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### Why Curie temperature is bigger for smaller lattice in 2D Ising model

Why numerical simulation gives seemingly wrong result? The exact expression for $T_c$ is given by $$T_c=\frac{2J}{\ln\left(\sqrt{2}+1\right)}\approx 2.269\,J,$$ where we assume that $k_B=1$. This ...
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The most general rule for uncertainty propagation comes from a derivative of the function. If you sketch a graph of a function, you can see how this works. For a function $f(x)$, if $\frac{\partial ... • 6,866 5 votes ### Conceptual misunderstanding in Buoyant forces Suppose you have two setups. One has layers of oil and water. The other has "layers" of water and water. In the second case, the ice floats higher than it would without the upper layer. So ... • 36.3k 5 votes Accepted ### Uncertainty on the sum of two non-commuting operators Instead of giving a rigorous argument based on the representation of Heisenberg-Weyl group, I suggest a physically minded argument which could be made rigorous however. (I henceforth assume$\hbar=1$) ... 5 votes ### Uncertainty on the sum of two non-commuting operators It does not matter that$Q$and$P$do not commute, if they are self-adjoint, so is your$E$, and the square of a self-adjoint operator is a non-negative operator: $$\langle \psi \vert E^2 \vert \psi\... • 122k 4 votes Accepted ### Statistical mechanics of vibrating string You can use the equipartition theorem since in fourier space the hamiltonian is a sum of uncoupled quadratic harmonic oscillators:$$E[y(x,t)] = \sum_{n=1}^N\left(\dfrac{M/2\dot A_n^2}{2} + \frac{n^2\... • 856 3 votes Accepted ### Clarification Needed for The Klein-Gordon Field Acting on the Vacuum State (Peskin and Schroeder) I think your confusion is because you have confused the vacuum state$\lvert 0 \rangle$with the zero vector. The two are not the same thing; the first one is a non-zero vector in the Fock space with ... • 47.2k 3 votes ### Conceptual misunderstanding in Buoyant forces Consider a vessel that is so big that the volume of the ice is negligible compared to the volume of water and of oil. This means that the height of the interface between water and oil is the same ... 3 votes ### Conceptual misunderstanding in Buoyant forces The thing that really matters is the relative pressure variations surrounding the ice, not the absolute pressure. When you put oil above the interface in place of air, the pressure relative to the ... • 32.8k 3 votes ### Bar suspended by three vertical ropes This is a canonical example of an underdefined problem (the number of unknowns (3 tensions) is less than the number of conditions (torque=0 and total force=0). You've tried to come up with a third ... • 3,451 3 votes Accepted ### The origin of energy density formulas in Maxwell's electromagnetism It's important to realize Poynting's theorem, stated in terms of the fields, is valid regardless of what you call$u = \frac{1}{2}\vec E\cdot\vec D + \frac{1}{2}\vec H\cdot\vec B$. In other words, you ... • 11.9k 3 votes Accepted ### Does this question require any calculus to solve? The multiple-choice answers are an editing mistake. No one measures distances in joules. You suggest in a comment that perhaps the intention is to integrate over the motion to find work done. But you ... • 86.3k 3 votes Accepted ### Solution to Laplace equation in spherical coordinates and Legendre polyomials Is there any physical reason that enforces us to use the polynomials instead of the general series solution? Yes. Legendre functions where$l$is not an integer diverge at$x=1$or$x=-1$or both, ... • 6,879 3 votes Accepted ### Is it possible to analytically find the eigenvalues and eigenvectors of the following matrix? Not really an answer, more of a trail-map. Your Toeplitz matrix is also manifestly hermitian, so $$M= R~{\mathbb I} +N,$$ for hermitian traceless N which determines all real eigenvalues and ... • 60.5k 3 votes ### What is the average distance$\langle r \rangle$of a particle in an ideal gas in a potential from the origin? Your calculations look fine, but notice that you could've calculated$\langle r \rangle$directly from the partition function since $$\langle r \rangle=-\frac{1}{\beta}\frac{\partial \log Z}{\partial \... • 1,550 2 votes ### Linear Momentum conservation Since there is a hinge at the end of the ruler, linear momentum is never conserved because the hinge is applying force on the ruler when acted by a force. However, angular momentum is conserved about ... 