z^2}$:
$$
(x^2 + z^2)... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/332695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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A geometrical calculation in Fresnel's paper "Memoir on the diffraction of light" 1819 It is a geometrical problem which I find difficult to solve reading the Fresnel's paper "Memoir on the diffraction of Light".
According to the figure Fresnel sets $z$ as the distance of the element $nn'$ from the point $M$---- (I sup... |
First, we have that the triangles $RAB$ and $RTC$ are similar, so that
$$TC=RC\,\left(\frac{AB}{RB}\right)=(a+b)\,\left(\frac{c/2}{a}\right)=\frac{(a+b)c}{2a}\,.$$
This means
$$\begin{align}
RF&=\sqrt{FC^2+RC^2}=\sqrt{(FT+TC)^2+RC^2}=\sqrt{\left(x+\frac{(a+b)c}{2a}\right)^2+(a+b)^2}
\\
&=(a+b)\,\sqrt{1+\left(\frac{x}{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/422721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Geodesics for FRW metric using variational principle I am trying to find geodesics for the FRW metric,
$$
d\tau^2 = dt^2 - a(t)^2 \left(d\mathbf{x}^2 + K \frac{(\mathbf{x}\cdot d\mathbf{x})^2}{1-K\mathbf{x}^2} \right),
$$
where $\mathbf{x}$ is 3-dimensional and $K=0$, $+1$, or $-1$.
Geodesic equation
Using the Christof... | I think the equations may be consistent after all. First a solution to the EL equation for $\mathbf{x}$ also satisfies the geodesic equation:
Starting with the EL equation I have above:
$$
\frac{d}{d\tau} \left[ a^2\left(\mathbf{x}' + \frac{K(\mathbf{x} \cdot \mathbf{x}')\mathbf{x}}{1-K\mathbf{x}^2}\right) \right] = \f... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/475877",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
X-ray diffraction intensity and Laue equations My textbook, Solid-State Physics, Fluidics, and Analytical Techniques in Micro- and Nanotechnology, by Madou, says the following in a section on X-Ray Intensity and Structure Factor $F(hkl)$:
In Figure 2.28 we have plotted $y = \dfrac{\sin^2(Mx)}{\sin^2(x)}$. This functio... | On applying L'Hôpital's rule, we get $ y = \frac{2Msin(Mx)cos(Mx)}{2sin(x)cos(x)}$. Again applying L'Hôpital's rule $\frac{sin(Mx)}{sin(x)} = M$, giving $y=M^{2}$.
Just here, it is proved that $\frac{sin^{2}(Mx)}{sin^{2}x}$ has maxima at $ x = n\pi$. Here n is any integer and not just even integers. In $\frac{sin^{2... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/526415",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
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Number degrees of freedom for Sphere on a inclined plane How many generalized coordinates are required to describe the dynamics of a solid sphere is rolling without slipping on an inclined plane?
What I think is that there two translational degrees of freedom and three rotational degrees of freedom. But I'm not able t... |
you have one holonomic constraint equation which is $~z=a~$ and two nonholonomic
constraint equations
the relative velocities componets , at the contact point between the sphere and the incline plane are zero thus:
$$\begin{bmatrix}
\omega_x \\
\omega_y \\
\omega_z \\
\end{bmatrix}\times \begin{bmatrix}
0 \\
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/603105",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
} |
Expanding the Graphene Hamiltonian near Dirac points upto second order term I was trying to solve the Graphene Hamiltonian near the Dirac points upto the second order term for the nearest neighborhood points.
So expanding the function near the Dirac Point, we get$$g(K+q)=\frac{3ta}{2}e^{i\pi/3}(q_x-iq_y)+\frac{3ta^2}{8... | Something seems to be off with the energy term here... Your last term goes as $\cos\theta$.
Let's try again :)
$$
H = \begin{pmatrix}
0&-t f_\mathbf{q}
\\
-tf_\mathbf{q}^*&0
\end{pmatrix}\,,
$$
where $f_\mathbf{q} = 1 + e^{i\mathbf{q}\cdot\mathbf{d}_1}+ e^{i\mathbf{q}\cdot\mathbf{d}_2}$ and $\mathbf{d}_1$ and $\mathbf{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/659577",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Spin $\frac{3}{2}$ representation in Georgi's book? Georgi's book Lie Algebras in Particle Physics 2ed equation 3.32 lists the spin operators in the spin $\frac{3}{2}$ representation as:
$$J_1=\left(
\begin{array}{cccc}
0 & \sqrt{\frac{3}{2}} & 0 & 0 \\
\sqrt{\frac{3}{2}} & 0 & 2 & 0 \\
0 & 2 & 0 & \sqrt{\frac{3}{2}... | There is a typo in the book's equation, and there doesn't appear to be an easily accessible online errata. If one follows the formulas the book gives for $J_\pm$: $$J_+=\frac{1}{\sqrt{2}}\left(J_1+i J_2\right)$$
$$J_-=\frac{1}{\sqrt{2}}\left(J_1-i J_2\right)$$
$$J_{-,m'm}\frac{\sqrt{\left(s-m\right) \left(m+s+1\right)}... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/212553",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
How to find the logarithm of Pauli Matrix? When I solve some physics problem, it helps a lot if I can find
the logarithm of Pauli matrix.
e.g. $\sigma_{x}=\left(\begin{array}{cc}
0 & 1\\
1 & 0
\end{array}\right)$, find the matrix $A$ such that $e^{A}=\sigma_{x}$.
At first, I find a formula only for real matrix:
$$\ex... | Observe that
\begin{equation}
\sigma_{z} = \begin{pmatrix}1&0\\0&-1\end{pmatrix} = \exp(B) = \sum_{r=0}^{\infty} \frac{B^{r}}{r!}
\end{equation}
with
\begin{equation}
B = i\pi\begin{pmatrix}2m&0\\0&2n+1\end{pmatrix},
\end{equation}
where $m,n\in\mathbb{Z}$.
Next, notice that
\begin{equation}
\sigma_{x} = U \sigma_{z} ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/225668",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 3,
"answer_id": 2
} |
Relation between Spin 1 representation and angular momentum and $SO(3)$ This is a naive question. It occurred to me while studying in detail the the Spin 1 angular momentum matrices.
The generators of $SO(3)$ are
$J_x=
\begin{pmatrix}
0&0&0 \\
0&0&-1 \\
0&1&0
\end{pmatrix} \hspace{1cm} J_y=\begin{pmatrix}
0&0&1 \\
0&0&... | The two representations are unitarily equivalent to each other, except for an overall factor of $i$.
To be clear, I'll write $J$ and $\tilde J$ for the generators in the two different representations. One representation is
$$
J_x = \left(
\begin{matrix} 0&0&0\\ 0&0&-1 \\ 0&1&0\end{matrix}
\right)
\hskip1cm
J_y = \... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/438522",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 1,
"answer_id": 0
} |
How compute the mass of AdS-Schwarzschild by ADM mass formula? I want to compute the mass of AdS schwarzschild by ADM mass formula but I could not find where I am wrong.
AdS schwarzschild line element is :
$$
ds^2 =-f dt^2 +\frac{dr^2}{f} +r^2 d\sigma^2_{d-1}
$$
where:
$$
f=k+\frac{r^2}{L^2}-\frac{\omega^{d-2}}{r^{d-2}... | I think the issue is that you’re using a formula for the ADM mass which assumes unit lapse. Consider a spherically symmetric spacetime of the form
\begin{align}
{\rm d}s^2 & = -f(r) \, {\rm d}t^2 + h(r)\, {\rm d}r^2 + r^2 ({\rm d}\theta^2 + \sin^2\theta\, {\rm d}\varphi^2)
\end{align}
From the outward pointing unit nor... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/716943",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Matrix operation in dirac matrices If we define $\alpha_i$ and $\beta$ as Dirac matrices which satisfy all of the conditions of spin $1/2$ particles , and $p$ is the momentum of the particle, then how can we get the matrix form
\begin{equation}
\alpha_i p_i= \begin{pmatrix} p_z & p_x-ip_y \\ p_x+ip_y & -p_z \end{pma... | It's just a matrix manipulation. Let $\sigma_i$ pauli matrices.
\begin{equation}
\alpha_i p_i= \begin{pmatrix} 0& \sigma_i \\ \sigma_i & 0 \end{pmatrix} p_i .
\end{equation}
$ \alpha_i p_i= \begin{pmatrix} 0& p_1 \sigma_1 \\ p_1\sigma_1 & 0 \end{pmatrix} + \begin{pmatrix} 0& p_2 \sigma_2 \\ p_i\sigma_2 & 0 \end{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/48044",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
The potential and the intensity of the gravitational field in the axis of a circular plate Calculate the potential and the intensity of the gravitational field at a distance $x> 0$ in the axis of thin homogeneous circular plate of radius $a$ and mass $M$.
Could anybody describe how to calculate this? Slowly and in det... | First let us calculate the potential for a ring of radius $a$ at a distance $x$ from the center along the axis
Potential due to an infinitesimal mass element $dm$ will be $$\frac{-Gdm}{\sqrt{a^2+x^2}}$$
Potential due to the ring is then $$\int{\frac{-Gdm}{\sqrt{a^2+x^2}}}=\frac{-G}{\sqrt{a^2+x^2}}\int{dm}=\frac{-Gm}{\s... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/62637",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
How to write the equation of a field line of an electrostatic field? How can we write the equations of a line of force between two charges, say $q$ and $q'$?
