Recent Questions - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2023-10-03T18:38:04Z https://physics.stackexchange.com/feeds https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/782967 1 Does the energy-momentum tensor inside Einstein's field equation include gravitational stress-energy? K. Pull https://physics.stackexchange.com/users/319518 2023-10-03T18:26:38Z 2023-10-03T18:35:57Z <p>The <a href="https://en.wikipedia.org/wiki/Einstein_field_equations" rel="nofollow noreferrer">Einstein field equations</a> <span class="math-container">$$R_{\mu\nu} - \dfrac{1}{2}Rg_{\mu\nu} = \kappa T_{\mu\nu}$$</span></p> <p>relate the space-time curvature <span class="math-container">$R_{\mu\nu}$</span> to the stress-energy <span class="math-container">$T_{\mu\nu}$</span> present in the system. I wondered whether <span class="math-container">$T_{\mu\nu}$</span> included the stress-energy due to gravity?</p> https://physics.stackexchange.com/q/782965 -1 Does galactic evolution approximate the distribution of the prime numbers due to the 2nd law? Alexander Alleavitch https://physics.stackexchange.com/users/379593 2023-10-03T18:16:38Z 2023-10-03T18:16:38Z <p>So, this question occurred to me after watching these two videos in quick succession and then doing some additional lit search, I apologize if my terminology is imprecise/inaccurate since this isn't my field but hopefully I can communicate my meaning here:</p> <p><a href="https://www.youtube.com/watch?v=Jm7omDy5_38" rel="nofollow noreferrer">https://www.youtube.com/watch?v=Jm7omDy5_38</a></p> <p><a href="https://www.youtube.com/watch?v=EK32jo7i5LQ" rel="nofollow noreferrer">https://www.youtube.com/watch?v=EK32jo7i5LQ</a></p> <p>We just learned that our models of galactic evolution were incorrect due to data from the JWST, and we currently don't know why galaxies form into spirals. What if galaxies actually just tend towards this shape over time because they are actually approaching exactly this distribution of the prime numbers when mapped onto polar coordinates? It even makes sense; every star has a gravitational influence on every other star in the galaxy, you would expect that their orbits would be influenced by resonance, they would naturally distribute based on approximating the distribution of the prime numbers which is necessarily the most chaotic pattern/the orientation with the highest entropy. Perhaps galaxies form spirals simply because they are following the 2nd law of thermodynamics, and the fact that you see four-armed spiral galaxies is exactly because of PNT sorting all the stars into four arms. In which case &quot;dark matter&quot; has actually always just been the 2nd Law. There may even be some relationship to the different spiral and elliptical galaxies we see and the shape of this plot of the prime numbers at different scales: at more &quot;zoomed out&quot; approximations you see a more elliptical shape and at more &quot;zoomed in&quot; approximations you see spirals with arms of different densities.</p> https://physics.stackexchange.com/q/782963 0 Speed of hitting the ground [closed] codproe https://physics.stackexchange.com/users/379607 2023-10-03T18:03:11Z 2023-10-03T18:33:43Z <p>Projectile is fired with initial speed of 20m/s at an angle of 30 degrees from the building with 15m height. Find speed of hitting the ground.</p> <p><span class="math-container">$v_x=v cos \alpha =17.32 \frac{m}{s}$</span></p> <p><span class="math-container">$v_y=v sin \alpha = 10 \frac{m}{s}$</span></p> <p><span class="math-container">$-h=v_y t - \frac{gt^2}{2}$</span></p> <p><span class="math-container">$t=2.8s$</span></p> <p><span class="math-container">$v_{1x}=v_x = 17.32 \frac{m}{s}$</span></p> <p><span class="math-container">$v_{1y}=gt = 27.488\frac{m}{s}$</span></p> <p><span class="math-container">$v_1=\sqrt{v_{1x}^2+v_{1y}^2}=32.473 \frac{m}{s}$</span></p> <p>Could you please help me with this? Is my result of <span class="math-container">$v_1$</span> (speed of hitting the ground) correct?