Q stringlengths 18 13.7k | A stringlengths 1 16.1k | meta dict |
|---|---|---|
Why does $H_2$ have $C_V$=$7/2 R$ at high temperatures, while the total number of degrees of freedom is 6? The two hydrogen atoms have 6 degrees of freedom in total. Of them, $3$ contribute to translation, $2 $contribute to rotation and $1$ contribute to the vibration.
I know that the vibrations motion is frozen at low... | The elastic potential energy of the bond is another degree of freedom which contributes to the heat capacity.
| {
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Arithmetic of Hamiltonian in canonical transformation I have the following Hamiltonian:
$$ \mathcal{H} = \frac{p^2}{2m} + V(q-X(t)) + \dot{X}(t)p, $$
and I make the usual canonical transformation for the momentum:
$$ p \rightarrow p' = p + m\dot{X},$$
and complete the square, which should give the following:
$$ \mathc... | The Hamiltonian transforms according to the following rule:
$H^\prime = H - \partial_t f$, where $f=f(q,t)$ (1).
We can find this function by using that:
$p = p^\prime-\partial_qf=p^\prime-m \dot{X}$.
So we see that $f=m \dot{X}q +c , c \in \mathcal{R}.$ (2).
Plug equation (2) into equation (1) gives:
$H^\prime=H-\pa... | {
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Electric field of an infinite sheet of charge I am trying to derive the formula for E due to an infinite sheet of charge with a charge density of $ \rho C/m^2$. I assumed the sheet is on $yz$-plane. I used Coulomb's law to get an equation and integrated the expression that over $yz$-plane. But, I have not succeeded in ... | Method 1 (Gauss’ law):
Just simply use Gauss’ law:
$$\int_{\partial V} \vec{E} \cdot \vec{da} = \frac{Q}{\epsilon_0}.$$
A pillbox using Griffiths’ language is useful to calculate $\vec{E}$. The pillbox has some area $A$. And due to symmetry we expect the electric field to be perpendicular to the infinite sheet. Imagi... | {
"language": "en",
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Significance of Compton scattering being second order? In the Compton scattering equation for changes in wavelength, for small angles the equation is second order in the angle. Is there any significance to this?
To me it seems to say that,if a photon takes a smooth path with no abrupt turns, then it shouldn't have muc... | The expected probability density is
\begin{equation*}
\langle|\mathcal{M}|^2\rangle=
2e^4\left(
\frac{\omega}{\omega'}+\frac{\omega'}{\omega}
+\left(\frac{m}{\omega}-\frac{m}{\omega'}+1\right)^2-1
\right)
\end{equation*}
The Compton formula is
\begin{equation*}
\frac{1}{\omega'}-\frac{1}{\omega}=\frac{1-\cos\theta}{m}
... | {
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Dresselhaus linear and cubic terms I've been trying to understand Dresselhaus effect, described here.
I've been looking up references to find when the cubic term becomes more dominant than the linear term and vice versa.
For example, in this paper ( or on Arxiv ), they give $ H_{so} = (\beta-\alpha)p_y \sigma_x + (\bet... | This is VERY late.. but neither of these are the cubic term (there is no explicit $p^3$). These are both the result of the $<p_z^2>$ term; i.e. the linear term. And the confinement direction is assumed to be the z-direction.
For a zincblende structure, in its full glory, the Dresselhaus Hamiltonian is:
$H_D = \frac{\ga... | {
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Nuclear physics and half life of a radioactive element Half-life for certain radioactive element is
5 min. Four nuclei of that element are observed
a certain instant of time. After five minutes
Statement-1: It can be definitely said that two
nuclei will be left undecayed.
Statement-2: After half-life i.e.5minutes,... | Imagine the following experiment:
I have two buckets; in one bucket there are N balls. Every 5 minutes, I take each ball in turn; I toss a fair coin, and if it comes up "heads" I put the ball in the other bucket. If it comes up "tails", I discard the ball.
How many balls will there be in the second bucket after five mi... | {
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In a vacuum can a cooler body radiate Infrared radiation to a warmer body? I mentioned vacuum, because I want to discount the effects of conduction or convection. I simply want to know if some of the infrared radiation(IR) goes from the cooler body to the hotter body? How does each body know how much to radiate at any ... |
can a cooler body radiate Infrared radiation to a warmer body?
Yes, the cooler body will radiate, according to its temperature, as you've mentioned, and some of this radiation energy could be absorbed by the warmer body.
This will depend on the percentage of the cooler body radiation the warmer body is exposed to an... | {
"language": "en",
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How can a transformer produce a high voltage and a low current? I understand that in ideal transformers, power is conserved. Because of this the product of voltage and current in the secondary winding is a constant.
This means that voltage and current are inversely related, which seems unintuitive because they are dire... |
I understand that in ideal transformers, power is conserved. Because
of this the product of voltage and current in the secondary winding is
a constant.
This isn't true. The expression of power conservation for an ideal transformer is
$$V_s\cdot I_s = V_p\cdot I_p$$
There is no requirement for $V_s\cdot I_s$ to b... | {
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Why not free electrons in atom doesn't radiates em waves\photons? Why not free electrons in atom doesn't radiates em waves\photons, although they move with acceleration? Like 1s electron of Titan, it doesn't emits em waves, yes? Why?
| It's because the classical model of an atom as a little solar system simply doesn't work for atoms. That was the message of the quantum physicists starting with Bohr (1913) and Heisenberg et al (1925).
We don't know what an atom "looks like" inside. Hard little balls going in orbits? Um, we cannot watch them. Some quan... | {
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Max Born's statistical interpretation of the wave function How did Max Born derive the probability of finding a particle between $x$ and $x+dx$ at instant $t$?:
$$ \left |\psi(x,t)\right|^2dx$$
Was this result mathematically derived? Or is it just a postulate, like the Schrödinger equation itself?
| Wave Mechanics was heavily influenced by classical electromagnetic waves, and in classical EM, the intensity of a wave is proportional to the square of the field's strength. Luckily, this turned out to be true in QM as well.
| {
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Woodworking clamps, does force add up? I was watching a woodworking video about glue, and the guy was clamping two pieces of wood together using a total of 8 clamps. He argued that by doing so he would apply 8 times the maximum force of 150N (a property of the clamp), resulting in 1200N in total.
I think he's wrong. I ... |
And so the clamping force on any given spot on the board will never
exceed the max. force of the clamp.
This would, quite obviously, be the case, if we assumed that the clamps are evenly distributed around a circle.
Under these conditions, due to symmetry, any redistribution of the reaction force, which, in total,... | {
"language": "en",
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How to put $c$ back into relativistic equations? Many books set the speed of light $c=1$ for convenience. For example, Weinberg in his textbook "Gravitation and Cosmology" (though $G$ is still left as a constant):
$$\begin{align}
\mathrm{d}\tau^2 &= \mathrm{d}t^2 - R^2(t)~[f(r,\theta,\phi)] \tag{11.9.16} \\
t &= \frac{... | Setting c = 1 essentially causes quantities with dimension of time to be expressed in units of length. If you know the length dimension of a calculated quantity, you can re-express it in terms of time by plugging in the corresponding amount of factors of $c$, which just act as conversion factors.