As an example, you may consider the simpler case of two opposite charges $+q$ and $-q$, and focus on the field line emerging out of $+q$ by making an angle of $\t... | Let a charge $+q$ be at the point $(a, 0)$ and a charge $-q$ be at the point $(-a, 0)$. Then the electric field at a point
$(x, y)$ is
\begin{equation}\tag{e1}\label{e1}
\vec{E} = q\vec{r}\left(\frac{1}{r_1^3} - \frac{1}{r_2^3}\right) - qa\hat{e}_x\left(\frac{1}{r_1^3} + \frac{1}{r_2^3}\right),
\end{equation}
where $\v... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/277835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "11",
"answer_count": 4,
"answer_id": 2
} |
Partitioning the kinetic energy into components in relativity In classic physics, kinetic energy is defined as
$$
KE = \frac{1}{2}m v_x^2 + \frac{1}{2}m v_y^2 + \frac{1}{2}m v_z^2
$$
So, by defining $ KE_x = \frac{1}{2} m v_x^2 $ , $ KE_y = \frac{1}{2} m v_y^2 $, $ KE_z = \frac{1}{2} m v_z^2 $, we can know the contrib... | What you are looking for is a function of $f(v_i)$ such that $f(v_x) + f(v_y) + f(v_z) = KE$.
In the case of Newtonian physics, $f(v_i) = \frac{1}{2}mv_i^2$.
In the case of special relativity, it is impossible to find any such function.
Proof by contradiction: Suppose that such an $f$ did exist. Now imagine three objec... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/422050",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 0
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Analytical expressions for acceleration due to zonal harmonics of a gravitational field? Wikipedia's Geopotential_model; The deviations of Earth's gravitational field from that of a homogeneous sphere discusses the expansion of the potential in spherical harmonics. The first few zonal harmonics ($\theta$ dependence onl... | Let's look at @mmeent's comment suggesting that the spherical coordinates used in the linked Wikipedia article set the polar angle equal to zero at the equator rather than the pole.
where spherical coordinates (r, θ, φ) are used, given here in terms of cartesian (x, y, z) for reference
While that link shows $\theta =... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/547131",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
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Contradiction between Force and Torque equations A thin uniform rod of mass $M$ and length $L$ and cross-sectional area $A$, is free to rotate about a horizontal axis passing through one of its ends (see figure).
What is the value of shear stress developed at the centre of the rod, immediately after its release from t... | Your initial calculations are correct. The pin force is indeed $\tfrac{m g}{4} $ a fact that kind of surprised me the first time I encountered this problem.
Your idealization on the second part is where things were missed. I am using the sketch below, and I am counting positive directions as downwards (same as gravity)... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/718414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
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Why is equivalent resistance in parallel circuit always less than each individual resistor? There are $n$ resistors connected in a parallel combination given below.
$$\frac{1}{R_{ev}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}+\frac{1}{R_{4}}+\frac{1}{R_{5}}.......\frac{1}{R_{n}}$$
Foundation Science - Physics (c... | We can prove it by induction. Let
$$
\frac{1}{R^{(n)}_{eq}} = \frac{1}{R_1} + \cdots+ \frac{1}{R_n}
$$
Now, when $n=2$, we find
$$
\frac{1}{R^{(2)}_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \implies R_{eq}^{(2)} = \frac{R_1 R_2}{R_1+R_2} = \frac{R_1}{1+\frac{R_1}{R_2}} = \frac{R_2}{1+\frac{R_2}{R_1}}
$$
Since $\frac{R_1}{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/137527",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 4,
"answer_id": 2
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Contradiction between Force and Torque equations A thin uniform rod of mass $M$ and length $L$ and cross-sectional area $A$, is free to rotate about a horizontal axis passing through one of its ends (see figure).
What is the value of shear stress developed at the centre of the rod, immediately after its release from t... | I think you can obtain the results like this (your approach)
I ) obtain $~F_H~,a_{CM}~,\alpha~$ with
sum of the forces at the center of mass
$$M\,a_{CM}=M\,g-F_H$$
sum of the torques at the center of mass
$$I_{CM}\,\alpha=\frac{F_H\,L}{2}$$
and the kinematic equation
$$\tan(\alpha)\approx\alpha=\frac{2\,a_{CM}}{L... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/718414",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
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Derive $\frac{\mathrm{d}}{\mathrm{d}t}(\gamma m\mathbf{v}) = e\mathbf{E}$ from elementary principles? It is experimentally known that the equation of motion for a charge $e$ moving in a static electric field $\mathbf{E}$ is given by:
$$\frac{\mathrm{d}}{\mathrm{d}t} (\gamma m\mathbf{v}) = e\mathbf{E}$$
Is it possible t... | If I understand your question correctly, you can show it by the next way.
First of all, use some expressions from Special relativity, which are Lorentz transformations for force $\mathbf F $, radius-vector $\mathbf r$ and speed $\mathbf v$ ($\mathbf u$ is the speed of inertial system):
$$
\mathbf r' = \mathbf r + \Gamm... | {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 6,
"answer_id": 5
} |
Calculating the expectation value for kinetic energy $\langle E_k \rangle$ for a known wave function I have a wavefunction ($a=1nm$):
$$\psi=Ax\exp\left[\tfrac{-x^2}{2a}\right]$$
for which I already calculated the normalisation factor (in my other topic):
$$A = \sqrt{\frac{2}{a\sqrt{\pi a}}} = 1.06\frac{1}{nm\sqrt{nm}... | In your problem, you need integrals of kind :
$I_{2n} = \int x^{2n} e^{- \large \frac{x^2}{a}} ~ dx$
Note first that $I_0 = (\pi)^\frac{1}{2} (\frac{1}{a})^ {-\frac{1}{2}}$
Now, it is easy to see that there is a reccurence relation between the integrals :
$$I_{2n+2} = - \frac{\partial I_{2n}}{\partial (\frac{1}{a}) } ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/72950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
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Calculating angle of min deviation of prism Two rays incident with angle 40 and 60 on one face of equilateral triangular prism the angle of deviation are equal .find angle of minimum deviation?
| The graph of angle of deviation vs angle of incidence is a $U$ shape. The fact that the angle of deviation is the same for these two rays means that a ray which is incident at $40$ degrees to the normal will emerge at $60$ degrees to the normal. This allows us to find the refractive index.
$$\sin i_1 = \sin 40 = n\sin... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Matrix representation of spin-2 system? I am surprised no one has asked this before, but what is the matrix representation of a spin-2 system?
Also, what are the equivalent of the Pauli matrices for the system?
| The irreducible representation of $su(2)$ corresponding to spin 2 is 5-dimensional. One possible choice of explicit $5\times5$ matrices for spin-2 angular momentum is
$$J_1=\left(
\begin{array}{ccccc}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & \sqrt{\frac{3}{2}} & 0 & 0 \\
0 & \sqrt{\frac{3}{2}} & 0 & \sqrt{\frac{3}{2}} & 0 \\
0... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/468129",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How the equation of a projectile represents a parabola? I am not able to prove that equation of motion of a projectile is parabola. The book simply says the given below is the equation of a parabola but does not clarify it
$$y= {\tan\theta}x - \frac{g}{2(u\cos\theta)^2}x^2$$
But equation of parabola says
(i) $y^2=4ax$... | None of the four options seem like the proper definition for a parabola. Mabe if you show that $y \sim x^2$ it would suffice.
Specifically, the vertex form of a parabola is $y = a (x-x_0)^2 + d$, where $a$, $x_0$ and $d$ are constants. Bring the given equation in this form to show it is indeed a parabola.
*
*Equate t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/745055",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Calculating the expectation value for kinetic energy $\langle E_k \rangle$ for a known wave function I have a wavefunction ($a=1nm$):
$$\psi=Ax\exp\left[\tfrac{-x^2}{2a}\right]$$
for which I already calculated the normalisation factor (in my other topic):
$$A = \sqrt{\frac{2}{a\sqrt{\pi a}}} = 1.06\frac{1}{nm\sqrt{nm}... | My statistical physics professor call those ones Laplace integrals $I(h)$.
$$I(h)=\int_{0}^{\infty}x^{h}e^{-a^2x^2}dx$$
Note that
$$\int_{-\infty}^{\infty}x^{h}e^{-a^2x^2}dx=2I(h) $$
some values
$$I(0)=\frac{\sqrt{\pi}}{2a}, I(1)=\frac{1}{2a^2}, I(2)=\frac{\sqrt{\pi}}{4a^3},I(3)=\frac{1}{2a^4}, I(4)=\frac{3\sqrt{\pi}}{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/72950",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 1
} |
What is the "associated scalar equation" of equations of motion? In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as:
$$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$
Fine. Then it says the "associated scalar equation" is:
$$r''=-\frac{\mu}{r^2}+\f... | The "associated scalar equation" is just the formula for the time evolution of the scalar magnitude of the displacement, $r$, rather than all its vector components. It really only makes sense to write such an equation if the right-hand side can be expressed in terms of $r$ only, and not $\mathbf{r}$. Then you can use i... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/132287",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Solution to invariance equation when deriving Lorentz transformation Requiring that the spacetime interval (1 spatial dimension) between the origin and an event $(t,\, x)$ stays constant under a transformation between reference frames:
$$c^2t^2-x^2= c^2t'^2- x'^2$$
we want to find a solution to that equation, i.e. exp... | The different types of Lorentz transformations can be categorized using group theory, but in this case, I think one can convince themselves with a bit of algebra.