</p> https://physics.stackexchange.com/q/782961 0 Need Recommendations for Nuclear Reaction Modelling Software Arsen Argandykov https://physics.stackexchange.com/users/379606 2023-10-03T17:47:22Z 2023-10-03T17:47:22Z <p>Working on a project where I need to model nuclear reactions with different fuel pellet shapes. Does anyone have recommendations for software that's good for this? Any advice or experiences are also welcome!</p> https://physics.stackexchange.com/q/782960 1 Is Monopolium Theoretically Stable? Hokon https://physics.stackexchange.com/users/373601 2023-10-03T17:41:12Z 2023-10-03T17:41:12Z <p><em>Monopolium</em> is the hypothetical composite of either (1) a north-south pair of monopoles, or (2) a magnetic monopole that has captured an atomic nucleus, analogous to how atomic nuclei capture electrons. My question concerns (2) and to be specific, the capture of an Aluminum-27 nucleus.</p> <p>In <em>Monopole ‘83</em>, it is discussed and mathematically shown on <a href="https://link.springer.com/chapter/10.1007/978-1-4757-0375-7_30" rel="nofollow noreferrer">pgs. 333-37</a> that (theoretically) Al-27 nuclei can be captured by monopoles, with a resulting ionization energy of ~560 keV.</p> <p><strong>Question:</strong> Is this configuration (theoretically) stable, and if so, can this kind of monopolium be able to form a bond with another “atom” of monopolium, in a similar way to how ordinary atoms can form molecules, as a <em>stable</em> monopolium “molecule”?</p> https://physics.stackexchange.com/q/782959 0 Why is a conserved quantity under a certain transformation the generator of that transformation? abc https://physics.stackexchange.com/users/311113 2023-10-03T17:41:10Z 2023-10-03T17:41:10Z <p>I would like to know if there is any explanation at the level of a first introdutory course in quantum mechanics of the fact that if we have a conserved quantity under a certain transformation, then that quantity is the generator of that transformation. For example we can prove using Noether's theorem that the momentum is conserved under a spatial translation. But why does it imply that the momentum operator is the generator of the translation transformation? I would like to stress the fact that I'm looking for an undergraduate level explanation, if there's any, like of an introductory course to quantum mechanics.</p> https://physics.stackexchange.com/q/782957 0 How to recognize Feynman diagrams from the $S$-matrix expansion? Filippo https://physics.stackexchange.com/users/355699 2023-10-03T17:35:36Z 2023-10-03T17:39:12Z <p>I'm studying scattering processes in QED and one usually have to compute first of all the Scattering matrix <span class="math-container">$$\hat{S}=T\biggl (\exp\{-i\int d^{4}x:\bar{\psi}(x)\gamma_{\mu}\hat{A}^{\mu}(x)\hat{\psi}(x):\}\biggr)=\sum_{n=0}^{\infty}\hat{S}^{(n)}.$$</span> One could show that the first order term of the expansion is useless since each term is kinematically not allowed, however the second-order is the physical relevant one and is given by (exploiting the Wick Theorem)</p> <p><span class="math-container">$$\hat{S}^{(2)}=\frac{(-ie)^{2}}{2!}∫ d^{4}x_{1}d^{4}x_{2}T\biggl(:\hat{\bar{\psi}}(x_{1})\gamma_{\mu}\hat{A}^{\mu}(x_{1})\hat{\psi}(x_{1})::\hat{\bar{\psi}}(x_{2})\gamma_{\mu}\hat{A}^{\mu}(x_{2})\hat{\psi}(x_{2}):\biggr)=$$</span></p> <p><span class="math-container">$$=\frac{(-ie)^{2}}{2!}∫ d^{4}x_{1}d^{4}x_{2}:\hat{\bar{\psi}}_{1}\gamma_{\mu}\hat{A}^{\mu}_{1}\hat{\psi}_{1}\hat{\bar{\psi}}_{2}\gamma_{\mu}\hat{A}^{\mu}_{2}\hat{\psi}_{2}:(a)$$</span></p> <p><span class="math-container">$$+\frac{(-ie)^{2}}{2!