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Confused about tensor notations of how vector and covectors act on each other

First, you can shorten things considerably by not changing coordinates and then changing back on the second term. $\delta^i_j=v^iv_j=(v^i(w_l)w^l)v_j=v^i(w_lw^l)v_j=v^inv_j=n\delta^i_j$ We can also ...
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Confused about tensor notations of how vector and covectors act on each other

I replaced the repeated $j$ in OP's question with $l$ to make the equations clearer. OP's derivation is as follows. \begin{aligned} \delta^i_j &= v^i v_j \\ &= \left(v^i (w_l) w^l \...
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Showing that the distance between two adjacent planes in a crystal is $d=2π/|\vec{G}|$

The solution occurred to me while I was writing the question, so I answer myself. Let $\vec{R}=n_1\vec{a_1}+n_2\vec{a_2}+n_3\vec{a_3}$ be the position of a point $P$ in the plane $(hkl)$, and let $\...
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4 votes

Clarification on clock synchronization passage from Kleppner and Kolenkow

You are not interpreting the described Newtonian procedure correctly. The moon clock is one second ahead of the earth clock so that seen from the earth the two tell the same time. But seen from the ...
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  • 577
5 votes
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Angle between the two vectors

The arcsine function on a calculator or computer will only return values between -90° and 90°. You have to add 180° to your answer of -18.5 to get the correct result of 161.5. Arccosine returns ...
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Multiplying non-constant acceleration by a constant on every point - would the total path be multiplied by the same constant?

If a(t) is multiplied by c the integral over a(t) and c*a(t) you can easily compare, the same with again taking the integral. So what is the problem?
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1 vote
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Prove that every component of angular momentum commutes with $f$

Commuting with $L$ is equivalent to being invariant under rotations. The quantities $r^2$, $r\cdot p$ and $p^2$ are all rotationally invariant, as is any function of them. That is all that is needed.
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Problem trying to graphically represent a 2D vector given angle and intensity

The angle of a vector is usually measured from the positive x-axis to the vector, with clockwise angles counting as positive. Thus a vector in the same direction as the positive y-axis has an angle of ...
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1 vote

Understanding conservation of energy in a pulley problem

I think you can argue as follows. We have a tension T acting on both masses. The variation of kinetic energy for $m_1$ is $$ KE^1_{f} - KE^1_{i} = \int_0^h (- m_1 g + T) \ dx, $$ while for $m_2$ we ...
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1 vote

Force diagram - Circular motion

I don't understand that because if N is the normal contact force, it should be perpendicular to the wall In this case they are just using the variable $N$ to represent the total contact force with ...
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If there is no net force, but the mass can change, can momentum remain unchanged?

In Newtonian mechanics and relativistic mechanics both B and C are true, and A and D are false. If we define, net force as the derivative of momentum for a body we have: $$\frac{\text{d}\boldsymbol{p}}...
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1 vote

Let $Q(t,\vec x)$ solve $\partial_t^2 Q = \nabla^2Q$. Why $\partial_t^2Q = 2 (\partial_r + r^{-1})\partial_{t-r}Q$?

The impeccable answer of @CWPP identifies the problem and leads to the straightforward correct answer; I'm only writing this as a footnote with an explicit expansion of his point to avoid the ...
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Probability of Different States - Canonical Ensemble - Partition Function

"Ideal gas" means independent particles. When particles are independent you can study one particle and you know what all other particles do on average. Mathematically, the $N$-particle ...
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3 votes
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Let $Q(t,\vec x)$ solve $\partial_t^2 Q = \nabla^2Q$. Why $\partial_t^2Q = 2 (\partial_r + r^{-1})\partial_{t-r}Q$?

You are not transforming the differentials correctly. $\partial/\partial t|_r=\partial/\partial u|_r$ and $\partial/\partial r|_t=\partial/\partial r|_u-\partial/\partial u|_r$.
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  • 577
1 vote

Tong QFT Problem set 2, question 6: Normal ordering of angular momentum operator

Okay, here is what I get $$\int d^3\vec{x}\ x^jT^{0k}=\int d^3\vec{x}\int \not{d}^3\vec{p}\int \not{d}^3\vec{p}' \omega_p p^{'k}\Big[ a_px^je^{-i(p+p')\cdot x}a_{p'}- a_px^je^{-i(p-p')\cdot x}a_{p'}^{...
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  • 1,585
1 vote

Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end

Alternative approach (if you are interested) is to note that in simple harmonic motion the mass will be in equilibrium at the mid-point of the motion, when the extension is $x=2.5$ cm. At this point, ...
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3 votes
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Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end

Yes, it is the force $F=mg$ that causes the length of the spring to elongate, however, the condition $F=k\Delta x$ gives you the elongation $\Delta x$ of the spring at which the upward force from the ...
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2 votes

Can't figure out my mistake in solving the problem of spring hanged from the ceiling and a weight attached at its end

The problem with your approach is that you are assuming the spring to be in equilibrium. The reality is the spring actually undergoes Simple Harmonic Motion. The maximum extension of the spring occurs ...
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  • 270
3 votes

Why does a stream of falling water get narrower at the bottom?