This amounts to doing ... | {
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Value of the invariant $R_{\mu \nu}F^{\mu \nu}$ Is there a simple way to find the value of $R_{\mu \nu}F^{\mu \nu}$ (where $R_{\mu \nu}$ is the Ricci tensor and $F^{\mu \nu}$ is the electromagnetic tensor), knowing that it is an invariant?
Inputting the definitions of $R_{\mu \nu}=R^{\lambda}_{\mu \lambda \nu}= \partia... | $R_{\mu \nu}F^{\mu \nu}$ is simply zero. No computations are needed to see this. Just note that $R_{\mu \nu}$ is a symmetric tensor while $F^{\mu \nu}$ is antisymmetric.
| {
"language": "en",
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The definition of the hamiltonian in lagrangian mechanics So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}$$
And then author affirms that this quantity is constant and... | In your first equation, the Hamiltonian is defined in terms of the Lagrangian via an operation called a Legendre transformation. They are most commonly seen in thermodynamics and classical mechanics, and are used to convert functions of one variable into functions of another variable without disturbing the physics. Sev... | {
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Power factor in Transamision What is normally the power factor in the transmssion.? Is there some standard or guidelines for that.?
Also, the low factor is not desirebale as it leads to more power loss. That makes sense. They say power factor can be improved adding a capacitor in parallel. Why in Parallel.?
| Power factor is a control variable used by electric utility transmission system control software (Energy Management System) to provide voltage control in the transmission system. Voltage sags on a node can be lifted up using added capacitor banks that are switched in or out of the circuit in real-time by EMS software.... | {
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Do compact symplectic manifolds play a role in physics? In classical mechanics, the phase space is the cotangent bundle of the configuration space, and it is a symplectic manifold, but not compact.
Do compact symplectic manifolds have physical meaning? Or just of mathematical interest?
| Every Calabi-Yau manifold, being Kähler, is symplectic. Compact Calabi-Yau manifolds play an important role in string theory, though their symplectic structures did not initially seem to play an important role (for as far as I know).
However, one context in which these do play a role is in homological mirror symmetry, ... | {
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What direction does electric current flow in when the voltage drop is negative? We were given the below diagram, and we needed to determine whether the unknown component is supplying energy into the system.
$\hspace{200px}$
I thought the unknown component was supplying energy since charge is flowing to a higher potenti... | I would have thought:
$$
\begin{align}
P &= V \cdot I \\[5px]
P &= -\frac{\mathrm{d}U}{\mathrm{d}t}
\end{align} \\ \, \\
\Delta U = -\int{P\cdot \mathrm{d}t} = - \int{V\cdot {I}\cdot \mathrm{d}t}
$$
$ V\cdot I > 0$ since the current is entering in the device through the positive reference of the voltage and I suppose... | {
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Is any particle's energy quantized?
In the above picture, the author is trying to summarize the correlation between particle and wave packet. In doing so, he assumes that frequency is related to energy as: $E=h\nu$. Is this apparent assumption correct?
Because as far as I know, the quantization is only for oscillator... | Any particle has an energy given by the relativistic equation for the energy:
$$ E^2 = p^2 c^2 + m^2 c^4 \tag{1} $$
In this equation the variable $m$ is the rest mass and $p$ is the (relativistic) momentum. The momentum is related to the de Brogie wavelength by:
$$ p = \frac{h}{\lambda} \tag{2} $$
For photons the rest ... | {
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Is the electron field of QED similar to the Higg's field? If I understand it (I probably don't...) the Higg's field is a field that permeates the universe, and its excitations are the Higg's bosons. It seems that this implies that the Higg's field is a property of the universe itself, and not generated by something lik... | The Higgs, electron and electromagnetic fields all permeate space. In QFT their excitations are indeed what we call particles.
They are different types of fields, however. Higgs is scalar, electron in spinor, and EM is vector.
| {
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Factors affecting Battery Voltage How do batteries produce a certain voltage, such as 1.5V or 9 V? From what I understand, battery EMF comes from oxidation of the anode, which releases electrons that can flow through a circuit. But how do batteries regulate that voltage? What is it about the chemical reaction that crea... | The voltage generated by an individual electrochemical cell will be in the approximate range of a fraction of a volt to a volt and a half for most cells and is determined by the particulars of the anode oxidation process, which vary from one metal to another as you point out.
However, to get 6 volts or 9 volts or 12 vo... | {
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Could black holes be formed by highly energetic gravitational waves? Could the gravitational waves released by two merging black holes contain enough energy to produce another black hole?
| Colliding gravitational waves can indeed form black holes, but the conditions for doing so are fairly strict. As it happens this has just been discussed in the preprint Black Hole Formation from the Collision of Plane-Fronted Gravitational Waves.
But it's exceedingly unlikely this would every happen. The gravitational ... | {
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Why isn't average speed defined as the magnitude of average velocity? Speed is usually defined as the magnitude of (instantaneous) velocity. So one could assume that average speed would be defined as the magnitude of average velocity. But instead it is defined as
$$s_{\textrm{average}} = \frac{\textrm{total distance t... | Given a velocity as a function of time, the speed as a function of time is the magnitude of the velocity at each point in time. The average speed is then the average of this magnitude, as it would be for any function of time - such as density or temperature. The question of whether the average magnitude of the velocity... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Magnetic field of a solenoid Why, when explaining the magnetic field of a solenoid, are the individual "quasi-rings" split up into two layers with opposite direction of current (see attached pictures; where the x-es and the dots indicate opposite directions). The original picture given clearly states the direction of t... | Just cut the solenoid in image 1 from a horizontal diameter and indicate the direction of current again in its wires
Note:- the dots mean current is coming out and the crosses mean its going inside
| {
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Why is it much more difficult to horizontally throw a toy balloon than a football? If you horizontally throw a sphere of radius $R$ it will feel, in this direction, a drag force due to air. Assume the drag is given by Stokes law, $F_D=6\pi\eta R v$, where $\eta$ is the air viscosity and $v$ is the horizontal speed. Thi... | For a balloon and a football (soccer ball for Americans) of the same size, shape, and initial velocity, the aerodynamic drag will be the same. The effect on the motion of these objects will be very different because of their different masses. A football has a mass of about 0.430 kg while a balloon has a mass of less th... | {
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Is the probability density of an electron in a hydrogen atom static? Probability densities are illustrated in text books and on Wikipedia as static pictures. Is the probability density of an electron within an isolated Hydrogen atom static or does it oscillate in some way?
| In this link, one can see how the plots you show are derived from the solution of the Schrodinger equation for the hydrogen atom. There are checkable options.
There is no time dependence in the wavefunctions, or the corresponding probability distributions for the given eigenvalues (seen by checking next to the solution... | {
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Reducing multi body system to a single body using reduced mass A two body system can be treated as a single body using reduced mass and the motion can be described using one generalized coordinate.
