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I know, the answer to this question may seem obvious: The resolution/magnification of an optical microscope is limited by the minimum wavelength one uses. This is due to the diffraction limit.
However, there are different types of optical microscopes. The classical type shines light on a sample and looks at the reflect... |
During big bang nucleosynthesis (BBN), deuterium has a lower binding energy per nucleon (~1.1 MeV) than the other similar nuclei, and so prevents heavy elements from forming until the temperature drops below about 0.1 MeV. Above 0.1 MeV, deuterium is unstable and will be broken apart by photodisintegration. This is the... |
The intensities should be equal no matter how a wave is represented. So clearly I think i'm making some elementary mistake, it seems they are not same :
$$
\Phi(x,t) = A_0 \cos{(kx-wt)} \\ \Phi(x,t) = A_o e^{i(kx-wt)} \qquad \text{complex notation}
$$
So in the above it is understood that the physical part of the wave ... |
I would like to use the Greens Function $G(t,t')$, satisfying \begin{equation} \int G^{-1}(t,s)G(s,t') ds = \left( \begin{array}{cc} \delta(t-t') & 0 \\ 0 & \delta(t-t') \end{array} \right) \end{equation}
where $$ G^{-1}(t,t') =
\begin{pmatrix}
-(3+\frac{d}{dt})\delta(t-t') & - 2 \pi t
\delta(t-t') \\
2 \pi t \delta(... |
We're currently learning about applying the TISE to one-electron (hydrogen) atoms in my intro to QM class. While reading about it in the textbook, I was a bit confused about radial probability density and distribution functions; mainly as my textbook uses them somewhat interchangeable (which I don't necessarily think i... |
Here is B-2 Spirit, a stealth bomber by Northrop Grumman. Another one is a depiction of a parabolic dish antenna receiver. As we knew, the B-2 is a stealth aircraft, which is not reflects the received electromagnetic (EM) wave signal (of course, there are several way to keep it stealth, such as reflect to another dir... |
The vector space of quantum states $|\psi\rangle$ is a hilbert space $\mathcal{H}$. Now, since the middle 20's of the past century, the quantization procedure states that one of the quantization requirements is the comutation relation, for the position and momentum operators:
$[\hat{q}(t),\hat{p}(t)]=i\hbar\hat{1} \ta... |
The Kinematic transport theorem is a very basic theorem relating time derivatives of vectors between a non rotating frame and another one that's rotating with respect to it with a uniform angular velocity.
I was trying to prove it for the special case of $3$ dimensions, and everything seems straightforward apart from t... |
Let the planes be $x = 0$, $y = 0$ and $z = 0$ and they all have the same potential $V_0$. the question is to find the potential in the region $x > 0$, $y > 0$ and $z > 0$.
I know that from the separation of variables, we can assume that potential function is given by:
\begin{align}
V(x, y, z) = X(x) Y(y) Z(z)
\end{al... |
I came upon a dynamics question in Physics 20. I solved the question, but there was another question in my head that has been bugging me. This is the question below.
I solved the question with relative ease, finding that the range of the mass is between 2.8kg and 9.2kg. However, I realized that as the mass decreased, ... |
My go-to resource for learning Electromagnetism thus far has been Griffiths, which has proven invaluable. On the section on induction, Griffiths makes a very clear distinction between what he refers to as "Faraday's Law", being that:
$\oint \vec E_{induced} \cdot \vec {dl} = -\iint \frac {\partial \vec B}{\partial t}\c... |
I was reading about the intermediate axis theorem and its mathematical proof. Typically one starts with the torque-free Euler's equations
$$
\begin{align}
0&=I_1\dot\omega_1 + (I_3-I_2)\omega_3\omega_2\\
0&=I_2\dot\omega_2 + (I_1-I_3)\omega_1\omega_3\\
0&=I_3\dot\omega_3 + (I_2-I_1)\omega_2\omega_1\\
\end{align}
$$
wi... |
I have the following exercise, and I don't understand, I can't find what this multiplication of sines means for a wave, I understand that it occurs in waves with similar amplitude.
Two vehicles are approaching along the same route, traveling in opposite directions at speeds $v_1$, and $v_2$ respectively. Both vehicles... |
In curved spacetime, the Lorentz factor is different than that in flat spacetime.