Let's write our transformation in matrix notation. Representing our event as a column vector, $\begin{pmatrix} ct \\ x \end{pmatrix}$, the transformation you... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/461374",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Massless Kerr black hole Kerr metric has the following form:
$$
ds^2 = -\left(1 - \frac{2GMr}{r^2+a^2\cos^2(\theta)}\right) dt^2 +
\left(\frac{r^2+a^2\cos^2(\theta)}{r^2-2GMr+a^2}\right) dr^2 +
\left(r^2+a^2\cos(\theta)\right) d\theta^2
+ \left(r^2+a^2+\frac{2GMra^2}{r^2+a^2\cos^2(\theta)}\right)\sin^2(\t... | It's simply flat space in Boyer-Lindquist coordinates. By writing
$\begin{cases}
x=\sqrt{r^2+a^2}\sin\theta\cos\phi\\
y=\sqrt{r^2+a^2}\sin\theta\sin\phi\\
z=r\cos\theta
\end{cases}$
you'll get good ol' $\mathbb{M}^4$.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/593675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 0
} |
Evaluating volume integral for electric potential in an infinite cylinder with uniform charge density Suppose I have an infinitely long cylinder of radius $a$, and uniform volume charge density $\rho$. I want to brute force my way through a calculation of the potential on the interior of the cylinder using the relation... | I drop the constant and focus on the integral, also the prime sign:
$$
\Phi(\mathbf{x})
= \int_{-\infty}^{\infty} dz
\int_0^{2\pi} d\phi
\int_0^a r dr
\frac{1}{\sqrt{x^2 + r^2 - 2xr\cos\phi + z^2}}
$$
The integral is divergent. To remove the divergence is to change the reference point of the potential from $x=\infty$ t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/616430",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Are any points at rest on a globe spinning both horizontally and vertically? Suppose there was a globe which could be spun about polar and horizontal axis. It is first spun about the polar axis, then promptly spun about the horizontal axis. Would any points be at rest?
| the components of a point on sphere surface are:
\begin{align*}
&\mathbf{R}=\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}=
\left[ \begin {array}{c} \cos \left( \theta_0 \right) \sin \left( \phi_0
\right) \\ \sin \left( \theta_0 \right) \sin \left(
\phi_0 \right) \\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/722642",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 1
} |
Velocity from velocity potential I have this homework question and I get a different answer to the solutions.
In Cylindrical polar coordinates $(r,\theta,z)$, the velocity potential of a flow is given by:
$$\phi = -\frac{Ua^2r}{b^2-a^2}(1+\frac{b^2}{r^2})cos\theta$$
Find the velocity.
I get the velocity as:
$$v = (-\... | For a flow in polar coordinates, the stream function $\phi$ leads to the velocities as
$$
v_r=\frac{1}{r}\frac{\partial\phi}{\partial\theta}\qquad v_\theta=-\frac{\partial\phi}{\partial r}
$$
and not $v_r=\partial_r\phi$ and $v_\theta=\partial_\theta\phi$. Thus,
$$
v_r=\frac{1}{r}\frac{\partial}{\partial \theta}\left(-... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/79835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Gauge-fixing of an arbitrary field: off-shell & on-shell degrees of freedom How to count the number of degrees of freedom of an arbitrary field (vector or tensor)? In other words, what is the mathematical procedure of gauge fixing?
| In this answer, we summarize the results. The analysis itself can be found in textbooks, see e.g. Refs. 1 & 2.
$\downarrow$ Table 1: Massless spin $j$ field in $D$ spacetime dimensions.
$$\begin{array}{ccc}
\text{Massless}^1 & \text{Off-shell DOF}^2 & \text{On-shell DOF}^3 \cr
j=0 & 1 & 1 \cr
j=\frac{1}{2} & n & \f... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/100995",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "17",
"answer_count": 2,
"answer_id": 0
} |
Infinitesimal generator flux of Lorentz trasformations in spacetime I'm considering the following matrixs which I know that they form a flux of Lorentz trasformation in spacetime.
I want to know how to calculate the infinitesimal generator of this flux. Unfortunately I have no particular knowledge of Lie algebra for t... | I'm a bit suspicious of the 22 entry of the matrix you write down ,
$$M=\begin{pmatrix}
\frac{4- \cos(\rho)}{3} & \frac{2- 2\cos(\rho)}{3} & 0 & -\frac{\sin(\rho)}{\sqrt{3}} \\
\frac{2\cos(\rho) - 2}{3} & \frac{4- \cos(\rho)}{3} & 0 & \frac{2\sin(\rho)}{\sqrt{3}}\\
0 & 0 & 1 & 0 \\
-\frac{\sin(\rho)}{\sqrt{3}} & -\fr... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/366097",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to derive the Klein-Nishina formula from the Dirac equation? I'm looking for the simplest demonstration of the Klein-Nishina formula, from the Dirac equation without the field described as a quantum operator:
https://en.wikipedia.org/wiki/Klein%E2%80%93Nishina_formula
Consider $\psi$ as a "classical" spinor field (... | In the center of mass frame, let $p_1$ be the inbound photon, $p_2$ the inbound electron, $p_3$ the scattered photon, $p_4$ the scattered electron.
\begin{equation*}
p_1=\begin{pmatrix}\omega\\0\\0\\ \omega\end{pmatrix}
\qquad
p_2=\begin{pmatrix}E\\0\\0\\-\omega\end{pmatrix}
\qquad
p_3=\begin{pmatrix}
\omega\\
\omega\s... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/416155",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 1,
"answer_id": 0
} |
Peskin & Schroeder's way of showing $Z_1=Z_2$ via integration by parts
I am trying to follow Peskin & Schroeder's textbook on Renormalization. I tried a few ways but this does not match with the textbook.
First equation (10.43 )in Peskin is given
\begin{align}
\delta_2 = -\frac{e^2}{(4\pi)^{\frac{d}{2}}}
\int_0^1... | First equation (10.43 )in Peskin is given
\begin{align}
\delta_2 = -\frac{e^2}{(4\pi)^{\frac{d}{2}}}
\int_0^1 dx \frac{\Gamma\left(2-\frac{d}{2}\right)}{\left( (1-x)^2 m^2 + x
\mu^2 \right)^{2-\frac{d}{2}}}
\left[ (2-\epsilon) x - \frac{\epsilon}{2} \frac{2x(1-x)m^2}{\left( (1-x)^2
m^2 + x \mu^2 \right)} (4-2... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/581281",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Massless Kerr black hole Kerr metric has the following form:
$$
ds^2 = -\left(1 - \frac{2GMr}{r^2+a^2\cos^2(\theta)}\right) dt^2 +
\left(\frac{r^2+a^2\cos^2(\theta)}{r^2-2GMr+a^2}\right) dr^2 +
\left(r^2+a^2\cos(\theta)\right) d\theta^2
+ \left(r^2+a^2+\frac{2GMra^2}{r^2+a^2\cos^2(\theta)}\right)\sin^2(\t... | A reference which answers this is Visser (2008). It discusses the limits of vanishing mass $M \rightarrow 0$, and rotation parameter $a \rightarrow 0$. Your example is in $\S5$. Visser comments "This is flat Minkowski space in so-called “oblate spheroidal” coordinates...", as described in a different answer here.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/593675",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 3,
"answer_id": 1
} |
Parametric representation of the motion of a particle in context of kruskal coordinates In "N. Straumann - General Relativity" talking about the Kruskal continuation of the Schwarschild solution it is considered for radial timelike geodesic the following:
$$d\tau=(\frac{2m}{r}-\frac{2m}{R})^{\frac{-1}{2}}dr$$
with the ... | Correct me if I am wrong, but I think it should be
$$\text{d}\tau = -\left(\frac{r_s}{r}-\frac{r_s}{R}\right)^{-\frac 12}\text{d}r$$
with $r_s = 2M$.