Can this concept be somehow used to reduce a body of say 3 or more particles to a single body?
| Not in general, no. The "three-body" problem does not even have closed-form solutions in general, let alone a simplification like this that would make it easy. It's not clear how you intended to use continued fractions here, but I don't see how that would help.
See, e.g., https://en.wikipedia.org/wiki/Three-body_prob... | {
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$\sqrt{-1}$ coefficient in a function In a simple harmonic oscillator with $\ddot{x} = -\omega^2x$, it can be shown through differentiation that one solution can be given by $\dot{x} = i \omega Ae^{i \omega t}$. What does the factor of $i$ do here? What effect does it have on velocity?
| The terms in a differential equation representing an oscillating system may be out of phase. Such equations can be represented by a “phase diagram” Each term in the equation can be considered as representing one component of a vector rotating around the origin in a 2D “phase space”. The phase space can be chosen as an... | {
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The theater, the actors and the 'graviton' In my perception of the universe, there's the theater which is the 'spacetime' and the actors meaning the 'particles'.
If i got it right, GR claims that the 'actors' effect the 'theatre' by 'bending' it.
Also in order to combine GR and QM, some beleive in the existence of the ... | The theater is still needed in either case simply because we observe and measure the theater. The difference is whether or not the theater is built by the actors.
In GR the theater arises dynamically from the distribution of the actors’ stress energy tensors. In QM the theater is a static background that is added sepa... | {
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Interaction-Picture Field as Solution of Klein-Gordon Equation I am following a problem in a QFT textbook (Srednicki) which asks us to show that the interaction-picture field
$$\phi_I(\textbf{x},t)=e^{iH_0 t}\phi(\textbf{x},0)e^{-iH_0 t}$$
obeys the Klein-Gordon equation. This is implicit in the rest of the text, but ... | We take the derivatives of the interaction-picture field and simplify to obtain the Klein-Gordon equation.
$\phi_I(\textbf{x},t)=e^{iH_0 t}\phi(\textbf{x},0)e^{-iH_0 t}$
$\frac{\partial \phi_{I}}{\partial t} = iH_{0} e^{iH_{0}t}\phi(\textbf{x},0)e^{-iH_{0}t} + e^{iH_{0}t}\phi(\textbf{x},0)(-iH_0)e^{-iH_0 t}$
$\frac{\pa... | {
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Confusion about how an electron gun works I'm a little unclear about the charge balance aspect of an electron gun. Referring to this diagram and similar diagrams I've seen, what I don't get is wouldn't the target of the electrons have to be connected to the positive anode so that the electrons fired at a target can b... | This is an Electrical Engineering question. The target is usually not in the hole (opening) of the anode. This is because, first, the hole is small. Second, you often want to be able to manipulate the electron beam, like what people do in the CRT TV. So the target is usually at the right end of your diagram, where your... | {
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"source": "stackexchange",
"question_score": "7",
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Is continuum mechanics a generalization or an approximation to point particle mechanics? Newtonian Mechanics is usually presented as a theory of point particles (and forces). My impression of the status of continuum mechanics is that it is mostly taken as an approximate description for certain situations, where many pa... | My guess is that point particles were initially introduced as approximations to avoid the effect of mass distribution in inertial and gravity phenomena description. Spring - mass systems usually only offer a very crude model of solids, although they can sometimes be used as a first approach (e g. phonons in crystals).... | {
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Confusion about the meaning of steady current I am trying to learn some elementary EM, but I have some confusion about the basic concepts of steady current.
Suppose I have a wire of uniform cross section area. The current is always flowing from left to right.
I imagine that I can cut a segment of this wire (with the ar... | $\int\int_S \mathbf{J}\cdot d\mathbf{A}$ represents the current $(\frac {dq} {dt})$ flowing through a cross-section area. It is not "the net change in charge out of this segment".
The "net change in the charge" would be equal to the difference between the current flowing in (through the left cross-section) and the curr... | {
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} |
Gravitational Time dilation for motion in a gravitational field An astronaut is travelling towards a black hole in a space ship traveling at 0.8 C. The guy’s position is not fixed relative to the black hole. He is travelling towards it from a distance. How to calculate the gravitational time dilation in this case, as t... | A radial free fall to a Schwarzschild black hole from rest a distance $R$ in geometrized units is given by the geodesics
$$ \tau=\dfrac{R}{2}\sqrt{\dfrac{R}{2M}}\left(\arccos\left(\dfrac{2r}{R}-1\right)+\sin\left(\arccos\left(\dfrac{2r}{R}-1\right)\right)\right) $$
And
$$ t=\sqrt{\dfrac{R}{2M}-1}\cdot\left(\left(\dfrac... | {
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"timestamp": "2023-03-29T00:00:00",
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Why can you assume the current remains the same after changing resistance? In my physics textbook there is a question about a circuit:
In question b) We find out the current in the circuit is 0.003A. This is before the voltmeter is added.
In question c) a voltmeter is added that affects the p.d across the first 1k res... |
In the mark scheme they say R(from B to A) = 666.7 ohms which is means they did 2/0.003. How is it known the current doesn't change at all, or am I missing something?
The given solution is incorrect.
If we connect the voltmeter as shown and measure 2 V, then we know that
$$ 6 \frac{R'}{R'+1000} = 2$$
where $R'$ is t... | {
"language": "en",
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Why does work depend on distance? So the formula for work is$$
\left[\text{work}\right] ~=~ \left[\text{force}\right] \, \times \, \left[\text{distance}\right]
\,.
$$
I'm trying to get an understanding of how this represents energy.
If I'm in a vacuum, and I push a block with a force of $1 \, \mathrm{N},$ it will move... | Often it is important to know if a given formula is a simplification of a more general equation and, when you encounter a conceptual problem, check the general formula. In this case it is a simplification of this formula:
$$W=\int_S F\cdot ds $$
Where $S$ is the path over which we are interested in the work and $ds$ is... | {
"language": "en",
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Definitions of $\vec{B}$ and $\vec{H}$ From here, I have got the definition of $\vec{H}$. However even in wikipedia and other sites, I cannot find a definition for $\vec{B}$ which shows its similarity with $\vec{H}$.
Similarity:
I know for free currents, $\vec{B}$ and $\vec{H}$ are same. I also know outside the magnet,... | I would say the more "fundamental" field is actually the magnetic field $\vec B$. The definition of $\vec H$ from this is then
$$\vec H=\frac{\vec B}{\mu_0}-\vec M$$
Where $\vec M$ is the magnetization of the medium your magnetic field is in.
Both are useful depending on the context. Since the magnetization in free sp... | {
"language": "en",
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Interefernce pattern for points where condition for constructive and destructive interference is not met I get the creation of alternate dark and bright fringes in the double slit light interference experiment. Where I am confused is there will be points on the screen where condition for constructive as well as destru... | It's because of the way your eyes work.
The intermediate areas are actually intermediate. The boundary between the light and dark areas is not a sharp line. But your eye doesn't pick up on that.