Is there any expression that gives the Lorentz factor for any arbitrary metric tensor?
|
In Comments on indeterminism and undecidability the abstract reads:
"In a recent paper 1, it has been claimed that the outcomes of a quantum coin toss which is idealized as an infinite binary sequence is 1-random. We also defend the correctness of this claim and assert that the outcomes of quantum measurements can be ... |
What I understand by Pascal's principle is that the pressure that when there is an increase in pressure at any point in a confined fluid, there is an equal increase at every other point in the container. However, while this should make intuitive sense to me, the notion clashes in my head when I consider also the formul... |
Suppose we have a BeamSplitter (BS, 50:50) and on one of the input sides we have a single photon, and on another side, we have a vacuum then we get the output as $|\psi \rangle_{out} = \frac{1}{\sqrt2}(|1\rangle_{3}|0\rangle_{4}+ |0\rangle_{3}|1\rangle_{4})$. Where we consider output as 3 and 4 and input as 1 and 2. Cl... |
It is very confusing that 1st law be used in inertial frame. For 2nd law we have many different sayings.
On what condition is 2nd law be used in inertial observer?
|
In his Quantum Theory of Fields, volume I, section 11.2, Weinberg gives an estimate of the shift of the energy levels of hydrogenic orbits due to vacuum polarization.
First he obtains the contribution to the potential:
\begin{equation}
\Delta V(r) = \frac{e_1 e_2}{(2\pi)^3}\int d^3 q \; e^{i \mathbf{q} \cdot \mathbf{r}... |
I know its the most basic question, but I do understand that cathode rays are beam of electrons and its not 'light'.
But why can't we see any 'reflected' light from the electron beam i.e if we put (shine) some light on the electron beam then why dont the electrons reflect that light and hence become visible (just like ... |
I'm working on my thesis which is about synthesis of iron oxide nanoparticles from a plant leaf extract and testing the particles activity on bacteria to measure their toxicity towards it. The thing is we didn't get any results and it's concerning because it should give a somewhat noticeable antibacterial effect, so af... |
My question is can we say that the uncertainty in electromagnetic wave due to bandwidth theorem is behind the uncertainty in other interaction, or it is inherent to a particle, because in particle's own frame of reference its spatial momentum is 0 and spatial position is constant,
Is the uncertainty in quantum mechanic... |
Shouldn't the ammeter between A and B read zero because A and B are maintained at the same potential, and for current to flow, a potential difference is required?
On the other hand, the current that entered the resistances has to return to the circuit through AB, which means that the ammeter's reading will be a non-zer... |
I am trying to derive the adiabatic theorem when my time-dependent Hamiltonian is stochastic and I have a few questions. Usually, one starts with the Schrödinger equation
\begin{equation}
i\frac{d |\psi(t)\rangle}{dt}=H_{\rm eff}(t) |\psi(t)\rangle
\end{equation}
My problem is two-fold: On one side, my Hamiltonian is n... |
This is a question from Heat and Thermodynamics by Zemansky and Dittman:
"From the differential equation for the thermodynamic potential A(T, V), derive
expressions for pressure P, entropy S, internal energy U, heat capacity at constant
volume $C_v$, heat capacity at constant pressure $C_p$, volume expansivity, and iso... |
So I was thinking about the fundamental reason why rotation operators don't commute so I started thinking about Euler rotations. I tried the experiment of the rotating frame, first around the x-axis then the y-axis; and the other around the y-axis then the x-axis.
The difference in the directions of a vector in z direc... |
I'm struggling to work out the high temperature limit of the Einstein model for heat capacity. The model is $$C_{v}=\frac{3N\hbar^{2}\omega^{2}}{4kT^{2}\sinh^{2}\left(\frac{\hbar \omega}{2kT}\right)}$$
The high temperature limit is for $kT \gg\hbar\omega$ which and my working out is that in the limit that $T \to \infty... |
I am trying to find the maximum and minimum electron energy for the calculation of the beta spectrum. I understood the minimum energy of the electron is the Q value. What is the maximum energy of the electron in any beta decay?
|
According to this preprint, it seems that there are topologies (like the Klein bottle and the torus) that break some symmetries (like the Lorentz and translation invariances).