This would be consistent for an infalling motion with the changes in the proper time and radius when compared to the changes in the parameter $\eta\,$ ($\text{d}r <0$ whi... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/621721",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Quantum Mechanics: Does $\vec{A} \cdot \vec{p} = \frac{1}{2} \vec{B}\cdot\vec{L}$? In a quantum mechanics question context, I noticed the need to prove that for a constant magnetic field $\mathbf{B}$, The vector potential $\mathbf{A}$ and the angular momentum operator $\mathbf{L}$, satisfy:
$$\mathbf{A}\cdot\mathbf{p} ... | This identity doesn't hold for non constant magnetic fields. For instance with:
$$\mathbf{A} = \frac{1}{2}B_0 k_0 ( -y^2, x^2) \Rightarrow \mathbf{B} = B_0 k_0 (0,0,x+y)$$
We got on one the hand:
$$ A_j p_j + p_j A_j = \frac{1}{2} B_0 k_0 ( -y^2 p_x + x^2 p_y - p_x y^2 + p_y x^2) = B_0 k_0 (x^2 p_y-y^2 p_x)$$
Where as:... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/741354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Is there a better, faster way to do this projectile motion question? The question is
In a combat exercise, a mortar at M is required to hit a target at O, which is taking cover 25 m behind a structure of negligible width 10 m tall. This mortar can only fire at an angle of 45 degrees to the horizontal, but can fire she... | Whether this is any faster is debatable, but you could do it this way:
The trajectory of the shell is symmetric, so $M$ firing a shell at $O$ is the same as $O$ firing a shell at $M$. So all you have to do is consider $O$ firing the mortar at 45° and ask what is the minimum velocity required to clear the wall. So, if $... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/86736",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
How to derive the relation between Euler angles and angular velocity and get the same form as mentioned in the bellow figure How to derive the relation between Euler angles and angular velocity and get this form:
$$
\left.
\begin{cases}{}
P \\
Q\\
R \\
\end{cases}
\right\}=
\left[
\begin{array}{c}
1&0&-\sin\Theta\\
0&\... | How to derive the relation between euler angles and angular velocity
\begin{align*}
&\text{The equations to calculate the angular velocity $\vec{\omega}$ for a given transformation matrix $S$ are: } \\\\
&\left[_B^I\dot{S}\right]=\left[\tilde{\vec{\omega}}_I\right]\left[_B^I S\right]\,\quad \Rightarrow
\left[\t... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/420695",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Commutation relations for creation, annihilator operators $a_\mathbf{p}^\dagger, a_\mathbf{p}$ Write the field $\phi$ and momentum $\pi$ in terms of creation and annihilation operators $a_\mathbf{p}^\dagger, a_\mathbf{p}$
$$
\phi(\mathbf{x}) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2\omega_\mathbf{p}}} [a_\mathbf{p}... | The only dependence on $\mathbf{x}$ that remains is in the exponent factor. Integrating it we get $\delta$-function,
\begin{equation}
\int d^3x\, e^{-i\mathbf{x}\cdot(\mathbf{p}+\mathbf{q})}=(2\pi)^3\delta^{(3)}(\mathbf{p}+\mathbf{q})
\end{equation}
That means that we can replace $\omega_\mathbf{q}$ with $\omega_{-\mat... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/503383",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Why do sharp edges of a metallic conductor have more charges than flat edges? charges accumulate on sharp edges more than flat edges.
| Like charges on the surface of a conductor repel each other. Let us consider two identical charges $q$ in points A and B on the surface of a conductor in an area where the curvature of the surface is $R$, so OA=OB=R (see the picture). Let us assume that the length of arc AB is $l$, and $l\ll R$. Then angle $\alpha=l/R$... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/700384",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 1
} |
4-momentum and an $y$ component of momentum I have 2 coordinate systems which move along $x,x'$ axis.
I have derived a Lorentz transformation for an $x$ component of momentum, which is one part of an 4-momentum vector $p_\mu$. This is my derivation:
$$
\scriptsize
\begin{split}
p_x &= mv_x \gamma(v_x)\\
p_x &= \frac{m... | Well this is how $p_y$ part of a four-momentum is put together.
\begin{equation}
\scriptsize
\begin{split}
p &= m v \gamma(v)\\
&\Downarrow\\
p_y &= m v_y \gamma(v) = m v_y \gamma \left( \sqrt{v_x^2 + v_y^2 + v_z^2}\right) = m v_y \gamma \left( \sqrt{v_x^2 + 0 + 0}\right) = m v_y \gamma(v_x) =\\
&= \frac{m v_y'}{\gamm... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/45811",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How to calculate the intensity of the interference of two waves in a given point? There are two different point sources which produce spherical waves with the same power, amplitude, ω, wavenumber and phase.
I can calculate the intensity of each wave in a point:
$$
I_1 = P / (4 \pi r_1^2)
$$
$$
I_2 = P / (4 \pi ... | I found the solution, but forgot to add it here. But since a considerable number of people has seen the question, the answer could be useful.
Notation
*
*$r_1$: distance between the given point and the focus 1.
*$r_2$: distance between the given point and the focus 2.
*$A_1$: amplitude in the given point of the wa... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/54556",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
George Green's derivation of Poisson's equation I was reading George Green's An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, and I got confused on one step in his derivation of Poisson's Equation. Specifically, how does Green obtain conclude that:
$$\delta\left(2\pi a^... | Let's first derive the value of $V$ inside the small sphere:
$$
V_\text{sphe} = \rho\int\frac{\text{d}x'\text{d}y'\text{d}z'}{r'},
$$
Where the sphere is sufficiently small such that $\rho$ can be considered constant. We can orientate the axes such that $p$ lies on the $z'$ axis. In spherical coordinates, the integral ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/217835",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Approximating an expression for a potential In a problem which I was doing, I came across an expression for the potential $V$ of a system as follows $$V = k\left(\frac{1}{l - x} + \frac{1}{l + x}\right)\tag{1}\label{1}$$ where $k$ is a constant, $l$ and $x$ are distances and $l \gg x$. Now I went to find an approximate... | We have
$$
x^2 \ll l^2
\implies \frac{2x^2}{l^2}
\ll 2
\implies \frac{2x^2}{l^2} + 2
\approx 2
$$
So,
$$
V
\approx \frac{k}{l} \bigg(2 + \frac{2x^2}{l^2}\bigg)
\approx 2\frac{k}{l}
$$
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/495163",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 0
} |
Minimum of the free energy in Landau-Ginzburg theory I'm reading David Tong's lectures and this page:
I understand how to get the solution $m_0$ when $m$ has no spatial dependence. But I do not understand how one can find the solution when $m = m(x)$
$m = m_0\tanh(\sqrt{\frac{-a}{2c}} x)$
In particular, where does tan... | You need to solve differential equation:
$$
\frac{d^2m}{d x^2} = \frac{am}{c} + \frac{2b m^3}{c}
$$
We multiple this equation by $\frac{dm}{dx}$ (if $\frac{dm}{dx}=0$ we obtain 2 vacuum solutions):
$$
\frac{dm}{dx}\frac{d^2m}{d x^2} = \frac{dm}{dx}\frac{am}{c} + \frac{dm}{dx}\frac{2b m^3}{c}
$$
$$
\frac{1}{2}\frac{d}{d... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/528152",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Parameterizing a universe with non-zero curvature, some matter, no dark matter, and no dark energy Defining $\Omega_i$ by $\rho_i (t_0) = \Omega_i \rho_{c_0}$, we can obtain the below equality.
$$H^2 = H_0^2 \left(\frac{\Omega_r}{a^4} + \frac{\Omega_m}{a^3}+\frac{\Omega_k}{a^2} + \Omega_\Lambda\right)$$
What is the mea... | The Friedmann equation for these models can be written
$$ \dot{a}^2 = H_0^2(\frac{\Omega_{m}}{a} + 1 - \Omega_{m}) $$
For a universe with both matter and nonzero curvature, we have
$$ \frac{\dot{a}^2}{a^2} = H_0^2(\Omega_m a^{-3} + \Omega_k a^{-2}) \implies (\frac{da}{dt})^2 = H_0^2(\Omega_m a^{-1} + \Omega_k) $$
There... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/652741",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Evolution of Euler's angles in time The general motion of a rigid body over time can be determined in the body frame by solving Euler's equations, selecting the principal axes for the body axes. Also, Euler's angles can be used to transform from the rotating body frame to an inertial space frame. But, in my physics me... | Euler Equation
\begin{align*}
&\mathbf{I}\,{\dot{\omega}}+\mathbf\omega\times \left(\mathbf{I}\,\mathbf\omega\right)=\mathbf\tau\tag 1
\end{align*}
and the kinematic equation
\begin{align*}
&\mathbf{\dot{\phi}}=\mathbf{A}(\mathbf{\phi})\,\mathbf{\omega}\tag 2
\end{align*}
all vector components and the inertia tensor ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/707276",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do I calculate the perturbations to the metric determinant for 3º order? From the post How do I calculate the perturbations to the metric determinant?,
I'm trying to calculate the expansion of the metric's determinant $\sqrt{-g}$ up to 3rd order. I saw in another post the procedure until further notice. but I'm not... | Following the calculations from the post you reference, note that all they have done is Taylor expand the logarithm and then the exponential. For that, we need to know the following Taylor identities:
\begin{align}
\exp(x)=&1+x+\frac{1}{2!}x^2+\frac{1}{3!}x^3+\dots\\
\log(1+x)=&x-\frac{1}{2}x^2+\frac{1}{3}x^3+\dots
\en... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/725670",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Divergence not defined I’m currently working on the practice problems in Introduction to Electrodynamics by Griffiths. I got confused by the solution to this problem.
What does “ill-defined divergence” even mean? I understand how and when to use delta function, but I don’t understand how divergence is not defined.
| I think we can use
$$
\nabla \cdot (\psi \vec{a}) = \vec{a} \cdot \nabla \psi + \psi \nabla \cdot \vec{a}
$$
to see what's happened for $n \lt -2$.
$$\begin{align*}
\nabla \cdot (r^{-3} \hat{r})
&= \left( \frac{1}{r^2} \hat{r} \right)
\cdot \nabla \frac{1}{r}
+ \frac{1}{r} \nabla \cdot \frac{1}{r^2} \hat{r} \\
&= -... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/738354",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Vector space of $\mathbb{C}^4$ and its basis, the Pauli matrices How do I write an arbitrary $2\times 2$ matrix as a linear combination of the three Pauli Matrices and the $2\times 2$ unit matrix?