What you see is intensity. Leaving out the details, intensity is proportional to the sine squared of something-or-other. (It ... | {
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Which field Passes Through the Polarization Film? Why Isn’t the Perpendicular Field Stopped? When one EM field is aligned so that it can pass through a polarizing lens the other field (E or B) is 90 degrees out. Is only one of the EM fields affected by a polarizing lens or film? How is it that one field is stopped yet ... | Both fields are equally affected by the polarizing rods/wires, it is just more conventional to visualize a small wire as being directly interfering with the E-field that is parallel with it than the B-field that is perpendicular to it.
You would never ask this question if had been taught to view the B-field not as a v... | {
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Fresnel's Equations Range of Validity Do Fresnel's equations ever break down at extremely small length scales?
I am wondering if I can apply Fresnel's equations to a very thin film (~10-20nm) at an interface with air with a free-space wavelength in the micron range. I know Fresnel's equations are derived from Maxwell'... | The Fresnel equations are based on the plane-wave solutions of the electromagnetic field. As such, there is nothing that forces them to break down for films that are thinner than the wavelength (and, indeed, they are essential in describing anti-reflectant coatings that are films of about $\frac14$-wavelength thickness... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Simple DC motor with one bar magnet https://www.instructables.com/id/The-Simple-DC-Motor/
I'm struggling to get my head around how this motor works.
Let's say for the sake of argument that the field lines are running from the LHS of the bar magnet round to the right and that current flows through the coil in a CW dire... | You are right that if the magnet's field lines run from left to right in the illustration, the coil will not turn. The type of magnet shown is magnetized in the vertical direction, so that the field lines exit the top and bottom faces. When a current flows through the coil, the coil roll rotate so that the magnetic f... | {
"language": "en",
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Calculating expectation values over an observable I have a question pertaining to a seemingly (to me) arbitrary, yet necessary distinction one must make when calculating the expectation value of, say, momentum $\hat p$:
I forgot to ask her, but my lecturer made it very clear that when computing this, that the following... | As long as the operator you're calculating the expectation value of is an observable, it will be Hermitian and (hence) the expectation value must be real.
So you might as well take the complex conjugate of the entire expression; it will not change the result.
If the operator is not Hermitian, it does matter on which ... | {
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Showing that the exterior surface charge density of two parallel plates with thickness are equal I have some problems with the next exercise. It states:
Two infinite conducting parallel plates I and II, with thickness $t_1$ and $t_2$ respectively are separated by a distance $L$ from its nearer faces. The surface charg... | The key is to consider a point inside one of the conductors
Now, we know the field must be $0$ in the conductor. We also know that the field due to an infinite sheet of charge is given by
$$E=\frac{\sigma}{2\epsilon_0}$$
Each surface forms an infinite sheet of charge. So if you add up the field from each (taking direct... | {
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Damped oscillations and generalized friction I'm reading damped oscillations from the book Classical Mechanics by Landau and Lifshitz, quoting from the text -
"There exists, however a class of problems where motion in medium can be
approximately described by including certain additional terms in the
mechanical equ... | The paragraph may possibly be interpreted along these lines:
1) Compare an object being pushed on the surface of a rough table with an object being pushed while immersed in fluid.
Both objects experience an opposing frictional force. For the immersed object, frictional force is dependent on velocity. For the table, fr... | {
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Irreducible decomposition of Lorentz tensors with Young tableaux I want to understand the irreducible decomposition of Lorentz tensors by using Young tableaux. Let me start with a trivial example. Suppose we work in $n=4$ dimensions, and that we have a rank 2 homogeneous tensor $T_{ab}$. Doing the Young tableau, we fin... | Another alternative for irreducible decompositions is by the aid of Lorentz group projectors as explained in https://www.mdpi.com/2218-1997/5/8/184 and sections 3 and 4 there.
| {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Simple question about change of coordinates Suppose we have two coordinate systems (Cartesian and spherical)
$$x^{\mu} = (t,x,y,z)$$
$$x'^{\mu'} = (t',r,\theta,\phi)$$
where $r= \sqrt{x^2 + y^2 + z^2} , \theta = \cos^{-1}(z/r), \phi = \tan^{-1} (y/x)$. My question is, in general, what are the components of a vector $A_... | Your transformation matrix:
I will ignore the "t" coordinate
\begin{align*}
&\text{The position vector for a sphere is: } \\
&\vec{R_s}=
\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}=
\left[ \begin {array}{c} r\cos \left( \vartheta \right) \cos \left(
\varphi \right) \\ r\cos \left( \vartheta ... | {
"language": "en",
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Under what circumstances are general relativistic coordinate transformations physically meaningful? Although the field equations of GR are covariant under arbitrary coordinate transformations, such as the transformation given by Dirac (in Princeton Landmarks pp 34) that eliminate the singularities in the Schwartzschild... | The coordinate transformation is never physically meaningful. Certain coordinate systems have some specific meaning.
The coordinates are just sets of arbitrary numbers with some convenient mathematical properties (a smooth and invertible map) but they are otherwise arbitrary. Coordinate transformations are just bookke... | {
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Lorentz Velocity Transform With Tensor Notation So I'm attempting to prove the Lorentz Velocity tranform:
$${v_x}' =\frac{v_x-u}{1-v_xu/c^2} $$
using tensor notation. In this case obviously $\beta = u/c$ and $\gamma=(1-\beta^2)^{-1/2}$. The velocity transform tensor can be represented as
$$\Lambda = \begin{pmatrix}
\... | 4-velocity is defined as
$$
u^{\mu}=\frac{dx^\mu}{d{\tau}}
$$
$\tau$ does not change under transformation. It is the proper time.
so in your formula $$v_{x'}=c\partial_{0'}x^{1'}$$ is not the correct way. Do you see the point?
| {
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Why can we see the cosmic microwave background radiation? This radiation (CMBR) is said to have its origin at the surface of last scattering that exposed itself when the big bang universe had expanded for less than a million years.
In order to see radiation from a source, one has to be on its future light cone. In a un... | One has to keep remembering that in the Big Bang model, the (0,0,0,0) is located in all points of the present day universe. Each of us is sitting at the center of the universe.
As the universe expanded all points expanded away from each other.
Light that decoupled from matter at 380.000 years after the Big bang, deco... | {
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UV Preconditioning test I wanna build a solar panel with a new material and I wanna to test the endurance of the material against UV light. I've found in internet a Standard for UV tests. It says it has to be irradiated with 15kwh/m² from 280nm to 385nm. Since I just have a 8w UV lamp I decided to do some scaling to th... | Does your 8W lamp put out 8W of light in that band of wavelengths? or does it use 8W of electricity? With most lamps, the Wattage rating tells you how much electricity it uses, which can be much greater than the amount of optical power that it puts out. That becomes even more of a concern if you are only interested in ... | {
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Non-acceleration and 0 force If a mass is moving at the rate of 30/ft. per minute for 5 minutes on a straight line and it strikes a second stationary mass and effects a change of position to this second mass, then we know from F=ma that the force is 0 since the acceleration is 0. Then why do we say because it is 0 that... | Hey when the collison occur the force will act on it this force will persist until they are in contact which is given by
F =(m1v1-m1u1)/t=(m2v2-m2u2)/t=impulse/t
Here v1 v2 are final velocities and u1 u2 are initial velocities t is the time upto which force acts or the time of contact
Here we can see that force is n... | {
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Once a black hole is formed, is there anything other than Hawking radiation which shortens its life? Hawking radiation is supposed to very slowly evaporate a black hole (terms and conditions apply :] ).