Is this right? Can they break more symmetries (like the Poincaré, diffeomorphism and CPT symmetries)?
|
Given a coil moving through a time-independent B field (such that we don't have to worry about induced E fields), we know by the flux law that an emf will be induced over the the conducting loop. If a current flows due to this emf, an additional magnetic force due to this additional motion of charge will act on the coi... |
I'm a bit lost with the following derivation:
\begin{equation}
\begin{aligned}
\left\langle p^{\prime}|x| p^{\prime \prime}\right\rangle & =\int\left\langle p^{\prime}|x| x^{\prime}\right\rangle\left\langle x^{\prime} \mid p^{\prime \prime}\right\rangle d x^{\prime}=\int x^{\prime}\left\langle p^{\prime} \mid x^{\prime... |
Can a pendulum produce harmonic frequencies? Like could I detect harmonic frequencies if I had a sensor on the pendulum?
|
Fermat's Principle is the statement that a ray will follow a minimum-time path between a point, A, to a point, B.
So, if I have a block of material of high refractive index, so that it slows the light considerably inside, the least-time path should avoid going through this block. In the picture below, in the topmost pa... |
I am looking into the section of the book by Peskin and Schroeder in which they connect the $S$-matrix to probabilities.
They start by considering the in state, which is a two-particle state
\begin{equation}
|\phi_{A}\phi_{B}\rangle_{in} \equiv \int \frac{d^3 k_A}{(2\pi)^3}\frac{1}{\sqrt{2E_{k_A}}}\phi_A(k_A) \int \fra... |
In the book Cosmology by Daniel Baumann, the author states that the critical density of the universe at the current time is:
$$\rho_{crit,0}=\dfrac{3H_0^2}{8\pi G}=1.9\cdot 10^{-29}h^2\text{grams}\cdot\text{cm}^{-3}$$
I understand where the theoretical expression comes from, but not the numerical result. My first probl... |
In the article https://arxiv.org/pdf/2009.01937.pdf the term "Information catastrophe" is explained. Suppose the later proposed experiment by this author https://aip.scitation.org/doi/10.1063/5.0087175 turns out to be correct, then the thought experiment in the first article by the author Melvin M. Vopson would also be... |
Bell predicted predetermined (nonlocal) choice as the criteria for a super-deterministic universe.
...our belief that we are free to choose to do one experiment rather
than another, absolutely predetermined - The Ghost in the Atom: A Discussion of the Mysteries of Quantum Physics, by Paul C. W. Davies and Julian R. Br... |
There are many different ideas and concepts related to design of fusion reactors and handling plasma confinement, but what about designing one around the principles of a maelstrom or a tornado? Is that something that has been researched? Any benefits as opposed to other reactor designs you think?
|
Typically, one defines a vielbein $e_\mu^a$ that translates between curved indices $\mu$ and local Lorentz indices $a$, in the sense that
$$g_{\mu\nu}=\eta_{ab}e_\mu^a e_\nu^b,$$
where $\eta$ is the Minkowski metric. Usually, one takes the Minowski metric to be
$$\eta=\text{diag}(-1,+1,...,+1),$$
but, in principle, it ... |
In my physics class, we use coefficients of static and kinetic friction to calculate the forces of friction. I was wondering if there are materials such that they have very high coefficients of static friction but very low coefficients of kinetic friction.
$$ F _{k} = F _{n} \cdot \mu _{k} $$
$$ F _{sMax} = F _{n} \cdo... |
In the book "Group theory and it's Applications to the Quantum Mechanics of atomic spectra " by Eugene P. Wigner
in chapter 4 The elements of quantum mechanics it is written
Consider a many dimensional space with as many coordinates as the system considered as position coordinates. Every arrangement of the positions o... |
Consider three cases that I've drawn. These are taken from my mechanics book. The ellipses are pulleys and the blocks are some weights and strings connect everything. The weight of the strings and pulleys as the friction between pulley and strings are neglected. In the third case there is a friction between the incline... |
In section 10.3 of Weinberg's Volume 1 in deriving LSZ reduction Formula, the author says,
We also define a 'truncated' matrix element $M_l$ by
$$\int d^4 x_2 \cdots e^{-q_2x_2} <\textbf q \sigma| T\{A(x_2)\cdots\}\Omega> = N^{-1} (2\pi)^{-3/2} \sum_l u_l(\textbf q, \sigma) M_l(q_2,\cdots).\tag{10.3.4}$$
Here $ u_l(\... |
There is no 'correct' inertial reference frame according to relativity. Objects are only 'in motion' relative to an arbitrary inertial reference frame. So let us take the following example. A person on Earth jumps. They are now moving at 1 m/s up relative to Earth. As per $K =\frac{1}{2}mv^{2}$, assuming they weigh 70 ... |
I was asking a question about sustainability of reversible computers against entropy. I referenced Landauer's principle to frame my question and someone told me in the comments that Landauer's principle is generally false. So I went digging on the web and eventually found a paper titled Information entropy and thermal ... |
I see that the partition function is associated to the way particles are "partitioned" among energy levels. The equipartition theorem divides (partitions) average energy among degrees of freedom allowed to the particles.