}∫ d^{4}x_{1}d^{4}x_{2}&lt;0|T\biggl(\hat{\psi}_{1}\hat{\bar{\psi}}_{2}\biggr)|0&gt;:\hat{\bar{\psi}}_{1}\gamma_{\mu}\hat{A}^{\mu}_{1}\gamma_{\mu}\hat{A}^{\mu}_{2}\hat{\psi}_{2}:(b)$$</span></p> <p><span class="math-container">$$+\frac{(-ie)^{2}}{2!}∫ d^{4}x_{1}d^{4}x_{2}&lt;0|T\biggl(\hat{\bar{\psi}}_{1}\hat{\psi}_{2}\biggr)|0&gt;:\gamma_{\mu}\hat{A}^{\mu}_{1}\hat{\psi}_{1}\hat{\bar{\psi}}_{2}\gamma_{\mu}\hat{A}^{\mu}_{2}:(c)$$</span></p> <p><span class="math-container">$$+other\hspace{0.5cm}contractions$$</span></p> <p>and in my notes there's written that those two terms (b),(c) are responsible for the following processes:</p> <p><a href="https://i.stack.imgur.com/TQasB.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/TQasB.png" alt="enter image description here" /></a></p> <p>How can I see clearly this fact? Is it correct to split <span class="math-container">$\hat{\psi}=\hat{\psi}^{(+)}+\hat{\psi}^{(-)}$</span> as well as <span class="math-container">$\hat{A}_{\mu}=\hat{A}_{\mu}^{(+)}+\hat{A}_{\mu}^{(-)}$</span> and expand their product inside the normal ordering? I was thinking that using the normal ordering with the correct signs may cancel some terms and we are left exactly with those processes represented above. Does it make sense?</p> https://physics.stackexchange.com/q/782956 1 Mass of ice block needed to cool area shrimp https://physics.stackexchange.com/users/379596 2023-10-03T17:33:21Z 2023-10-03T17:33:21Z <p>I intend to grow king oyster mushrooms inside a small greenhouse tent. The problem is the mushrooms prefer a temperature of approximately 60F. I'm trying to avoid needing to run an air conditioner to do this.</p> <p>This is the tent <a href="https://rads.stackoverflow.com/amzn/click/com/B0769FYDRX" rel="nofollow noreferrer" rel="nofollow noreferrer">https://www.amazon.com/dp/B0769FYDRX/</a></p> <p>Assume the room the tent is in will be kept about 72F. The tent is 27&quot; x 19&quot; x 63&quot;. Assuming the required ice would fit in the tent, I'd like to keep a few containers of that size full of ice in a freezer and swap them out when necessary, preferably once or twice a day. The ice would be on the top rack (I imagine that would be the best spot as the cold air would fall) and probably need to sit on a tray to collect condensation.</p> <p>What mass of ice would I need, inside a (presumably plastic) container on a tray, to keep this tent 12F cooler? If I'm missing any necessary information, I'll update my question, or you can make assumptions based on common household conditions</p> https://physics.stackexchange.com/q/782955 1 Noether currents for $SU(2)$ symmetric Lagrangian Fernando Garcia Cortez https://physics.stackexchange.com/users/280421 2023-10-03T17:01:12Z 2023-10-03T17:31:25Z <p>I'm currently reading Bilenky's Introduction to the Physics of Massive and Mixed Neutrinos.</p> <p>In chapter 3, section 3.2, we discuss the following Lagrangian: <span class="math-container">$$\mathcal{L}_0=\overline{\psi }(x)(i\gamma ^{\alpha }\partial _{\alpha }+m)\psi (x)\tag{Equation 3.2}$$</span> Where <span class="math-container">$\psi$</span> is a doublet of the <span class="math-container">$SU(2)$</span> group: <span class="math-container">$$\psi (x)=\begin{pmatrix} \psi ^{(+1)}(x) \\ \psi ^{(-1)}(x) \\ \end{pmatrix} \tag{Equation 3.1}$$</span> Where, of course, each of the components is a spin 1/2 field.</p> <p>It is clear that the Lagranian is invariant under the global phase <span class="math-container">$SU(2)$</span> transformation <span class="math-container">\begin{align} \psi '(x) &amp;= Uy \\ \overline{\psi} '(x) &amp;= \overline{\psi }U^{\dagger} \end{align}</span> Where <span class="math-container">$$U=\exp \left( i\frac{1}{2}\boldsymbol{\tau }\cdot \boldsymbol{\Lambda }\right) \tag{Equation 3.4}$$</span> We are then presented with the isovector current: <span class="math-container">$$j_i^{\alpha }=\overline{\psi }\gamma ^{\alpha }\frac{1}{2}\tau _i\psi \tag{Equation 3.5}$$</span> (For which, of course, we have <span class="math-container">$\partial _{\alpha }j_i^{\alpha }=0$</span>).