Apart from Hawking radiation, is there any mechanism or effect that can make a black hole cease to exist? Or once th... | Edit: It seems that the answer is no, as far as we can tell presently.
For relevance, I keep my previous answer, which raised some controversy, below.
--
After searching a little and asking around, I've also come across the Penrose process, and in the comment above Count Iblis mentioned Phantom energy accretion.
So,... | {
"language": "en",
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"Iron Core" in Inductive Charigng Inductive charging used for wireless charging often faces the hindrance of being too short ranged for many cases. There appear to be some workarounds such as using a capacitor to resonate them at the same resonant frequency. Please excuse the naiviety of the question, but when looking ... | At the high frequencies implied by your question the eddy current heating of the iron core and the hysteresis loss due to the rapid oscillation of the magnetic field in the iron core will result in the Q-value of the circuit being very small.
In other words the energy losses of an iron cored inductor would be too high.... | {
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Why doesn't using more appliances at home decrease the electricity bill? I know that at home the electric circuits are parallel, and this explains why if one appliance (eg bulb) fails, everything else continues to work, but if more devices are added in parallel to each other, their combined resistance should decrease, ... | To minimize your power consumption and save on electricity bills, you actually want to maximize your home's equivalent resistance, for the reason you mention. This is done when every appliance is turned off, because each turned-off appliance has (approximately) infinite resistance (up to some current leakage). Adding a... | {
"language": "en",
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What makes the disturbance in Electromagnetic waves move? I get that changing electric field will have a curly changing magnetic field and changing magnetic field will have curly changing electric field. So when we move a charge up and down, electric field will change, which will produce this changing magnetic field an... | I think that it is a natural property of the universe, just like how a disturbance of the water at one point in a lake spreads out to the whole lake.
| {
"language": "en",
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What if... you had a bowl of electrons? My chemistry teacher used to tell us that if you had a soup bowl with only electrons in it, the explosion could make you fly to Pluto. Was he right? Could this happen?
| A uniform sphere of radius $r$ and total charge $Q$ has an electric potential energy of $3Q^2/20\pi \epsilon_0 r$.
Let's say your bowl is like a sphere of radius 5 cm. If it's all water ( molecular weight 18$m_u$) and $\mu= 9/5$ of a mass unit for every electron, then the number of electrons is
$ N_e = 1000 \frac{4\pi ... | {
"language": "en",
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Instantaneous velocity So here’s a question I’ve been thinking of for a while. Suppose we say, “an object is having an instantaneous velocity along a particular direction ( say 10 m/s along the $x$-direction)” . Is it fair to conclude that it is traveling in a straight line along the $x$-axis? Well my opinion on this i... | Well a projectile(obliqe projectile moves with parabolic path and its velocity is not always along x axis. This happen only when it reaches its maximum height .At any time in the projectile motion its Velocity is given by v⃗ =v⃗ₓ+v⃗y where v⃗ₓ and v⃗y are the velocities along x axis and y axis respectively and this v⃗y... | {
"language": "en",
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"source": "stackexchange",
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Covariance matrix after projection on gaussian state I cannot understand the proof in Eisert article (https://arxiv.org/abs/quant-ph/0204052) about finding the covariance matrix of a state after projecting some of its modes onto a gaussian state.
We have the characteristic function of the remaining (unprojected) modes ... | As suggested by @flippiefanus, it’s a case of completing the square. (Or, alternatively, it is a well known formula for Gaussian integration, but this sounds a bit like cheating)
If I’m not mistaken, you made a typo in your first equation, which is equation (8) in the Eisert, Scheel, Plenio article. You should replace ... | {
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How to measure angles in Minkowsky space, and how do they transform? I want to know how an ordinary angle $\theta$ transforms under a Lorentz boost. For that purpose I consider a 4-vector given by
$$ a ^ \mu = ( t , \cos \theta , \sin \theta , 0 ) .$$
The angle I will analyze is the one between this 4-vector and the ... | Length measurement in the moving frame should be done at simultaneous times in the moving frame, so you need to do a Lorentz transformation fixing $t'= 0$.
This gives
$$ t = \beta \cos \theta $$
So
$$ a'^\mu = ( 0 , \frac {\cos \theta} \gamma , \sin \theta , 0 ) $$
So
$$ \theta ' = tan^{-1} ( \gamma \tan \theta ) $$
| {
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Confusion with regards to uncertainty calculations Let’s say we have a scenario of a ball being released from the top of the building. This can be modeled simply with the kinematics equation $S=ut +\frac{1}{2}at^2$, which reduced to $S=\frac{1}{2}at^2$. We are given $\Delta t, t, \Delta S, S$, are we are to find $a, ... | Aaron Stevens' answer above has already answered the question but I wanted to add something extra that might help others who see this question.
To calculate uncertainty for a multiplication or division, we add the fractional uncertainties in quadrature.
So, considering $S=\frac{1}{2}at^2$, we have
\begin{align}
\frac{\... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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How is momentum conserved in this example? Suppose a sticky substance is thrown at wall. The initial momentum of the wall and substance system is only due to velocity of the substance but the final momentum is 0. Why is momentum not conserved?
| The wall will move a little bit as well as exert a small force on whatever it's attached to, etc., etc., until you get to applying a force to the Earth. Everything else is so massive, so we can't see this happening. You are assuming an immovable wall, which is not physically the case.
| {
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"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
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Why do we use the RMS but not the fourth root mean quad? Why do we use the power of $2$? What is the relation between this and having the same heat energy in both AC and DC?
| The root mean square can be derived from something more general.
First lets look at the space of T-periodic, real functions. Its inner product is
$$\langle x,y\rangle = \int_0^T x(t)y(t) \mathrm{d}t.$$
Induced from this inner product, we can define a norm on this space:
$$\lvert\lvert x \rvert\rvert_2 = \sqrt{\langle ... | {
"language": "en",
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How does the Lorenz gauge eliminate the scalar component of the vector field? Wikipedia states that by using the Lorenz gauge, $\partial_\mu A^\mu=0$, we eliminate the scalar part (spin-0) of the vector potential that previously had spin-1 and spin-0 components${}^1$.