Equating the average energy based on the partition function, given by
$$\langle E\rangle = -\frac{\... |
Suppose I am charging a capacitor, so that a magnetic field is generated. This capacitor is placed inside a dielectric material with permittivity $\epsilon$ and then it is moved parallel to it's plates relative to the dielectric. How will the magnetic field change compared to the case when the capacitor is stationary?
|
Whether the speed of light can only be measured at the observer's place, in his local inertial frame of reference, that is, where the measurement is made. It is about the fact that the observer (that is, his measuring devices) are in his local reference frame, but he is measuring the speed of light in some distant refe... |
Electric charge does not affect spacetime. Mass curves spacetime. This means that if there are two particles A and B, which are spherically symmetrical and have the same mass, even if A is charged and B is uncharged, they both must curve the spacetime in the same way.
There is RN metric for non-rotating charged black h... |
I am confused of what is $R$ in torque is it the distance to the axis of rotation or to a specific point or origin of axis of rotation (pivot)? the forces should treated as lines?
|
Here is a thought experiment:
Since two parallel current carrying wires repel each other, does the same hold true for any element of a circular current carrying wire and thus the wire expands radially outwards?
The problem with this experiment is, that only infinite parallel wires repel each other and that the whole re... |
Some context:
An ideal quantum measurement (in Von Neumann's sense) is described by the following situation:
There is a hilbert space $\mathcal{H_S}$ with basis $|\sigma_i\rangle_{1\leq i \leq n}$ describing the system and a hilbert space $\mathcal{H_M}$ with basis $|\mu_i\rangle_{1\leq i \leq n}$ describing the measur... |
How to decompose the drag force in three components of spherical polar coordinates?
Since
$ F =\frac{1}{2}\rho C A v^2 $
or in vector form :
$\vec F = \vert F\vert \hat F = \frac{1}{2}\rho C A v^2 \hat v = \frac{1}{2}\rho C A v^2 \frac {\hat v}{\vert {\vec v}\vert} = \frac{1}{2}\rho C A v \hat v$
$\dot{r}= \vec v$ = $r... |
Is $B=μ_{0}nI$ for the magnetic field magnitude at the center of the solenoid axis only, or at any point inside the solenoid as long as it is far enough away from its edges as it is suggested by the figure down below?
|
In QED we have one vertex where one line is virtual and the other two are physical:
But recently I came across the so-called Deeply Virtual Compton Scattering in which, after the interaction of an electron with a proton, a physical photon appears, as is possible in QED? Or another field theory is used here?
I also... |
Hi I'm trying to understand basic physics but with a more formal scheme. I'm reading P.K.Kundu book of mechanical fluids. In page 90 he proves that stress tensor is symmetric. But first applies torques to a infinitesimal volume element. I cannot understand why the torque in z direction gives that relation, it does not ... |
I understand echoes are sound reflections. But if foam absorbs sound, and if it reflects off the wall, then that means the waves travel through the foam, and if sound waves get transmitted from a source through the foam and the wall into the next room, doesn't that mean in both cases the sound will have gotten absorbed... |
Consider the potential well with $$U(x) = \begin{cases}-\dfrac{\hbar^2}{m}\kappa_{0}\delta(x), & |x|<a\\+\infty, & |x|\geq a\end{cases}$$
I want to find $E_{n}$. First of all, $U(x)$ is even, can this be useful? What conclusion can one draw from this? Anyway, I started with $$\psi''(x) - \kappa^2\psi(x) = 0$$ when $x\i... |
Imagine a simple circuit consisting of a battery with potential $V$ connected to a resistor with resistance $R$ by a loop of copper wire with $0$ resistance. It is obvious that a current $I=\frac VR$ will flow through the loop. Now, say that the section of wire between the battery and the resistor bifurcates into two s... |
Every single-sided antenna I have looked basically radiates in all directions but has some type of material to reflect the radiated wave to create nulls.