Any example for the same might help ?
| A slow construction would go...
$$$$
$$
\begin{pmatrix}a&b\\c&d\end{pmatrix}
=
a\begin{pmatrix}1&0\\0&0\end{pmatrix}
+b\begin{pmatrix}0&1\\0&0\end{pmatrix}
+c\begin{pmatrix}0&0\\1&0\end{pmatrix}
+d\begin{pmatrix}0&0\\0&1\end{pmatrix}
$$
$$
\begin{pmatrix}1&0\\0&0\end{pmatrix}
=\frac{1}{2}
\begin{pmatrix}1&0\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 3,
"answer_id": 0
} |
Equivalent Rotation using Baker-Campbell-Hausdorff relation Is there a way in which one can use the BCH relation to find the equivalent angle and the axis for two rotations? I am aware that one can do it in a precise way using Euler Angles but I was wondering whether we can use just the algebra of the rotation group to... | As $\mathrm{SO}(3)$ is a connected group, $\exp(\mathsf{L}(\mathrm{SO}(3))) = \mathrm{SO}(3)$ and hence this should – in theory – work. Let us work in the fundamental representation of $\mathrm{SO}(3)$, that is orthogonal 3x3 matrices.
Assume you have a rotation $B$ acting first and a second rotation $A$, the resultin... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/29100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 1
} |
$N$ coupled quantum harmonic oscillators I want to find the wave functions of $N$ coupled quantum harmonic oscillators having the following hamiltonian:
\begin{eqnarray}
H &=& \sum_{i=1}^N \left(\frac{p^2_i}{2m_i} + \frac{1}{2}m_i\omega^2 x^2_i + \frac{\kappa}{2} (x_i-x_{i+1})^2 \right)\,, \qquad x_{N+1}=0\,,\\
&=& \... | (If I did not make obvious algebra errors) the elegant solution to this problem is to realize that the matrix
$$
\Lambda=\left(\begin{array}{lllr}
0&1&0\ldots&0\\
0&0&1&\ldots 0\\
\vdots&\vdots&\vdots&1\\
1&0&0\ldots&0
\end{array}\right)
$$
actually commutes with $H$ since $\Lambda$ basically maps $x_{i+1}$ to $x_i$.
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/209424",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "7",
"answer_count": 2,
"answer_id": 1
} |
Projectiles on inclined planes with coefficient of restitution
This is the problem I am currently attempting. So far, I've resolved the velocity parallel and perpendicular to the plane to get, perpendicular: $u \sin \theta$ upon launch and $-u\sin \theta$ on landing.
Parallel: $u \cos \theta - 2u \sin \theta \tan \alp... | @EDIT: SOLVED IT! :) The key part I was missing was that $e$ only acts on the component perpendicular to the slope (the y-component), i.e $u_{r,x} = v_x$ and $u_{r,x} = -e v_y$. Huge thanks to @Floris for spotting this, and all the help!!
Always start with a diagram :)
I tried solving this two ways: relative to vertic... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Non-zero components of the Riemann tensor for the Schwarzschild metric Can anyone tell me which are the non-zero components of the Riemann tensor for the Schwarzschild metric?
I've been searching for these components for about 2 weeks, and I've found a few sites, but the problem is that each one of them shows differen... | According to Mathematica, and assuming I haven't made any silly errors typing in the metric, I get the non-zero components of $R^\mu{}_{\nu\alpha\beta}$ to be:
{1, 2, 1, 2} -> (2 G M)/(r^2 (-2 G M + c^2 r)),
{1, 2, 2, 1} -> -((2 G M)/(r^2 (-2 G M + c^2 r))),
{1, 3, 1, 3} -> -((G M)/(c^2 r)),
{1, 3, 3, 1} -> (G M)/(c^2 ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/295814",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
} |
Scattering amplitude with a change in basis of fields Suppose I know the Feynman rules for the scattering process $\pi^j \pi^k \rightarrow \pi^l \pi^m$ where $j,k,l,m$ can be $1, 2$ or $3$. Define the charged pion fields as $\pi^\pm=\frac{1}{\sqrt{2}}(\pi^1 \pm i \pi^2)$ and neutral pion field as $\pi^0=\pi^3$. I would... | I assume your Lagrangian might have a term of the form $\bar{N}\vec{\pi}.\vec{\tau}\gamma^5N$, or something with $\vec{\pi}.\vec{\tau}$. One method is to expand the following and see how $\pi^+$ and $\pi^-$ come into your Lagrangian,
\begin{align*}
\vec{\pi}.\vec{\tau} &= \pi^1\sigma^1 + \pi^2\sigma^2 +\pi^3\sigma^3 \\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/402495",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Computation of the self-energy term of the exact propagator for $\varphi^3$ theory in Srednicki In M. Srednicki "Quantum field theory", Section 14 -Loop corrections to the propagator-, the exact propagator $\mathbf {\tilde \Delta} (k^2)$ is stated as
$$\frac{1}{i} \mathbf {\tilde \Delta} (k^2) = \frac{1}{i} \tilde \D... | It is very easy to integrate it with Mathematica. From Eq.(14.14) and Eq.(14.42) we have $D=x(1-x)k^2+m^2$ and $D_0=[1-x(1-x)]m^2$, so
$$
\frac\alpha2\int_0^1\text{d}x\ D\ln\frac D{D_0}=\frac\alpha{12}\left[4(k^2+m^2)-\sqrt3(k^2+2m^2)\pi+2\sqrt{\frac{(k^2+4m^2)^3}{k^2}}\arctan\sqrt{\frac{k^2}{k^2+4m^2}}\right].
$$
Ther... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/485832",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Issues with Feynman parameters As a sanity check, I have tried to evaluate a Feynman parameter integral, and have been unable to reproduce the textbook result. I wish to verify the identity
$$\frac{1}{ABC} = \int\limits_0^1\int\limits_0^1\int\limits_0^1dxdydz\frac{2\delta(x+y+z-1)}{[Ax + By + Cz]^3} ~\hat{=}~I.$$
We ca... | The problem is in the very first step sadly. When you resolve the $\delta$ function you are putting
$$
z = 1 - x - y\,.
$$
This will hold only when $x+y \leq 1$ because you are integrating only in the region $z \in [0,1]$. In other words, the zero of the $\delta$ function sometimes falls outside of the region of integr... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/551626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding common eigenvectors for two commuting hermitian matrices Let $A = \begin{bmatrix}
1 &0 &0 \\
0& 0& 0\\
0&0 &1
\end{bmatrix}$ and $B = \begin{bmatrix}
0 &0 &1 \\
0& 1& 0\\
1&0 &0
\end{bmatrix}$ the representation of two hermitian operators in a $(\phi_{1},\phi_{2},\phi_{3})$ basis. Find a commo... | Just use the matrix of the eigenvectors of B:
$$
U = \left( \begin{matrix}
1&1 &0 \\
0& 0&1\\
1& -1&0
\end{matrix} \right)
$$
With this matrix, you find that:
$$
U^{-1}AU = \left( \begin{matrix}
1&0&0 \\
0& 1&0\\
0& 0& 0
\end{matrix} \right)$$
and $$
U^{-1}BU = \left( \begin{matrix}
1&0&0 \\
0& -1&0\\
0& 0& 1
\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/723538",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Having trouble understanding Re-derivation of Young’s Equation I was going over a section 2.1 of this article (regarding Young's Equation and for some reason I wasn't able to derive E10 from E9 (see image below). I know it is more about the trigonometry of the problem, but when I tried to derive it myself it became cum... | The first parenthesis in E9 is simplified into
$$
\begin{aligned}
&\sin^2\theta + \cos\theta \frac{-(1 - \cos\theta)(2 + \cos\theta)}{1 + \cos\theta} \\
= &(1 - \cos^2\theta) - \cos\theta \frac{2 - \cos\theta - \cos^2\theta}{1 + \cos\theta} \\
= &1 - \cos\theta \frac{\cos\theta(1 + \cos\theta) + (2 - \cos\theta - \cos^... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/697730",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Verification of the Poincare Algebra The generators of the Poincare group $P(1;3)$ are supposed to obey the following commutation relation to be verified:
$$\left[ M^{\mu\nu}, P^{\rho} \right] = i \left(g^{\nu\rho} P^{\mu} - g^{\mu\rho} P^{\nu} \right)$$
where $M^{\mu\nu}$ are the 6 generators of the Lorentz group and ... | Consider $$M_{31} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -i & 0 \\
0 & 0 & 0 & 0 & 0 \\
0 & i & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0
\end{pmatrix} \text{ and } P_0 = -i \begin{pmatrix}
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/127690",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What does this imaginary number mean for time and velocity? As some have pointed out in the chat, perhaps the question that I should have asked is, am I really integrating for velocity? My integration might be misleading in that it integrates for something but probably not velocity. Velocity could not be in the same di... | Complex velocity doesn't make sense in physics so you have to choose the parameters $a,b,c$ so you don't get an imaginary velocity.
\begin{align*}
& \sqrt{a v^4- b v^2+\frac{c}{a}}\quad\Rightarrow\quad a v^4- b v^2+\frac{c}{a} \ge 0\\
&v^2 \mapsto x\quad\Rightarrow\\ &g_1=a x^2- b x+\frac{c}{a} \ge 0\\
&g_1=(x-\ta... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/411362",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
The Electromagetic Tensor and Minkowski Metric Sign Convention I am trying to figure out how to switch between Minkowski metric tensor sign conventions of (+, -, -, -) to (-, +, +, +) for the electromagnetic tensor $F^{\alpha \beta}$.