However, this excellent Phys.SE answer by @Acciden... | The easiest way to see it is to write it in momentum space
$$p_\mu A^\mu = 0$$
In the center of mass frame, where the spin components are well defined, you have
$$\bar p_\mu=(M,0,0,0) \implies \bar p_\mu A^\mu =M A^0 = 0 \implies A^0 = 0$$
| {
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Conceptual question in thermodynamics about isothermal processes If a process is isothermal then Δ U is zero. So ΔQ is non zero. But isn't ΔQ=nCΔT. Implying that ΔQ is zero. Where am I wrong? (Considering ideal nature)
| In freshman physics, we learned the $Q = nC\Delta T$, but in thermodynamics, we change the definition a little to reflect our understanding that, while Q depends on process path, C is supposed to be a physical property of the material experiencing the temperature change, independent of process path. So we redefine C, ... | {
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Why position and momentum operators are both continuous spectrum while angular momentum is discrete? We know that position $\hat{r}$ and momentum $\hat{p}$ are both continuous spectrum operators, i.e.
$$\hat{r}|r'\rangle=r'|r'\rangle, \quad \hat{p}|p'\rangle=p'|p'\rangle.$$
But the angular operator $\hat{L}=\hat{r}\tim... | In general for a bounded state, its energy spectrum is discrete. Free particles on the other hand has continuous energy spectrum. This is true for a rotating motion as it is also bounded, i.e. the starting point is exactly equal to the end point upon rotation by $2\pi$, therefore its eigenvalues are also discrete.
| {
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How to relate random matrix theory with Quantum mechanics approach In Quantum mechanics we first built Hamiltonian and then find its eigenvalues and eigenvectors. How I can relate this with random matrix theory?
| Random matrices were suggested as Hamiltonians in very complex systems from nuclear physics. Here, you had a lot of different energy levels appearing in your spectra, but unlike with, say, the hydrogen atom, it was no longer clear how to label them in a systematic way. So rather than trying to derive or propose an expl... | {
"language": "en",
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"timestamp": "2023-03-29T00:00:00",
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Specifying the initial nonequilibrium distribution $f(\textbf{r},\textbf{v},t)$ in Boltzmann equation? Within the single relaxation time approximation, the collision term in the Boltzmann equation is approximated as $$\Big(\frac{\partial f}{\partial t}\Big)_{\rm coll}=-\frac{(f-f_{\rm eq})}{\tau}$$ where $f\equiv f(\te... | Note that this is really a non-linear equation, because $f_{eq}$ depends on the 0'th moment (particle number=chemical potential), 1st moment (mean velocity) and 2nd moment (mean energy=temperature) of $f$. As a result, establishing convergence is not entirely trivial. Also note that $f$ has non-trivial $x$ dependence, ... | {
"language": "en",
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What is the experimental evidence that the nucleons are made up of three quarks? What is the experimental evidence that the nucleons are made up of three quarks? What is the point of saying that nucleons are made of quarks when there are also gluons inside it?
|
What is the experimental evidence that the nucleons are made up of
three quarks?
Some strong pieces of evidence for the quark model of the proton and the neutron, not stated in another answer, are the magnetic moment of the proton and the magnetic moment of the neutron, which are consistent with the quark model and... | {
"language": "en",
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Is Quantum Mechanics Compatible with Conservation of Information? What is exactly the law of conservation of information? In quantum mechanics we have truly random outcomes in experiments, but doesn't this randomness mean that new information is produced and the law of conservation of information is violated?
| Any conservation law -- energy, momentum, you name it, holds only in an isolated system. If a system interacts with its environment, then neither energy nor information associated with the system will be conserved. Of course, you can consider the system and its environment together as an isolated system, to which the c... | {
"language": "en",
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What part of special relativity is factored in in relativistic redshift velocity equations? While doing research into redshift equations (doppler redshift and cosmological redshift), both types of redshift had two equations for finding recession velocity: a 'non-relativistic' equation and a 'relativistic' equation. I l... | The simplest case is the "ordinary" relativistic Doppler redshift or blueshift due to relative motion.
Considering the relativistic effect (redshift or blueshif), it is necessary to take into account the time dilation. At relativistic velocities all processes in moving source or moving observer run slower, thus a sourc... | {
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Frustrated Ising model Consider a 2D Ising model with nearest neighbour, and second nearest neighbour interactions
$\mathcal{H}= -J_1\sum_{\langle ij\rangle}\sigma_i \sigma_j-J_2\sum_{\langle\langle ik\rangle\rangle}\sigma_i \sigma_k$
where $\sigma =\pm 1$. And $|J_1|=|J_2|$
For $J_1>0$ and $J_2<0$ the system is frustr... | This model (assuming a square lattice) has been studied in the literature for a while. K Binder and DP Landau Phys Rev B, 21, 1941 (1980) discuss the three possibilities for the ground state: ferromagnetic (F), antiferromagnetic (AF), and superantiferromagnetic (SAF). Basically one considers the likely structures and p... | {
"language": "en",
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Global destructive Interference and conservation of energy As an engineer I see it like this. Imagine I send a wave and then I send another wave in phase shift to cancel that wave. Unless I am sending the wave from exactly the same point in both instances, then I will not have perfect destructuve interference everywher... | You are right about what happens in reality. But physical models are simple, and generally, they don't care about practical nuances like this.
So, according to any physical model, even when you emit two waves like that (from the same point, with precisely the phase difference that you want), the mathematical background... | {
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Why are tall block stacks so hard to make? Consider a stack of wood chips: each 0.5cm thick and 2x2 cm in length and width. There are 200 of them all stacked on each other. For some reason they all instantly fall. Evwn though their centre of gravity is at the centre of the stack and they have the extra added help of ... | An additional factor that I haven't seen mentioned yet is the fact that the environment acts upon a structure such as that. Errant air currents would have a deleterious effect on such a stack the higher it went. Even a tiny breeze would topple a perfect stack if it was beyond a certain height.
| {
"language": "en",
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Is the relative speed of light really invariant, irrespective of the motion of the observer? If 3 observers are on a planet which 100 light years from a star, and the star goes supernova, if one observer moves towards the star and one moves in the opposite direction, each observer will see the explosion at a different ... | the concise answer: by moving the observers you changed the distance to the supernova, not the speed of light in your experiment.
| {
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What is the most useful to learn out of complex analysis and differential equations for undergraduate studies in physics? Next year I'm planning to start on my bachelor's in physics, however, I have already started taking some undergraduate courses in mathematics and next semester I will have to choose between complex ... | As an undergraduate, having a decent understanding of complex numbers, rather than complex analysis, goes far. Complex analysis - complex functions and integrals of complex functions in the complex plane and power series of complex functions and the multiple of theorems, etc... - is not very useful in an undergraduate ... | {
"language": "en",
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Work done by a gas In the expression for work done by a gas,
$$W=\int P \,\mathrm{d}V,$$
aren't we supposed to use internal pressure?