I'm wondering if it's possible to construct a radiator which is truly single sided? Ex a horn-antenna but which doesn't achieve directivity by reflection or cancelati... |
I'm reading Chapter 18 (BEC and Superfluidity) of Girvin and Yang and ran into some confusion.
Let $|\alpha\rangle = e^{-|\alpha|^2}e^{\alpha b^†_0}|0\rangle $, where $\alpha$ is just a complex number. Note that $b_0|\alpha\rangle = \alpha|\alpha\rangle $. Now suppose we do a $U(1)$ symmetry transformation with $b_{\... |
The centripetal force on Earth is constantly exposing Earth to the acceleration. Why can't we feel this change of direction?
|
I am trying to prove an analogue to Fermat's principle that light rays travel along the path of least time for a static metric. First, I want to show that a null geodesic for a static metric (i.e. all metric components are independent of coordinate time and $g_{\mu 0} = 0 \hspace{0.4cm} \forall \mu \neq 0$) can be writ... |
In QFT fields are Hermitian operators. And observables are represented by operators. I am confused are fields also observables?
|
I am a cognitive science graduate student who began to study MRI with a little physics background. Please forgive me if I don't use proper verbs or prepositions.
I think I understood slice selection and frequency encoding. Let's assume that we want to get axial images. In slice selection, gradient magnetic field is app... |
Van de Waals (VdW) forces are intermolecular forces that are for example, due to spontaneous polarisation effects between atoms. As far as I understand, they occur between any type of atom/molecule/material, it does not matter if it’s an insulating or conducting material, if there are filled electronic shells etc.
One... |
Looking into textbooks, I got the impression in renormalization perturbation theory one adds
counterterms to the Lagrangian to cancel terms (usually integrals) that are infinite.
My question is, could one just ignore the infinite integrals (setting them to zero, so to say),
and get the same results as by adding counter... |
For Onsager relation $\sigma_{ij}(B)=\sigma_{ji}(-B)$, why should one flip the sign of the $B$-field for reciprocity?
In the case of antiferromagnetism, with zero net magnetic moment, can the Onsager relation be used $\sigma_{ij}(B)=\sigma_{ji}(-B)$ directly instead $\sigma_{ij}(B,M)=\sigma_{ji}(-B,-M)$?
|
This post is a version of a post with identificator Meta PSE 12909 that I've asked in recent past years on Meta Physics Stack Exchange Asking if the following post could be suitable for the main site Physics Stack Exchange. These first paragraphs are edited verbatim from the post deleted by a moderator on MathOverflow ... |
Im talking about the de Broglie equation-
wavelength $\lambda =\frac{ h} {p}$
I studied a bit about Davisson-Germer experiment that provided conclusive evidence for wave nature of electrons and more importantly their results corresponded with de broglie equation.
Now, experimentally measured values may fluctuate but va... |
Let me preface this by saying that I'm not too code-savvy, which is probably why I need to ask this here. As a first step, I want to find a way to generate 2-particle irreducible (2PI) Feynman diagrams in Mathematica or Python for a specific theory, the Feynman rules of which I can give as an input, or even just for qe... |
Given a $\phi^4$ theory in $d<4$
$$S_{\Lambda} = \int d^dx \left[\frac{1}{2}(\partial_i \phi)^2 + \frac{1}{2} \mu_0^2 \phi^2 + \Lambda^{d-4} \tilde{g}_0 \phi^4 \right]\,,$$
the corresponding RG flows in the coupling space are qualitatively like the ones shown in the picture. The question is whether the high Temperatur... |
In my notes I wrote that the FLRW metric is reported as
$$ ds^2 = -dt^2 + a(t)^2( d\, \chi^2 +f_k(\chi)^2 d\,\Omega^2 )$$
with
$d\,\Omega^2= d\theta^2 + \sin{\theta}^2 d\phi^2$ and
$$f_k(\chi)=
\begin{cases}
k^{-\frac{1}{2}} \sin{ \sqrt{k}\chi},& k>0,\quad \text{open universe}\\
\chi & k=0,\quad \tex... |
I hope this is an appropriate question for this forum. It is one I have struggled with for a while.
