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Results tagged with homework-and-exercises
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user 215452
Applies to questions of primarily educational value - not only questions that arise from actual homework assignments, but any question where it is preferable to guide the asker to the answer rather than giving it away outright. Please READ THE GUIDANCE IN META before asking homework-like questions.
3
votes
1
answer
79
views
Stability of Chemical Reactions [closed]
Given the following reactions:
$$A + X \xrightarrow{k_{1}} 2 X$$
$$Y + X \xrightarrow{k_{2}} 2 Y$$
$$Y \xrightarrow{k_{3}} B$$
I was able to write the following rate equations for the concentrations:
…
- 653
1
vote
1
answer
76
views
Action of $M_{\mu \nu}$ on local operators $\mathcal{O}(x)$
I'm following the TASI Lectures on the Conformal Bootstrap by David Simmons-Duffin.
Let $M_{\mu \nu}$ be the conserved charge operator associated with rotations. The action of said operator on local o …
- 653
2
votes
1
answer
177
views
Conserved charge operator CFT
In David Simmons-Duffin's TASI lectures on conformal bootstrap, there is a discussion about conserved charges and operators, which is as follows:
Given $\epsilon = \epsilon^{\mu}(x) \partial_{\mu}$, …
- 653
3
votes
2
answers
459
views
Euclidean propagator expression for massless particle
Let $\Delta_F(\tilde{x})$ denote the Feynman propagator in the Euclidean variable $\tilde{x}$, in $D$ dimensions,
$$\Delta_F(\tilde{x}) = \int \frac{\text{d}^D\tilde{p}}{(2\pi)^D}\frac{e^{i\,\tilde{p} …
- 653
1
vote
Accepted
Lorentz boost of Dirac spinor
Thanks to @G. Smith and @mike stone, I've come to a solution.
Expanding in Taylor Series,
$$ S(\delta_x) = e^{\frac{\delta_x}{2}
\begin{pmatrix}
0 & \sigma_x \\
\sigma_x & 0
\end{pmatrix}} = \sum_{n …
- 653
1
vote
1
answer
1k
views
Lorentz boost of Dirac spinor
Let $\psi_\vec{0}^+$ be a Dirac wavefunction describing a motionless particle,
$$\psi_\vec{0}^+(x) = \sqrt{2m} \begin{pmatrix}
\chi \\
0
\end{pmatrix} e^{ip \cdot x}$$
where $p = (m, \vec{0})$. Acte …
- 653
1
vote
Matrix Representation of Lorentz Group Generators
Thanks to @Charlie and @Cosmas Zachos I was able to find the correct answer.
It simply suffices to develop the sum
$$\frac{\omega^{\alpha \beta}}{2}\left(J_{\alpha \beta} \right)^{\mu}{}_{\nu} = -\del …
- 653
1
vote
2
answers
1k
views
Matrix Representation of Lorentz Group Generators
Let $\Lambda^{\alpha}{}_{\beta}$ denote a generic Lorentz transformation.
Then, an infinitesimal transformation can be written like
$$\Lambda^{\mu}{}_{\nu} = \delta^{\mu}{}_{\nu} + \omega^{\mu}{}_{\n …
- 653
2
votes
1
answer
275
views
Transition amplitude integral and causality
I was trying to prove that Quantum Mechanics violates causality.
To do that, I started by computing the transition amplitude between the fixed position $x_0$ and an arbitrary position $x$, during a ce …
- 653
1
vote
Accepted
Master equation for reproduction and mutual annihilation process
After searching in several books, I found out that the expression is, indeed, correct. However, some authors prefer to absorve the $\frac{1}{2}$ in the definition of $\lambda$.
For future reference, a …
- 653
2
votes
1
answer
69
views
Master equation for reproduction and mutual annihilation process [closed]
I was solving some exercises regarding the Master Equation and couldn't solve the following problem.
Consider a population with individuals $A$. This population can suffer the following processes:
i) …
- 653
3
votes
1
answer
377
views
Ising model 2D and mean field theory
Consider the 2D Ising model. Now, let's divide it into 4-spins blocks and treat the interaction inside each block exactly, while applying the mean-field approximation to the interaction between blocks …
- 653
1
vote
0
answers
62
views
Renormalization theory in system with two coupling constants
Suppose a system has 2 coupling constants, $t$ (temperature) and $h$ (applied field). Let $K_{1}$ and $K_{2}$ be coupling constants, such that
$$\left[ \begin{array}{c} K_1 \\ K_2 \end{array} \right] …
- 653
1
vote
Accepted
Fourier transform of linear response function
Thanks to user110971 in the comments, I think I've managed to find the solution.
According to Wikipedia, the Convolution Theorem states that if $f$ and $g$ are two functions, then $f \ast g$ denotes t …
- 653
1
vote
1
answer
168
views
Fourier transform of linear response function
I was studying Linear Response Theory from 'A modern course in statistical physics' by Reichl, and some doubts came up.
The response function is defined as
$$<\alpha(t)>_{F} = \int_{-\infty}^{+\infty} …
- 653