Moreover work done by gas is the work done by the force exerted by the gas, but everywhere I find people using external pressure instead of internal pressure.
| The work done by the gas on the piston is
$$W_1 = \int P_{\text{int}} \, dV$$
where $P_{\text{int}}$ is the pressure of the gas right next to the piston. This is just a mild rephrasing of the definition of work. The work done by the piston on the outside is
$$W_2 = \int P_{\text{ext}} \, dV$$
where $P_{\text{ext}}$ is ... | {
"language": "en",
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Solution of diffusion equation with spherical sink I hope this question is not too basic, but I have no experience with partial differential equations and would like to ask for some hints on how to solve the following problems:
The visual idea is to describe the diffusion of some dilute chemical around a spherical sink... | For steady state diffusion in an unbounded region toward a spherical sink of radius $r_0$, we have $$4\pi r^2 D\frac{dc}{dr}=Q$$where Q is the diffusion rate of the species toward the sink (a constant). The solution to this equation is $$c=c_{\infty}-\frac{Q}{4\pi rD}$$where $c_{\infty}$ is the concentration far from ... | {
"language": "en",
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The charge of the electron before measurement Knowing that electrons do not have a definite position before they are being measured, how can their charge be described before the measurement? Where is the charge? Does it make sense to talk about their charge before measurement?
| The charge and mass of an electron are natural constants. All electrons have the same charge. The position of an electron is not constant at all. Where is the charge? Wherever the electron is. The statement that a particle "does not have a definite position" before being measured of course means that the position of it... | {
"language": "en",
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Determining the acceleration of the Universe from a single star? It Occurs to me we might be able to find an entirely independent method of determining the Universe's acceleration using a single source.
If one was to watch a single high source consistently one should be able to simply simply watch for the change in it'... | The idea of measuring the change in redshift over time of a distant galaxy has been around since at least the 1960s. Unfortunately, it remains far beyond our technical capabilities. In a previous post, I derived the equation for $\dot{z}$:
$$
\dot{z} = (1+z)H_0 - H\!\left(\!\frac{1}{1+z}\!\right),
$$
where $H(a)$ is th... | {
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Are uncertainties higher than measured values realistic? Whenever I measure a positive quantity (e.g. a volume) there is some uncertainty related to the measurement. The uncertainty will usually be quite low, e.g. lower than 10%, depending on the equipment. However, I have recently seen uncertainties (due to extrapolat... | If your extrapolated results have more than 100% uncertainty, which is possible, it just means that either you sample data was unrepresentative of the population, or that your extrapolation is wrong. Depending on what your experiment is, a linear extrapolation might lead to vastly incorrect results.
I'm not sure what ... | {
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Is the classification of (Symetry Protected) Topological Order for 3 band models different than for two band models? I was reading this article: https://arxiv.org/abs/1512.08882 on the 10 fold way which gives a nice explanation of the possible topological phases for each of the symmetry classes. The example explanation... | That is a good question. Yes, the classification theory will be different, but the answer will be much, much more complicated than the Ten-Fold Way. There are mathematical frameworks (e. g. the K-theoretic framework by Freed and Moore) which are powerful enough to investigate the question you have, but keep in mind tha... | {
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Why are there only odd numbered harmonics in one closed end resonant tube?
*
*Why do we only have odd numbered harmonics at one-end closed tubes, however, if we do a frequency spectrum we have some periodic spikes between the odd harmonic spikes, just like the picture below shows.
*What do these "even" spikes mean?... | *
*Hint: You have to count nodes & antinodes of a single closed end pipe.
*The suppressed even harmonics presumably indicates that the actual "single closed end pipe" can behave partly like a second open end, i.e. an antinode.
| {
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Additive constant in Hamilton-Jacobi theory? In Hamilton-Jacobi theory Hamilton's principal function $S$ is a function of $n+1$ constants. But we take one of the $n+1$ constants as an additive constant. I don't get this step?
| The HJ equation is a non-linear first-order PDE for $S$ in $(n+1)$ variables $(q^1, \ldots, q^n, t)$, but the PDE does not depend on $S$ directly, only its derivatives. Therefore one additive integration constant $S\to S+\alpha_{n+1}$ is trivial.
For more information, see also this related Phys.SE post.
| {
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Can any body be uniform in the universe? If I take any body in the shape of a rod and stretch that, after it reaches breaking stress it breaks at one point.
Even though we apply the same the stress on each and every part of the rod it broke at one point. If it's uniform it should break at all points because breaking st... | You could consider it as one more demonstration of the underlying quantum mechanical frame keeping atoms and molecules bonded together. Quantum mechanics is a probabilistic theory, and which bond will "break" depends on the square of the wavefunction describing the rod, with a probability which manifests in this one i... | {
"language": "en",
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Where does the time asymmetry come from in Hawking Radiation? Taking General Relativity and Quantum Field Theory, Hawking predicts radiation emitted from a black hole.
Both GR and QFT are time CPT symmetric.
Taking just GR by itself, a black hole will stay the same forever.
Taking QFT by itself a vacuum will stay the s... | The generic answer to this kind of thing is that the asymmetry comes from the choice of boundary conditions. Here, I imagine the boundary conditions are chosen to be that there are only outgoing photons, no incoming ones.
The justification for this choice of boundary conditions would be that we're really talking about ... | {
"language": "en",
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Should loop choice affect induced electric field? Let's say we have a time varying magnetic field, but that it is uniform over a region, for instance $B(x,y)=(t+2)\hat{z}$ for $x,y,z\in[-5,5]$. Since we have a changing magnetic field, there will be an accompanying induced electric field given by $\oint\vec{E}\cdot \mat... | When there is a time-varying magnetic field, the electric field is no longer conservative, meaning that $\nabla \times \mathbf{E} \neq \mathbf{0}$. Thus, $\oint \mathbf{E} \cdot \text{d} \mathbf{l}$ will depend on the path chosen.
they're the same path just shifted
This is not true in general. The paths may be geomet... | {
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Finding the Eigenvalues and Eigenvectors of the Hamiltonian for three spin-1/2 particles coupled antiferromagnetically Problem
Given three spin-1/2 particles with the total spin operator $\vec{S}=\sum\limits_{i=1}^3 \vec{S}_i$ and its $z$ projection $S_z=\sum\limits_{i=1}^3 S_{z,i}$, and the Hamiltonian
$$H = J\sum\lim... | Note that
$$S_{1}S_{2}+S_{2}S_{3}+S_{3}S_{1}=\frac{1}{2}\left(S^{2}-S_{1}^{2}-S_{2}^{2}-S_{3}^{2}\right)$$
and that's what you need. Your calculation is almost correct (your last Hamiltonian is wrong), but longer than it should be.
| {
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Color and the absorption of light in quantum mechanics In an answer to a another question, the poster states without sources the following:
From a quantum mechanical perspective, all light scattering is a form
of absorption and re-emission of light energy. Photons don't bounce
off a surface.
If this is true, what... | Reflection is just some type of coherent scattering. Scattering happens because of electric dipoles oscillating and radiating their energy to all directions. But how do they oscillate? They absorb the energy of the electric wave, meaning the electric wave scattered is the result of energy being absorbed and re-emitted ... | {
"language": "en",
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Theoretical definition and pratical mesurement of differential cross section In Sakurai's book, the definition of differential cross section is:
$$d\sigma/d\Omega= transition \;rate / probability\; flux $$
However this def doesn't contain any information about the thickness of the material or the density of target part... | Correct, the scattering cross-section is a measure of the intrinsic probability for a given process. It doesn't know anything about the experimental conditions under which the process is actually observed (density of the target material, flux of incoming particles, etc.).