I read that in GR gravity is not a force, and that the apparent force we feel and can measure (eg with a free body test or accelerometer) standing on the Earth’s surface is a consequence of the surface’s outward acceler... |
So I was watching some videos. Over here Padmanabhan claims that in a zero cosmological constant universe setting $a =1 $ in the FLRW metric is quite non-trivial:
so there is a constant in the universe which you can determine
only if this $a_0$ is given but the Friedmann equations do not fix
$a_0$... Friedman equation... |
In (5+1) spacetime dimensions suppose we have a scalar field $a$ and a $3$-form field $$A=\frac{1}{3!}A_{[\mu_1\mu_2\mu_3]}dx^{\mu_1}\wedge dx^{\mu_2}\wedge dx^{\mu_3}.$$ $da$ is the exterior derivative of $a$. We denote the interior product of $da$ with $A$ by $\iota_{da}A$. Is it possible to write $d(\iota_{da}A)$ in... |
This is (at least for me) an entirely hypothetical question. How would a lower-dimensional band within a higher-dimensional band structure affect the overall conductive properties of a material? To understand what exactly I mean, assume the band structure of the following Hamiltonian:
$ H(\vec{k}) = \begin{pmatrix} 0 &... |
I need to accelerate a chunk of steel with a solenoid.
I calculated the B field in the bore of the solenoid, and it is > 3 T.
Most definitions of saturation place the saturation of steel around 1.5 or 2 T.
Here is my question:
I know that further increasing the magnetic field cannot store additional energy in the core.... |
Can Hamilton's principle, i.e. the principle of stationary action, be posed as an initial-value problem instead of the usual boundary-value problem and still produce the correct equations of motion?
For a concrete example, consider the classical harmonic oscillator where the Lagrangian functional is given as $$L\left[x... |
Almost all papers on plasma mentioned that plasma ions drag on a particle is in the direction is opposite to the velocity of the particle. But no one said anything about the other two components; are the other components zeroes? What about the velocity which is a vector and in any direction will always have three comp... |
ASTM D5032-19 describes how to generate and maintain a constant humidity by means of aquesous glycerine solutions. The procedure allows for the relative humidity to be calculated from the mass of the glycerine and the mass of the water in the solutioI am located in Johannesburg South Africa at around 2000m and in pract... |
I'm a bit confused imagine the following question:
X moves to Y ($T_0$ light years away) with speed $v$ and Y is stationary. Z observes X's trip from earth. How long X moves in its own reference frame and in Z's reference frame?
I am confused which formula to use $t'=\gamma t$ or $t'=\gamma\left(t-\frac{vx}{c^2}\right)... |
I'm having some trouble figuring out the formula for the transition line for a BEC, i.e. the function $P(v)$, where $P$ is the pressure and $v$ is the volume of the BEC. I've substituted
$$k_BT_c=\frac{2\pi\hbar^2}{m}\left(\frac{n}{g_{3/2}(1)}\right)^{2/3}$$
and
$$v_c=\frac{\lambda_T^3}{g_{3/2}(1)}$$
into the expressi... |
When we bring two waveguides close together we can see some coupling between the modes of the two waveguides and verify energy transfer. Can the same transfer happen in modes within a single waveguide? In a square or rectangular waveguide we have a degeneracy in the polarization of TE modes. Even with dimensions of sin... |
The particle current density can be defined as :
$$\textbf j(\textbf r,t)=\sum_{i=1}^{N} \textbf v_i\delta(\textbf r-\textbf r_i(t))$$
Its spatial Fourier transform :
$$\textbf j(\textbf k,t)=\sum_{i=1}^{N}\textbf v_ie^{-i\textbf k.\textbf r_i}$$
Now,considering the wave vector to be directed along x direction, longitu... |
I am trying to understand the image created when a coherent light source is incident on a diffraction grating as it is swept through the focal point of a lens. The situation is illustrated in the figure below,
where $P_1$ is the object (diffraction grating) plane, located at the lens focal point. $P_2$ is the image pl... |
I am reading the Scientific Background on the Nobel Prize in Physics in 2007, The Discovery of Giant Magnetoresistance. In Chapter 2, paragraph A it is written:
In the free atoms, the 3d and 4s atomic energy levels of the 3d transition elements are hosts for the valence electrons. In the metallic state these 3d and 4s... |
Consider a system composed of two spin-1/2 particles. The total spin operator is defined as
$$
\mathbf{S} = \mathbf{S_1} + \mathbf{S_2}
$$
We can write a common eigenbasis of the operators $\mathbf{S^2}, S_z, \mathbf{S_1^2}$ and $ \mathbf{S_2^2}$ in terms of the eigenbasis of the operators $S_{1z}, S_{2z}, \mathbf{S_1^... |
From multivariable calculus you can assure that the line integral of a conservative field around a closed loop is always going to be zero. It is well known that the electric field is a conservative field (or could we think of non-conservative electric fields?).