There will be an additional contraction term due to the fact that the water entering the stream sideways carries with it a velocity component in the x and y directions, but I do not know how to ...
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8 votes
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Why does a stream of falling water get narrower at the bottom?

Is my line of reasoning correct? Is there any incorrect assumptions that I made? Is this resoult coherent or even correct? Seems reasonable to me, at least to zeroth order. You are basically using: $$...
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1 vote
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Determine velocity vector on sloped surface

the components of the velocity vector given in ball system are $$\mathbf v_B= \left[ \begin {array}{c} v\sin \left( \beta \right) \\ v\cos \left( \beta \right) \\ 0 \end {array} \right] $$ to ...
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How do I calculate the mass of a moving proton?

Relativistic mass $= m_{rest}+ \frac{E_{kin}}{c^2}$ Relativistic mass$ = \frac{1}{\sqrt{ 1-v^2/c^2 }} * m_{rest}$ Relativistic mass of proton is good for: Calculating force needed to steer the ...
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5 votes

How do I calculate the mass of a moving proton?

Suppose we accelerate a proton to a speed $v$ then crash it into some large mass $M$ so that the proton comes to rest inside the large mass. Then we measure the speed $u$ that the large mass + proton ...
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Earth to Low Earth Orbit (LEO), gravity drag and potential energy

I suspect what is missing is the free velocity gained by launching in the same direction as the rotation of the Earth. The closer you are to the Equator, the more "free velocity" you can get ...
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Motion in one dimension. One car overtaking a different one

Just a small supplement to the solution in the book: With start time $t_0 = 0$ the truck and car reached a distance $s_{truck} = ut \ \ \ $ (0.1) and $s_{car} = 1/2 a t^2 \ \ \ $ (0.2) w.r.t. the ...
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1 vote

Motion in one dimension. One car overtaking a different one

From equation (1) we have $t =\frac s u$ Substituting for $t$ in equation (2) gives $s = \frac 1 2 \frac {as^2}{u^2}$ Rearranging this gives equation (3).
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2 votes

Help with an integral in Peskin & Schroeder - QFT

Hint: $$\begin{align} \int_{m}^{\infty}\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt} ~=~~~&\int_{m}^{-i\infty}\! dE~\sqrt{E ^2 - m ^2} e ^{-iEt}\cr ~\stackrel{E=-imz }{=}&-m^2\int_{i}^{\infty}\! dz~\sqrt{...
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Physical interpretation regarding heat equation

It implies that the net vertical flow of heat at any elevation y due to the T* part of the solution is zero, and that the net (downward) flow of heat (per unit distance into the paper) is constant at $...
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Help with an integral in Peskin & Schroeder - QFT

P&S do not employ the method of residues to reach the final conclusion you cite. Instead, what they do is simply to make a variable substitution of the form $E=\sqrt{p^2+m^2}$. So, then, if you ...
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2 votes

Physical interpretation regarding heat equation

One possible interpretation could come from noticing that: $$\frac{1}{L}\int_0^Lf(x)\,dx$$ is the mean value of $f$ on $[0,L]$. That woud make: $$\frac{1}{L}\int_0^L\frac{\partial T}{\partial x}(x,y^*)...
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Would water flow from the higher container to the lower one?

Shouldn't the pressure be the same in both buckets, meaning that the water does not flow? This is from a comment, and is the crux of the reasoning for the question. The answer below explains why ...
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2 votes

Would water flow from the higher container to the lower one?

I like the willingness to experiment! The result of the experiment is indeed expected. Basically, because there is a connection this is all one body of water. If the surface of a body of water is ...
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3 votes
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How to show that $\psi(x)=-A\exp(-\alpha^2x^2)$ satisfies TISE for $V(x)=\frac 1 2 m\omega_0^2x^2$?

Write the TISE as $$\frac{d^2\psi}{dx^2}=\frac{m^2\omega_0^2x^2-2mE}{\hbar^2}\psi$$ and observe that for large $x$, $2mE$ is negligible compared to $m^2\omega_0^2 x^2$. Therefore, you can write it as $...
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0 votes

How do you find find power dissipated by lightbulb when given voltage, all values for resistance (except the bulb), and a value for current?