For the convention of (+, -, -, -) I know the contravariant and covarient forms of t... | I use this way:
\begin{equation}\tag{1}
F_{ab} = \partial_a \, A_b - \partial_b \, A_a,
\end{equation}
where
\begin{equation}\tag{2}
A^a = (\phi, \, A_x, \, A_y, \, A_z), \qquad\qquad A_a = (\phi, - A_x, - A_y, - A_z).
\end{equation}
Then, we have:
\begin{align}
E_i &= \Big( -\, \vec{\nabla} \, \phi - \frac{\partial \... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/476673",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Landau-Lifshitz Equation of Motion for Triangular Heisenberg Antiferromagnet There is a paper (PhysRevB.95.014435) in which the dispersion relation for some Heisenberg model on the honeycomb lattice is derived from the Landau-Lifshitz equation:
\begin{align}
\frac{d S_i}{dt} = - S_i \times \mathcal H_{\rm eff}
\end{al... | I see two possible problems in your consideration.
*
*You've investigated perturbations of ferromagnetic ground state. When spin variations $\delta m$ are zeros, spins on three sublattices are the same:
$$
S_i = (0, 0, 1),\quad \forall i.
$$
*The Landau-Lifshitz equation is a nonlinear one. Effective field ${\cal H... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/589583",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Problem with the proof that for every timelike vector there exists an inertial coordinate system in which its spatial coordinates are zero I am reading lecture notes on special relativity and I have a problem with the proof of the following proposition.
Proposition. If $X$ is timelike, then there exists an inertial coo... | Actually, your matrix can be greatly simplified as
$$
M =
\begin{bmatrix}
\frac{1}{\sqrt{a^2 - p^2}} a & \frac{p}{\sqrt{a^2 - p^2}} & 0 & 0 \\
\frac{p}{\sqrt{a^2 - p^2}} & \frac{a}{\sqrt{a^2 - p^2}} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}
$$
since $(\mathbf{e}, \mathbf{q}, \mathbf{r})$ forms an ortho... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/592938",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 3,
"answer_id": 0
} |
Dimensional regularization: order of integration This is a two-loop calculation in dim reg where I seem to be getting different results by integrating it in different orders. I am expanding it about $D=1$. What rule am I breaking?
$$\int \frac{d^{D} p}{(2\pi)^D}\frac{d^{D}q}{(2\pi)^D}\frac{p^2+4m^2}{(q^2+m^2)((q-p)^2+m... | I'd say, when you set
$$
\int \frac{d^{D}q}{(2\pi)^D}\frac{1}{(q^2+m^2)((q-p)^2+m^2)}=\frac{1}{m}\frac{1}{p^2+4m^2}
$$
the correct answer actually has a $+\mathcal O(d-1)$ piece. The $p$ integral has $1/(d-1)$ divergences which, when multiplied by the missing subleading piece, leaves a finite contribution.
We can do th... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/619289",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 1,
"answer_id": 0
} |
Balmer proportionality How did Johannes Balmer arrive at
$$
\lambda \propto \frac{n^2}{n^2-4}, \quad (n=3,4,\dots),
$$
and then how did Rydberg mathematically derive
$$
\frac{1}{\lambda}=R\left(\frac{1}{n^2_1}-\frac{1}{n^2_2}\right)?
$$
I know $n$ stands for the shells but in the textbook, it doesn't define what $n$ is... | I recommend reading Balmer's original paper "Notiz über die Spektrallinien des Wasserstoffs" (1885).
Balmer took the known wavelengths of the visible hydrogen spectrum
($H_\alpha$, $H_\beta$, $H_\gamma$, $H_\delta$) as measured by
Ångström with high precision.
He recognized they are related by certain fractions.
$$\beg... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/734989",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
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Height of the atmosphere - conflicting answers Okay. I have two ways of working out the height of the atmosphere from pressure, and they give different answers. Could someone please explain which one is wrong and why? (assuming the density is constant throughout the atmosphere)
1) $P=h \rho g$, $\frac{P}{\rho g} = h = ... | Neither calculation is anything approaching physically realistic, but I guess you know that and you're just interested in why the two approaches give different answers.
Take your equation from your second method:
$$ 4\pi r^2 P = \rho V g $$
If the area is a flat sheet you have $V = Ah$ and $A = 4\pi r^2$, and substitut... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/33627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 0
} |
Derivation of Total Momentum Operator for Klein-Gordon Field Quantization I am studying the second chapter of Peskin and Schroeder's QFT text. In equation 2.27 and 2.28, the book defines the field operators:
$$
\phi(x) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{\sqrt{2 w_p}} (a_p + a^\dagger_{-p}) \, e^{ipx} \\
\pi(x) = \in... | From
$$\mathbf{P} = \int \frac{d^3p}{(2\pi)^3} \frac{\mathbf{p}}{2} \bigg( a_p a^\dagger_{p} -a^\dagger_{-p} a_{-p}\bigg)$$
you actually have $$
\mathbf{P} = \int \frac{d^3p}{(2\pi)^3} \left(\frac{\mathbf{p}}{2}a_p a^\dagger_{p} - \frac{\mathbf{p}}{2}a^\dagger_{-p} a_{-p}\right) = \int \frac{d^3p}{(2\pi)^3} \left(\fra... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/375585",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "14",
"answer_count": 1,
"answer_id": 0
} |
Proof of form of 4D rotation matrices I am considering rotations in 4D space. We use $x, y, z, w$ as coordinates in a Cartesian basis. I have found sources that give a parameterization of the rotation matrices as
\begin{align}
&R_{yz}(\theta) =
\begin{pmatrix}
1&0&0&0\\0&\cos\theta&-\sin\theta&0\\0&\sin\th... | For each of your 4D rotation matrix $~\mathbf R~$ if this equation
$$\mathbf Z^T\, \mathbf Z= \left(\mathbf R\,\mathbf Z\right)^T\,\left(\mathbf R\,\mathbf Z\right)$$
is fulfilled the rotation matrix $~\mathbf R~$ is orthonormal .$~\mathbf R^T\,\mathbf R=\mathbf I_4$
where
$$\mathbf Z= \begin{bmatrix}
x \\
y \\
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/652658",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 3,
"answer_id": 2
} |
Height of the atmosphere - conflicting answers Okay. I have two ways of working out the height of the atmosphere from pressure, and they give different answers. Could someone please explain which one is wrong and why? (assuming the density is constant throughout the atmosphere)
1) $P=h \rho g$, $\frac{P}{\rho g} = h = ... | The first formula is just a first order expansion in $1/r$ of the second formula which is thus the exact one.
The expansion is:
$$h = \frac{P}{\rho g} - \left( \frac{P}{\rho g} \right)^2 \frac{1}{r} + \frac{5}{3} \left( \frac{P}{\rho g} \right)^3 \frac{1}{r^2} + \ldots$$
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/33627",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
How to get "complex exponential" form of wave equation out of "sinusoidal form"? I am a novice on QM and until now i have allways been using sinusoidal form of wave equation:
$$A = A_0 \sin(kx - \omega t)$$
Well in QM everyone uses complex exponential form of wave equation:
$$A = A_0\, e^{i(kx - \omega t)}$$
QUESTION... | You asked about the second equation. See below:
$e^{ix}{}= 1 + ix + \frac{(ix)^2}{2!} + \frac{(ix)^3}{3!} + \frac{(ix)^4}{4!} + \frac{(ix)^5}{5!} + \frac{(ix)^6}{6!} + \frac{(ix)^7}{7!} + \frac{(ix)^8}{8!} + \cdots \\[8pt]
{}= 1 + ix - \frac{x^2}{2!} - \frac{ix^3}{3!} + \frac{x^4}{4!} + \frac{ix^5}{5!} - \frac... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/53005",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "5",
"answer_count": 3,
"answer_id": 1
} |
Types of photon qubit encoding How many types of qubit encoding on photons exist nowadays? I know only two:
*
*Encoding on polarization:
$$ \lvert \Psi \rangle = \alpha \lvert H \rangle + \beta \lvert V \rangle $$
$$ \lvert H \rangle = \int_{-\infty}^{\infty} d\mathbf{k}\ f(\mathbf{k}) e^{-iw_k t} \hat{a}^\dagger_{H... | Yeah there is a couple of 'em - off the top of my head I can think of:
\begin{align}
|\uparrow\;\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|\downarrow\;\rangle\;=\;
\begin{pmatrix}
0 \\ 1
\end{pmatrix}\\
|g\rangle\;=\;
\begin{pmatrix}
1 \\ 0
\end{pmatrix}\qquad&\qquad
|e\rangle\;=\;
\begin{pmatrix}
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/59731",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "6",
"answer_count": 2,
"answer_id": 1
} |
What is the depth in meters of the pond? A small spherical gas bubble of diameter $d= 4$ μm forms at the bottom of a pond. When the bubble rises to the surface its diameter is $n=1.1$ times bigger. What is the depth in meters of the pond?