</p> <p>How does Bilenky conclude equation 3.5 above? I'm familiar (yet not too experienced) with getting currents and I'm having some trouble getting this one.</p> https://physics.stackexchange.com/q/782953 0 Solving the Euler-Lagrange equation for with two unknown functions Avishai Barnoy https://physics.stackexchange.com/users/139896 2023-10-03T16:56:39Z 2023-10-03T17:34:22Z <p>I am trying to minimize the energy of a system (lipid membrane) given its energy density <span class="math-container">$f$</span>. I am getting a differential equation that I am not able to solve and I feel it is the result of me not fully understanding how to use the Euler-Lagrange method for minimization. I get a weird result of something that looks like two differential equations combined.</p> <p>The part I am struggling with is that <span class="math-container">$\theta$</span> is also a function of <span class="math-container">$r$</span>. Is it something that I don't understand about EL equation or there is something I am missing in the understanding of the physics of my system?</p> <p>My energy density is: <span class="math-container">$$f=\frac{\kappa_m}{2}\left(\frac{\theta}{r}+\frac{\partial\theta}{\partial r}- \frac{\partial t}{\partial r}-\frac{t}{r}\right)^2+\kappa_trt^2$$</span> <span class="math-container">$$F=2\pi\int_0^\infty{f(r)rdr}.$$</span> When I try to develop the EL equation for <span class="math-container">$t$</span> by <span class="math-container">$$\frac{\partial F}{\partial t}=\frac{d}{dr}\frac{\partial F}{\partial t'}$$</span> I get: <span class="math-container">$$r^2\frac{\partial^2\theta}{\partial r^2}+r\frac{\partial\theta}{\partial r}-\theta=r^2 \frac{\partial^2t}{\partial r^2}+r\frac{\partial t}{\partial r}-\left(1+\frac{\kappa_m}{\kappa_t}r^2\right)t.$$</span> Which I didn't manage to solve.</p> https://physics.stackexchange.com/q/782950 0 Question is about the maximum energy of the spring mass system Shubham https://physics.stackexchange.com/users/379591 2023-10-03T16:40:53Z 2023-10-03T16:40:53Z <p>I was asked this question in my exam and I searched for the solution but couldn't understand the solution can someone help me <a href="https://i.stack.imgur.com/H8OFl.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/H8OFl.jpg" alt="enter image description here" /></a></p> https://physics.stackexchange.com/q/782949 0 Maple sap science Matthew McCartin https://physics.stackexchange.com/users/379590 2023-10-03T16:37:17Z 2023-10-03T16:37:17Z <p>Maple syrup science claims that ice forms in fiber cells which are air filled, causing air to compress, which causes high positive pressures within vascular system, and hence exudation from tap holes and up to 200 kpa pressure. They claim ice forms from vapor distillation. This picture is incomplete.</p> <p>Vapor distillation in isolated fiber lumina cannot cause water migration up the tree toward ice formation, which is what it claims. Since fiber lumina are isolated, humid air condensation to ice cannot account for water uptake.</p> <p>If ice does form in small branching during freezing, where does the ice form? Can crystalline ice form in living cells without rupture or is it always amorphous ice? Can amorphous ice cause water uptake?</p> https://physics.stackexchange.com/q/782947 -1 Working of transistor Akshat Shrivastava https://physics.stackexchange.com/users/374466 2023-10-03T16:05:36Z 2023-10-03T18:26:00Z <p>For the operation of the transistor, the base-emitter junction is forward-biased and the Base collector junction is reversed-biased. Due to the concentration gradient, electrons from the N side (Emitter) go towards the P side (Base). Since the base is lightly doped and thin and due to the reverse biasing of the base-collector junction(electric field exits between the base and collector), the electrons coming from the N side (Emitter) go towards the N side (Collector) without much recombining with the hole in the P region (Base).