These are instead encoded in what is called th... | {
"language": "en",
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How can a particle in circular motion about a fixed point accelerate, if the point doesn't too? When a particle is performing uniform circular motion attached to a string about a fixed centre, at any instant of time its acceleration is directed towards the centre but the centre has no acceleration. But I was taught in ... |
accelerations of both the ends along the string is same if the string is not slacked
Now, I understand your problem.
If you have a string placed in the shape of s in vacuum and if you start pulling it from one end it finally becomes l i.e straight.Here you can say that string isn't slackened because acceleration of ... | {
"language": "en",
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How does Newton's corpuscular theory explain the speeding up of corpuscles when entering a denser medium? I can't find an explanation for this anywhere. Intuition would imply that the corpuscle would slow down. I mean a person running at a constant speed enters a crowd of people or a forest. The presence of obstacles w... | They were talking classically in analogy to sound wave. Sound wave travel faster in metal.
See this post. Did Newton argue that particles speed up when entering a more dense medium? and http://www.physics-and-radio-electronics.com/blog/corpuscular-theory-light/ under the title "Newton’s Corpuscular Theory Statement".
... | {
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Orbital parity of simple bound states in atomic and particle physics The parity operator commutes with the Hydrogen atom Hamiltonian. The energy eigenfunctions are parity eigenstates with orbital parity $(-1)^\ell$ which follows from the fact that $Y_{\ell m}(\theta,\phi)$ is an eigenstate of parity with parity eigenva... | Basically, yes to both.
When you solve the Schroedinger equation you use separation of variables, getting two equations, one for $r$ and one for $(\theta,\phi)$. The first involves the potential $V(r)$ but the second does not. So you end up with the $Y_{\ell m}(\theta, \phi)$ spherical harmonics whatever the form of $V... | {
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Are there any general results about the nodes of energy eigenfunctions in higher dimensions? A well-known result of quantum mechanics is that for a single particle in one dimension in a bounding potential $V(x)$ that goes to $+\infty$ as $x \to \pm \infty$, the energy eigenfunctions are discrete and the $n$th eigenfunc... | The result is actually applicable to 1d-equivalent motion, and as such is applicable to the radial part of the Schrodinger equation in any dimension if this radial part can be separated.
In general, the twist is that the equivalent 1d motion depends on the effective potential - in the case of a 3D central potential the... | {
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Why does gauge invariance in electrodynamics mean that there are redundant degrees of freedom? It is possible to choose different gauges in electrodynamics. I am familiar with two of them: Coulomb gauge and Lorenz gauge. Let us stick to the Coulomb gauge. It sets $$\nabla\cdot\vec{A}=0.$$ The wisdom is that with this c... | The existence of "superfluous" degrees of freedom is implies not so much by the gauge condition, but by the gauge transformations itself.
The point is that you four "numbers", i.e. functions, $\phi$ and $A_i$ are equivalent to four other functions, $\phi + \dot\alpha$ and $\vec A+\text{grad}\, \alpha$ for any function... | {
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Relativistic space rocks. Are they possible? I was thinking that the Universe is full of extreme events. Colliding galaxies, exploding stars, colliding planets (like in one of models of our Moon formation), colliding black holes...
Can it be possible, that such event would accelerate some rocks to let's say $0.1c$ ?
D... | Yes, if a rock is on a sling shot trajectory around a black hole, or a neutron star. However that speed would be relative to that black hole, or the neutron star. The speed of that rock can be very different relative to earth. I think spaghettification would not be an issue at .1c but actual calculations need to be don... | {
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How do we know quantum entanglement works no matter the distance? It is said quantum entanglement works regardless of distance. 2 particles can be entangled and information is shared instantaneously, even if they are lightyears away from each other.
But how do we know this still works with such a vast distance between ... | I consider the question of how far quantum entanglement works this way:
For as long as entanglement is considered a mysterious Quantum phenomenon and discussed in terms of wave functions or other such mathematical descriptions it is hard to think of it as something that can survive great distances across space (and tim... | {
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How many eyes are needed to see a four-dimensional world? If a four-dimensional world were to exist, how many eyes would a creature minimally need to see it (in three dimensions)?
Three? Four?
(Bonus question: how should these eyes be spatially configured?)
| You can already see four dimensions with the two eyes you've got. It's just that two of those dimensions happen to be the same, so you don't distinguish them. But it could be otherwise.
In addition to distances up/down and left/right, you can perceive distances ahead of you by the fact that the light rays reaching yo... | {
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Can a free electron accelerating in a gravitational field absorb photons? An 'free' electron accelerated in an electromagnetic field can both absorb and emit a photon. What about an election accelerating in a gravitational field?
Edit:
Some users have suggested that the question is a duplicate. However, my question ask... |
A 'free' electron accelerated in an electromagnetic field can both absorb and emit a photon.
Both electrons and photons belong to the table of elementary particles in the standard model of particle physics, i.e. are quantum mechanical entities and have to be modeled as such. Thus , an electron does not absorb a photo... | {
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What is the difference between Wien's Displacement Law for peak frequency vs peak wavelength? While doing research for a high school report I came across the fact that WDL actually has two forms, one for peak frequency and one for peak wavelength, and that these two forms are not the same and can not be used interchang... | The blackbody radiation curve presents the spectral intensity density, that is, the radiant power per unit of ... whatever unit you please. Watts per unit interval of wavelength or watts per unit interval of frequency.
The problem is that a unit interval of wavelength is not the same as a unit interval of frequency,... | {
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Why isn't the emissive power of a black body 1? My text book has a question which says that the emissive power of a black body isn't one but the answer states that the absorptive power is 1, considering that $$e=a \tag{Kirchoff's law}$$
and a black body is defined as an object which has $$e=1$$ Then why isn't $a=1$
... | The emissivity ($e$) of a perfectly black body is 1. Emissivity is the ratio of energy emitted by a body to the energy emitted by a black body. The emissive power is the energy emitted per unit area per unit time. The emissive power of a black body is not 1, and it varies with temperature.
It is true that emissivity = ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/440304",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "1",
"answer_count": 2,
"answer_id": 1
} |
Geodesics of anti-de Sitter space It is said that (p. 9), given the anti-de Sitter space $\text{AdS}_2$, let's say in the static coordinates
$$ds^2 = -(1 + x^2) dt^2 + \frac{1}{(1+x^2)} dx^2$$
Every timelike geodesic will cross the same point after a time interval of $\pi$. That is, if $(x_0, t_0) \in \gamma$, then $(x... | "Every timelike geodesic will cross the same point after a time interval of $\pi$" will be true if the half-period is $\pi$. You found the general solution for $x(\tau)$, namely
$$x(\tau)=A\sin\tau+B\cos\tau$$ or, alternately, $$x(\tau)=A\sin{(\tau-\tau_0)}.$$ When $\tau$ increases by $\pi$, $x$ does come back to what ... | {
"language": "en",
"url": "https://physics.stackexchange.com/questions/440470",
"timestamp": "2023-03-29T00:00:00",
"source": "stackexchange",
"question_score": "8",
"answer_count": 2,
"answer_id": 0
} |
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