Note: water's surface tension and density are $σ= 73 \times 10^{-3} \mbox{ N}$ a... | The surface tension of the bubble would help you to find out how much work the bubble is doing while it increases its volume.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/82920",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
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Apparent dimensional mismatch after taking derivative Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them.
Now, we wish to take the derivative of $u = log (\frac{x^2}{a^2})... |
Where am I going wrong?
Recall
$$\frac{d}{dx}f(g(x))=f'(g(x))g'(x)$$
with
$$f(\cdot) = \ln(\cdot) \rightarrow f'(\cdot) = \frac{1}{\cdot}$$
and
$$g(x) = \frac{a^2}{x^2} \rightarrow g'(x) = \frac{-2a^2}{x^3}$$
Thus
$$\frac{d}{dx}\ln\frac{a^2}{x^2} = \frac{1}{\frac{a^2}{x^2}}\frac{-2a^2}{x^3} = -\frac{2}{x}$$
An alter... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/113715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 0
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Force distribution on corner supported plane This question has been annoying me for a while. If you have a completely ridged rectangular plate of width and height x and y that is supported on each corner (A,B,C,D) and has force (F) directly in its center then I think the force on each corner support will be F/4.
What ... | I think I have a solution. Considering the corner forces $A$, $B$, $C$ and $D$ you have a system of 3 equations and 4 unknowns
$$\begin{align}
A + B + C + D & = F \\
\frac{y}{2} \left(C+D-A-B\right) &= 0 \\
\frac{x}{2} \left(A+D-B-C\right) & = 0
\end{align}$$
$$\begin{vmatrix}
1 & 1 & 1 & 1 \\
-\frac{y}{2} &... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/243626",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Simple Electric Field Problem Solve the Electric Field distance z above a circular loop of radius r. The charge/length = $\lambda$
The arc-length is 2$\pi$r. So the smallest portion of the circle is 2$\pi r \delta \theta$ and charge is therefore
\begin{align}
q&=2\pi r \delta \theta*\lambda
\\
R&=\sqrt{r^2+z^2}= \tex... | The length element should be $r d\theta$ not $2\pi r d\theta$. So the charge element is
$$dq=\lambda r d\theta$$
but not
$$dq=\lambda 2\pi r d\theta.$$
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/288654",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
How do one show that the Pauli Matrices together with the Unit matrix form a basis in the space of complex 2 x 2 matrices? In other words, show that a complex 2 x 2 Matrix can in a unique way be written as
$$
M = \lambda _ 0 I+\lambda _1 \sigma _ x + \lambda _2 \sigma _y + \lambda _ 3 \sigma_z
$$
If$$M = \Big(\begin{... | Let $M_2(\mathbb{C})$ denote the set of all $2\times2$ complex matrices. We also note that dim$(M_2(\mathbb{C}))=4$, because if $M\in M_2(\mathbb{C})$ and
$M=\left(
\begin{array}{cc}
z_{11} & z_{12}\\
z_{21} & z_{22} \\
\end{array}
\right)$, where $z_{ij}\in \mathbb{C}$,
then
$M=\left(
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/292102",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "13",
"answer_count": 4,
"answer_id": 0
} |
Zero mass Kerr metric When mass in Kerr metric is put to zero we have $$ds^{2}=-dt^{2}+\frac{r^{2}+a^{2}\cos^{2}\theta}{r^{2}+a^{2}}dr^{2}+\left(r^{2}+a^{2}\cos^{2}\theta\right)d\theta^{2}+\left(r^{2}+a^{2}\right)\sin^{2}\theta d\phi^{2},$$
where $a$ is a constant. This is a flat metric. What exactly is the coordinate ... | As mentioned in @Umaxo's comment,
according to Boyer-Lindquist coordinates - Line element:
The coordinate transformation from Boyer–Lindquist coordinates
$r,\theta,\phi$ to Cartesian coordinates $x,y,z$ is given
(for $m\to 0$} by:
$$\begin{align}
x &= \sqrt{r^2+a^2}\sin\theta\cos\phi \\
y &= \sqrt{r^2+a^2}\sin\the... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/490819",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 1
} |
Manipulation of the diffusive term in MHD induction equation I am trying to solve the magnetohydrodynamic (MHD) equations with a spatially varying resistivity, $\eta$. To remove some of the numerical stiffness from my finite volume approach, I am trying to get rid of these curl expressions with some vector calculus ide... | There are several vector/tensor calculus rules that will come in handy, so I will defined them here first (in no particular order):
$$
\begin{align}
\nabla \cdot \left[ \nabla \mathbf{A} - \left( \nabla \mathbf{A} \right)^{T} \right] & = \nabla \times \left( \nabla \times \mathbf{A} \right) \tag{0a} \\
\nabla \cdot... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/542898",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Finding Locally flat coordinates on a unit sphere I know this is more of a math question, but no one in the Mathematics community was able to give me an answer, and since physicists are familiar with General Relativity, I thought I might get an answer.
Imagine a unit sphere and the metric is:
$$ds^2 = d\theta ^2 + \cos... | starting with components of the unit sphere :
\begin{align*}
&\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}=\left[ \begin {array}{c} \cos \left( \phi \right) \sin \left( \theta
\right) \\ \sin \left( \phi \right) \sin \left(
\theta \right) \\ \cos \left( \theta \right)
\end {array} \right]
\end{align*}
from... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/718912",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Comparing Static Frictions In this figure, which of the static frictional forces will be more?
My aim isn't to solve this particular problem but to learn how is static friction distributed . Since each of the rough-surfaces are perfectly capable of providing the $-1N$ horizontal frictional force but why don't they ? T... | The contact forces with two blocks are $N_1 = m_1 g + m_2 g$ for the bottom block (to the floor) and $N_2 = m_2 g$ for the top block (to the 1st block).
The available traction is $F^\star_1 = \mu_1 (m_1+m_2)\,g$ and $F^\star_2 = \mu_2 m_2\, g$ or
$$ \begin{pmatrix}F_1^\star\\F_2^\star\end{pmatrix} = \begin{bmatrix}1&-... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/64780",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 4,
"answer_id": 2
} |
Moment of inertia of a hollow sphere wrt the centre? I've been trying to compute the moment of inertia of a uniform hollow sphere (thin walled) wrt the centre, but I'm not quite sure what was wrong with my initial attempt (I've come to the correct answer now with a different method). Ok, here was my first method:
Consi... | The mass of the ring is wrong. The ring ends up at an angle, so its total width is not $dx$ but $\frac{dx}{sin\theta}$
You made what I believe was a typo when you wrote
$$\text{d}m = \frac{M}{4\pi R^2}\cdot 2\pi \left(R^2 - x^2 \right)\text{d}x$$
because based on what you wrote further down, you intended to write
$$\te... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/109761",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 1,
"answer_id": 0
} |
Physical reason for Lorentz Transformation Seeing the mathematical derivation of the Lorentz Transformation for time coordinates of an event for two observers we get the term
$$t'=-\frac{v/c^2}{\sqrt{1-\frac{v^2}{c^2}}}x+\frac1{\sqrt{1-\frac{v^2}{c^2}}}t$$
Now how to make sense physically of the $t-\frac{vx}{c^2}$ fact... | The physical reason IS the constancy of the velocity of light... since I'm writing in a tablet the answer won't be complete, but expect to get you to the mathematical cross-road.
Constancy of velocity of light implies that
\begin{equation}
\frac{d|\vec{x}|}{dt} = c, \quad\Rightarrow\quad d|\vec{x}| = c\,dt.
\end{equa... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/131100",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Stability and Laplace's equation Consider four positive charges of magnitude $q$ at four corners of a square and another charge $Q$ placed at the origin. What can we say about the stability at this point?
My attempt goes like this. I considered 4 charged placed at $(1,0)$, $(0,1)$, $(-1,0)$, $(0,-1)$ and computed the p... | The Coulomb Potential is a solution to Laplace's equation in 3 dimensions. In 2 dimensions the equivalent solution is a logarithmic potential. You have written down the Coulomb potential for 4 charges but then treat the problem as 2 dimensional, which is causing your problems. To resolve this you need to add a load of ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/198094",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
What is the minimal G-force curve in 2-dimensional space? Given two parallel roads, which need to be connected, what shape of curve would produce the minimum overall horizontal G-force(s) on travelers?
Is it a $sin$ or $cos$ wave?
Is it a basic cubic function?
Is it something else?
I'm working on an engineering projec... | The answer is two arcs. One arc with a constant gee loading in one direction and then flipping to the opposite direction. This is called the bang-bang method, and it is no very smooth, but the gee forces never exceed the specified maximum.