</p> <p>Since we have a concentration gradient for electrons to flow from emitter to base then why do we need the base supply voltage in a transistor circuit?</p> https://physics.stackexchange.com/q/782946 0 Is the physics of the coalescence cascade of a water droplet completely understood? Eylul https://physics.stackexchange.com/users/379588 2023-10-03T16:01:53Z 2023-10-03T16:07:11Z <p><a href="https://www.youtube.com/watch?v=pbGz1njqhxU" rel="nofollow noreferrer">https://www.youtube.com/watch?v=pbGz1njqhxU</a></p> <p>On the link above you can see this process in motion. I am wondering if the physics of this phenomenon is completely understood?</p> https://physics.stackexchange.com/q/782943 0 Deriving the shape of a suspension bridge with a heavy load Eli Yablon https://physics.stackexchange.com/users/324864 2023-10-03T15:56:02Z 2023-10-03T15:56:02Z <p>You can use the Euler Lagrange equation to show how the shape of a suspended cable with no load is simply a catenary. However, if you suspend a much heavier load (i.e. a bridge) with the cable, the cable forms a parabolic shape. Does anyone know how to show this fact using the Euler-Lagrange equation?</p> https://physics.stackexchange.com/q/782942 0 accelerating body in a non-inertial frame of reference Nandu https://physics.stackexchange.com/users/375672 2023-10-03T15:49:33Z 2023-10-03T15:59:57Z <p>If a body is subjected to a force, can I find a non-inertial frame of reference in which the body is not accelerating?</p> https://physics.stackexchange.com/q/782941 0 If the two ends of a rope in equilibrium are pulled with forces of equal magnitude and opposite directions, why isn’t the total tension in the rope 0? student1928374 https://physics.stackexchange.com/users/319505 2023-10-03T15:44:12Z 2023-10-03T15:44:12Z <p>&quot;If the two ends of a rope in equilibrium are pulled with forces of equal magnitude and opposite directions, why isn’t the total tension in the rope zero?&quot;</p> <p>Can someone explain why the TOTAL tension is zero? Because I do get that the tension force is present and is therefore not zero, but I don't get why the TOTAL tension is zero.</p> https://physics.stackexchange.com/q/782940 0 Polaron vs Electron energy level taqiuddin yusri https://physics.stackexchange.com/users/379584 2023-10-03T15:34:59Z 2023-10-03T15:34:59Z <p>i just started my study in organic semiconductor. I realized to increase conductivity, polarons are formed in the polymer backbone. From what i read due to localised energy state level by polaron that can exist in the band gap, the band gap is narrowed.</p> <p>My question is if why is can polaron exist in band gap while electron can't though they are both almost identical. Also, do the formation polaron exist as somewhat of a 'bridge' to narrow the gap between conduction band and valence band?</p> <p>Helps or guides are really appreciated.</p> https://physics.stackexchange.com/q/782931 2 Lorentz Transformation or Time Dilation? Klegzart https://physics.stackexchange.com/users/360864 2023-10-03T14:12:37Z 2023-10-03T17:39:31Z <p>Say there is a spaceship really far away from Earth moving at 80% the speed of light (away from Earth-edited), at some point a radio signal is sent by observers on Earth. I need to be able to calculate how long it would take for the signal to reach the spaceship. Would calculating the time required for the signal to reach the spaceship in Earth's frame, which is pretty straight forward and then applying time dilation directly be valid? If not please explain what the issue behind the approach is. The other approach would be to apply Lorentz transformations, but honestly I don't get what exactly makes one approach right and other one wrong in certain scenarios.