With that, the integral
$$\oint_{\partial\Sigma}\vec{E}\cd... |
If we consider motion of a body in a central (gravitational) field the equation of motion would be
$$-\frac{\gamma m_1 m_2}{r^2}\frac{\vec{r}}{r}=m_1 \vec{r}^{\prime\prime}$$
where the origin of inertial frame of reference is placed in the center of gravity.
For two-body problem (also with gravitational force between t... |
It is well-known that the Poisson bracket can be recovered out of the Moyal bracket under the limit when $\hbar$ goes to zero $$\lim_{\hbar\rightarrow 0}\frac{1}{i\hbar} \lbrace f,g\rbrace_M=\lbrace f,g \rbrace_P.$$
This is easy to verify in the differential form of the brackets, but the Moyal bracket admit an integral... |
It seems a trivial question but I don't know something explicitly.
We suppose $\hat{H}=\frac{\hat{p}^2}{2 m}$ and we have a rotation defined by $\hat{R}_{\mathbf{n}}(\varphi)=\exp \left[-\frac{i \varphi}{\hbar} \mathbf{n} \cdot \hat{\mathbf{L}}\right]$.
It is said the $\hat{H}$ is invariant after rotation.
In my proof,... |
In Atiyah's formulation, a Topological Quantum Field Theory (TQFT), is a functor $Z:d\text{Bord}\to\text{Hilb}$. That is, $Z$ assigns:
\begin{align}
\text{Closed compact $(d-1)$-manifolds} &\to \text{f.d. Hilbert spaces} \\
\text{$d$-dimensional bordisms between manifolds} &\to \text{Unitary maps between corresponding ... |
I feel like this question has been asked in so many ways, but I am still unable to explain my personal experience based on previous answers. From other discussions:
You can always replace an off-center force by the same force centered plus a torque r×F.
Trajectory and acceleration of the COM are the same you would ge... |
I'm trying to understand a mathematical expression where it's like this:
$$
I' = I+ m \left( d^2 E - \mathbf{d}\otimes\mathbf{d}\right)
$$
where $I'$ is the new tensor flow, $I$ is the tensor flow, $m$ is mass, $d$ is the distance from the center of mass to the new origin of the center of mass (as a vector), and $E$... |
The square of the Gell-Mann matrices $\lambda_1$, $\lambda_2$, and $\lambda_3$ has the bizarre value $2/3 * I + \lambda_8 / \sqrt{3}$.
Is there a simple way to deduce the result from the fact that $\lambda_8$ is orthogonal to all other $\lambda_n$, and has trace $0$? (And maybe some other property.)
And a similar quest... |
For 2 timelike seperated events A and B you can find different inertial reference frames in which A and B happen in different order.But what if the entropy of a system after A happens is much higher than the entropy of the system after event B happens then could we observe in any reference frame the entropy of the sys... |
I am looking at this energy dispersion relationship that has the form
$$
\varepsilon = -\alpha k_y + \sqrt{\beta k_y^2 + \gamma k_x^2 }
$$
Here is my conundrum - I am able to find the minima of the energy dispersion (happens to be at (0,0)) and using the definition
$$
[M_{ij}]^{-1} = \left(\frac{1}{\hbar^2}{\frac{\par... |
Theory:
Let's look at qubits. A qubit is a abstract quantum system that stands for 1 property of a real physical quantum system (e.g. the position of a particle in 2 wells). A state of a qubit is formally described by a complex vector in $\mathbb{C}^2$ with length 1 (using the standard scalar product).
I know that "the... |
Does anyone know why Faraday's Law for the transformer $emf$ is sometimes written as showing the partial derivative of $B$ with respect to time INSIDE the integral, and sometimes $\frac{\mathrm{d}}{\mathrm{d}t}$ (not partial) is OUTSIDE the integral?
$$\int \frac{\partial B}{\partial t} \cdot \mathrm{d}S$$ vs $$\frac{\... |
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