You know the voltage across the 330 ohm resistor, therefore you can find the current through it. Then you can apply KCL at either of branching nodes to find the current through the branch with the ...
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1 vote

Kinetic energy and de Broglie wavelength of an electron

You're right: the parameter $\delta$ is not dimensionless. The question is telling you to choose units so that $E$ in eV gives you $\lambda$ in angstroms. A better way to phrase this is “please ...
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1 vote

When blowing into a coke bottle, do all ambient frequencies outside of a coke bottle exist in the bottle along with the Helmholtz resonant frequency?

The coke bottle is a selective filter which only admits & re-radiates the resonant frequency of the bottle. The mass of air inside the neck rolls off the higher frequencies which hence cannot ...
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3 votes

When blowing into a coke bottle, do all ambient frequencies outside of a coke bottle exist in the bottle along with the Helmholtz resonant frequency?

In any resonant cavity, there will be many transient frequencies that only persist for a very short period due to destructive interference. Only resonant frequencies will constructively interfere (i.e....
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2 votes

Calculation of Lagrangian from Hamiltonian $\frac{1}{2}(-i\partial_\phi -A)^2$

We start with the Lagrangian $$ L_M~=~\frac{m}{2}\left(\frac{d\vec{r}}{dt_M}\right)^2 + q \vec{A}\cdot\frac{d\vec{r}}{dt_M}-q\phi_M, $$ for a non-relativistic point particle in Minkowski space ...
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If there is no net force, but the mass can change, can momentum remain unchanged?

A mining cart is rolling on the tracks. Suddenly you drop a rock into the cart. No net force was ever exerted horizontally, yet the cart will slow down since it has to pull the newly added mass up to ...
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4 votes

If there is no net force, but the mass can change, can momentum remain unchanged?

The answer book is simply wrong. Both B and C are true. Within the context of Newtonian mechanics, I maintain that a “body” is rigid, and its mass is constant. The answer book claims the speed can ...
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If there is no net force, but the mass can change, can momentum remain unchanged?

Thinking of it, you are absolutely right... How can a body lose mass: In most cases we encounter, mass is lost from a body as smaller masses. Rocket propulsion, Evaporation, sand leaking from a ...
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If there is no net force, but the mass can change, can momentum remain unchanged?

So, in the question we should assume that the mass is allowed to change but all external forces cancel. How can the mass change with no external forces? Here is an example. The body spontaneously ...
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If there is no net force, but the mass can change, can momentum remain unchanged?

Translation $$\frac{d}{dt}(m\,\vec v)=\frac{d}{dt}(\vec p)=\vec 0\quad\Rightarrow$$ linear momentum $~\vec p~=$ constant and from the equations of motion $$m\,\frac{d\vec v}{dt}=\vec 0\quad\Rightarrow$...
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Is Wave function normalizable?

Unless you forgot to add some extra sentences that came before the question, this question is badly defined. So I'll assume that this particular wavefunction is defined on $\mathbb{R}^2$. Then it's ...
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Is Wave function normalizable?

Hint: Look up square-integrable functions. In Quantum mechanics, the wave function of a state has to be square integrable.
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1 vote
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Two questions concerning dirac delta function and Hamiltonian

The first is simple I think: $$\frac{d}{dE}\theta[E-H(x,p;V)]=\delta[E-H(x,p;V)]=\delta[H(x,p;V)-E]= \delta\Big[\frac{p^2}{2m}+\phi(x;V)-E\Big]$$ where in the third step I used the symmetric property ...
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Magnetic field lines of a hollow rod with current

Are you sure that you have quoted the problem correctly? It is true that externally the magnetic field is the same as that of a uniform conductor, but internally the fields differ. The equivalence of ...
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  • 577
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If there is no net force, but the mass can change, can momentum remain unchanged?

The question is wrong and it has some mistakes. Let me explain why? Option A isn't correct because the body can rotate or it can move with a constant velocity when all the external forces are taken ...
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2 votes

The Inner Product of In and Out Spins

Be careful in a complex vector space. $$\langle i|=\frac{1}{\sqrt{2}}\langle u|-\frac{i}{\sqrt{2}}\langle d|$$ so you forgot to take the complex conjugate. The result is immediate after that.
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0 votes
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2D Conformal Group

Hint: do the right thing. $$ (z,\bar z)\mapsto (r,\phi) , \\ r=\sqrt{z\bar z}, \qquad \phi= {1\over 2i}\log(z/\bar z),\\ \partial_z= (\partial_z r )~~\partial_r + (\partial_z \phi )~~ \partial_\phi ~~...
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