Given a path $y(x)$ the instantaneous radius of curvature at each x is
$$ \rho =... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/250905",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Product of two Pauli matrices for two spin $1/2$ In the lecture, my professor wrote this on the board
$$
\begin{equation}
\begin{split}
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})|++\rangle &= |++\rangle \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(\blacktriangledown)\\
(\vec{\sigma}_{1}\cdot\vec{\sigma}_{2})(|+-\rangle+|-+\rangle) &... | Your expression for:
$$(\vec \sigma_1 \cdot \vec \sigma_2) |+\rangle_1 \otimes |+\rangle_2=\vec \sigma_1 |+\rangle\otimes \vec \sigma_2 |+\rangle_2$$
Is wrong. It sould read:
$$(\vec \sigma_1 \cdot \vec \sigma_2) |+\rangle_1 \otimes |+\rangle_2=\sigma_{1x}|+\rangle_1\otimes \sigma_{2x}|+\rangle_2+\sigma_{1y}|+\rangle_... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/260916",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Angular velocity by velocities of 3 particles of the solid Velocities of 3 particles of the solid, which don't lie on a single straight line, $V_1, V_2, V_3$ are given (as vector-functions). Radius-vectors $r_1, r_2$ from third particle to first and second are given aswell. How could I find the angular velocity $w$ of... | The algebra is not especially nice, but it is just algebra. This is rigid body rotation, taking point 3 as the origin of coordinates,
so effectively
$$\mathbf{r}_1=\mathbf{R}_1-\mathbf{R}_3, \qquad
\mathbf{r}_2=\mathbf{R}_2-\mathbf{R}_3.
$$
We start as you suggested, and abbreviate
$$
\mathbf{v}_1=\mathbf{V}_1-\mathbf... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/431674",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
A certain regularization and renormalization scheme In a certain lecture of Witten's about some QFT in $1+1$ dimensions, I came across these two statements of regularization and renormalization, which I could not prove,
(1) $\int ^\Lambda \frac{d^2 k}{(2\pi)^2}\frac{1}{k^2 + q_i ^2 \vert \sigma \vert ^2} = - \frac{1}... | Let's just look at the integral
$$\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2}.$$
The other integrals should follow from this one.
Introduce the Pauli-Villars regulator,
$$\begin{eqnarray*}
\int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2}
&\rightarrow& \int \frac{d^2k}{(2\pi)^2} \frac{1}{k^2+\alpha^2} - \int... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/28194",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "4",
"answer_count": 2,
"answer_id": 0
} |
Is the spin 1/2 rotation matrix taken to be counterclockwise? The spin 1/2 rotation matrix around the $z$-axis I worked out to be
$$
e^{i\theta S_z}=\begin{pmatrix}
\exp\frac{i\theta}{2}&0\\
0&\exp\frac{-i\theta}{2}\\
\end{pmatrix}
$$
Is this taken to be anti-clockwise around the $z$-axis?
| For your example, we have $e^{i\theta S_z}\mathbf{S}e^{-i\theta S_z}=\begin{pmatrix}\cos\theta & -\sin\theta&0\\\sin\theta & \cos\theta&0\\0&0&1\end{pmatrix}\mathbf{S}$, with $e^{i\theta S_z}=\begin{pmatrix}e^{i\frac{\theta }{2}} & 0\\ 0 & e^{-i\frac{\theta }{2}}\end{pmatrix}$ and $\mathbf{S}=\begin{pmatrix}S_x\\ S_y\... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/91483",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "2",
"answer_count": 2,
"answer_id": 0
} |
Apparent dimensional mismatch after taking derivative Suppose I have a variable $x$ and a constant $a$, each having the dimension of length. That is $[x]=[a]=[L]$ where square brackets denote the dimension of the physical quantity contained within them.
Now, we wish to take the derivative of $u = log (\frac{x^2}{a^2})... | I think the second half of your derivative is wrong:
$ \frac{d}{dx} \log\left( \frac{a^2}{x^2}\right) = \frac{x^2}{a^2} \cdot \frac{d}{dx} \left(a^2 x^{-2}\right) = \frac {x^2} {a^2} \left(-3a^2\right) x^{-3} = \frac{-3 a^4}{x} $
which has the correct dimension.
| {
"language": "en",
"url": "https://physics.stackexchange.com/questions/113715",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 3,
"answer_id": 2
} |
Line element in Kruskal coordinates I try to calculate the line element in Kruskal coordinates, these coordinates use the Schwarzschild coordinates but replace $t$ and $r$ by two new variables.
$$
T = \sqrt{\frac{r}{2GM} - 1} \ e^{r/4GM} \sinh \left( \frac{t}{4GM} \right) \\
X = \sqrt{\frac{r}{2GM} - 1} \ e^{r/4GM} \co... | I don't think you can drive the line element with the jacobian $J$
The Kruskal-Szekeres line element
Beginning with the Schwarzschild line element:
\begin{align*}
&\boxed{ds^2 =\left(1-\frac{r_s}{r}\right)\,dt^2-\left(1-\frac{r_s}{r}\right)^{-1}\,dr^2-r^2\,d\Omega^2}\\\\
r_s &:=\frac{2\,G\,M}{c^2} \,,\quad
... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/407108",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 2,
"answer_id": 0
} |
Potential due to line charge: Incorrect result using spherical coordinates Context
This is not a homework problem. Then answer to this problem is well known and can be found in [1]. The potential of a line of charge situated between $x=-a$ to $x=+a$ ``can be found by superposing the point charge potentials of infinit... |
Adjusting from [1] $(b→a)$, the answer to the problem below is
$$\Phi{\left(r, \frac{\pi}{2},\pi\pm \frac{\pi}{2}\right)}
=
\frac{1}{4\,\pi\,\epsilon_o}
\frac{Q}{ a}
\left[
\ln{\left(\frac{ a+ \sqrt{a ^2 + r^2}}{ -a+ \sqrt{a ^2 + r^2}}\right)}
\right]$$
This is wrong by a factor of 2, in the original answer they... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/749536",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
The analytical result for free massless fermion propagator For massless fermion, the free propagator in quantum field theory
is
\begin{eqnarray*}
& & \langle0|T\psi(x)\bar{\psi}(y)|0\rangle=\int\frac{d^{4}k}{(2\pi)^{4}}\frac{i\gamma\cdot k}{k^{2}+i\epsilon}e^{-ik\cdot(x-y)}.
\end{eqnarray*}
In Peskin & Schroeder's bo... | The calculation of the propagator in four dimensions is as follows.
\begin{eqnarray*}
\int\frac{d^4 k}{(2\pi)^4}e^{-ik\cdot (x-y)}\frac{1}{k^2}
&=& i\int \frac{d^4 k_E}{(2\pi)^4}e^{ik_E\cdot (x_E-y_E)}\frac{1}{-k_E^2} \\
&=& \frac{-i}{(2\pi)^4} \left( \int_0^{2\pi}d\theta_3 \int_0^{\pi}d\theta_2 \sin \theta_2 \righ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/263846",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 4,
"answer_id": 3
} |
Does the unit hypercube in Minkowski space always have the 4-volume of 1? Suppose we have a unit hypercube in Minkowski space defined by the column vectors in the identity matrix $$ \mathbf I = \begin{bmatrix}
1 & 0 & 0 & 0 \\[0.3em]
0 & 1 & 0 & 0 \\[0.3em]
0 & 0 & 1 &... | In terms of matrix components, Lorentz transformations have matrices that satisfy
$$\eta = \Lambda^T \eta \Lambda$$
where $\eta$ is the Minkowski metric. Taking determinants of each side, we have
$$|\eta| = |\Lambda^T| |\eta| |\Lambda| = |\Lambda|^2 |\eta|$$
which implies that $|\Lambda| = \pm 1$. Since a transformatio... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/283186",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Hamiltonian for a magnetic field An atom has an electromagnetic moment, $\mu = -g\mu_B S$ where S is the electronic spin operator ($S=S_x,S_y.S_z$) and $S_i$ are the Pauli matrices, given below. The atom has a spin $\frac{1}{2}$ nuclear magnetic moment and the Hamiltonian of the system is
\begin{gather*}
H = -\mu ... | The Hamiltonain is calculated as
\begin{align}
H =& \, g \mu_B \, \left(B_x S_x + B_y S_y + B_z S_z\right) \, + \, \frac{1}{2}A_0 S_z = \\
=& \, \frac{g \mu_B}{2} \, \left(B_x \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} + B_y \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix} + B_z \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/440351",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "3",
"answer_count": 1,
"answer_id": 0
} |
Optimizing engines to produce a certain torque and net force Say we have $n$ engines sitting on a rigid body. Each engine has position $R_i$ and points in a certain direction and generates a force in that direction, $F_i$. The magnitude of that force ($k_i$) follows this constraint: $0 \leq k_i \leq m_i$. In other word... | The problem is one of least squares (until the point where the magnitude is capped).
Consider the target force vector $\vec{F}$ and the target moment vector $\vec{T}$ as the right-hand side $\boldsymbol{b}$ of a linear system of equations, and the vector $\boldsymbol{x}$ of $n$ force magnitudes is the unknowns.
$$ \mat... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/571117",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 1,
"answer_id": 0
} |
Subsets and Splits
Fractions in Questions and Answers
The query retrieves a sample of questions and answers containing the LaTeX fraction symbol, which provides basic filtering of mathematical content but limited analytical insight.