2 votes Accepted ### 3-dimensional 3-state Potts model critical temperature Small caveat: if you’re doing mean field, you lose any notion of dimension, you effectively replaced your lattice by a complete graph. It is therefore contradictory to talk about mean field and ... • 9,603 2 votes Accepted ### Obtaining Geodesic equation for Massive particles using Schwarzschild metric The second equation doesn't really come from the Euler-Lagrange equations (though one could probably derive it from them.) It's much easier to see from the definition of the particle's proper time:$$... • 47.2k 2 votes ### Two-dimensional Two-particle system You are told both$H_a$and$H_b$are simultaneously diagonalizable to diag($E_0,E_1\$) in their respective bases. So the full hamiltonian diagonalizes to the 4x4 matrix $$H^0= \operatorname{diag}(E_0,... • 60.5k 2 votes ### Can two normal 1D waves form a wave packet? The resulting wave will look like that pictured below, and I agree with mmesser314's answer that it would be unusual to call such a wave a "wave packet". That said, it does make sense to ... • 11.9k 2 votes Accepted ### Explanation of diffraction of a single light ray by Huygens' principle Huygens principle works when light is treated as a wave. Here, the diffraction patterns arise due to the interference of different points of the wavefront. However, when you talk about a single ray, ... • 2,010 2 votes Accepted ### Fourier transform of the Heisenberg antiferromagnetic model According to well known identity, You have$$ \sum_ke^{ika}\hat{a}_k\hat{a}_{-k} = \sum_k \cos(ka)\hat{a}_k\hat{a}_{-k} + i\sum_k \sin(ka)\hat{a}_k\hat{a}_{-k} $$Since the sum contains -k for each ... • 4,740 2 votes ### Voltage across a capacitor in a circuit "Am I supposed to find the voltage across each resistor first?" Yes, that's a good first step. [The quick way is to treat the left and right resistor combinations as potential dividers.] ... • 34.9k 2 votes Accepted ### Moment of inertia of a rolling cylinder The formula I=\frac{1}{2}mr^2 for the moment of inertia of a solid cylinder is only valid, if its density is homogeneous. But it is not valid anymore, if the density varies with radius. For example: ... • 36.4k 2 votes ### Explanation of diffraction of a single light ray by Huygens' principle I planned a comment, but after reading the existing ones, I think a direct answer addressing the main conceptual issue could be more appropriate. There is nothing like a straight-line, one-dimensional ... 2 votes ### Minimal Time for Quantum System to Reach Orthogonal State You have the right steps and final answer, but in reference to your question about showing the time you found to be the minimal time, realise that when you write \log(-1) = i\pi, you are making the ... 2 votes Accepted ### Minimal Time for Quantum System to Reach Orthogonal State Your solution is indeed correct and elegant! To find the minimal time in terms of the expectation value of the energy, you just need to compute: Calculate the expectation value of the energy at time: ... • 1,158 2 votes Accepted ### Using Galilean transformation to solve a question with a block-spring-block system As you anticipated, your argument breaks down because Block 2 is accelerating, so there is no inertial frame corresponding to "the frame of Block 2." To see that it must be accelerating, ... • 46.5k 2 votes Accepted ### Dirac Equation and the Klein-Gordon Equation It's simple if you think about how you change the summation variables. \gamma^\nu \gamma^\mu\partial_\mu\partial_\nu=\frac{1}{2}(\gamma^\nu \gamma^\mu\partial_\mu\partial_\nu+\gamma^\nu \gamma^\mu\... 2 votes ### What is the average distance \langle r \rangle of a particle in an ideal gas in a potential from the origin? Yes, the answer looks correct.$$ Z_1 = \int d^3\mathbf{p} d^3\mathbf{q}e^{-\beta\frac{\mathbf{p}^2}{2m}-\beta u(\mathbf{q})}= \int d^3\mathbf{p} e^{-\beta\frac{\mathbf{p}^2}{2m}} \int d^3\mathbf{q}e^{...
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