</p> https://physics.stackexchange.com/q/782902 0 Coupled quantum harmonic oscillator: Decomposition in terms of number basis representation for numerical implementation MrDerDart https://physics.stackexchange.com/users/217725 2023-10-03T09:41:25Z 2023-10-03T17:09:30Z <p>Consider the Hamiltonian of a coupled quantum harmonic oscillator</p> <p><span class="math-container">\begin{align} \hat{H}&amp;=\frac{1}{2m}(p_1^2+p_2^2)+\frac{m\omega^2}{2}(q_1^2+q_2^2)+\alpha(q_2-q_1)^2 \\&amp;=\frac{1}{2m}(p_1^2+p_2^2)+(\frac{m\omega^2}{2}+\alpha)(q_1^2+q_2^2)-2\alpha q_1q_2\tag{1} \end{align}</span> with <span class="math-container">$$[q_n,p_m]=i\delta_{nm}$$</span> and the other commutators as usual.</p> <p>We can do the following two transformations to diagonalise it: First, let <span class="math-container">\begin{equation} \begin{pmatrix} q_1\\q_2 \end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&amp;\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&amp;\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix} \tilde{q}_1\\\tilde{q}_2 \end{pmatrix},\qquad \begin{pmatrix} p_1\\p_2 \end{pmatrix}=\begin{pmatrix}\frac{1}{\sqrt{2}}&amp;\frac{1}{\sqrt{2}}\\-\frac{1}{\sqrt{2}}&amp;\frac{1}{\sqrt{2}}\end{pmatrix}\begin{pmatrix} \tilde{p}_1\\\tilde{p}_2 \end{pmatrix}. \end{equation}</span></p> <p>This transformation preserves the commutators, i.e. <span class="math-container">$[\tilde{q}_k,\tilde{p}_l]=i\delta_{kl}$</span>. Substituting this transformation, we obtain the Hamiltonian <span class="math-container">\begin{equation} \begin{aligned} \hat{H}=\frac{1}{2}\sum_{k=1}^{N=2}\frac{1}{m}\tilde{p}_k^2+m\omega_k^2\tilde{q}_k^2 \end{aligned} \end{equation}</span> where <span class="math-container">$\omega_1=\sqrt{\omega^2+\frac{4\alpha}{m}}, \,\,\,\omega_2=\omega$</span>.</p> <p>Secondly, we can define ladder operators <span class="math-container">$a_k,a_k^{\dagger}$</span> for <span class="math-container">$k\in\{1,2\}$</span>: <span class="math-container">\begin{equation} a_k=\frac{1}{\sqrt{2m\omega_k}}\left(m\omega_k\tilde{q}_k+i\tilde{p}_k\right),\qquad a_k^{\dagger}=\frac{1}{\sqrt{2m\omega_k}}\left(m\omega_k\tilde{q}_k-i\tilde{p}_k\right) \end{equation}</span> satisfying <span class="math-container">$[a_k,a_l^{\dagger}]=\delta_{kl}$</span>. Hence, <span class="math-container">\begin{equation} \tilde{q}_k =\frac{1}{\sqrt{2m\omega_k}}(a_k+a_{k}^{\dagger}), \qquad \tilde{p}_k=-i\sqrt{\frac{m\omega_k}{2}}(a_k-a_{k}^{\dagger}) \end{equation}</span></p> <p>This finally gives a Hamiltonian <span class="math-container">\begin{equation} \hat{H}=\sum_{k=1}^{N=2}\omega_k(a_k^{\dagger}a_k+\frac{1}{2})\tag{2} \end{equation}</span></p> <p>I want to implement this model numerically. Let's say I want to compute the interaction energy with corresponding Hamiltonian <span class="math-container">$\hat{H}_I=-2\alpha q_1 q_2$</span> for a global thermal state <span class="math-container">$\rho=e^{-\beta\hat{H}}/Tr[e^{-\beta\hat{H}}]$</span> at inverse temperature <span class="math-container">$\beta$</span>. This amounts to computing <span class="math-container">\begin{equation} E_I=Tr[\hat{H}_I\rho] \end{equation}</span> I can easily write the Hamiltonian <span class="math-container">$\hat{H}$</span> as a matrix in the number basis (and choosing some finite cutoff dimension <span class="math-container">$d$</span>, with mode number operator <span class="math-container">$\hat{n}_k=\hat{a}_k^{\dagger}a_k$</span>.</p> <p>To express the interaction Hamiltonian <span class="math-container">$\hat{H}_I$</span> as a matrix in this basis using ladder operators, I express it as <span class="math-container">\begin{equation} \begin{aligned} \hat{H}_I=-2\alpha q_1 q_2=\alpha(\tilde{q}_1^2-\tilde{q}_2^2)\\ =-\frac{\alpha}{2m}(-\frac{1}{\omega_1}(a_1^2+(a_1^{\dagger})^2+2a_1^{\dagger}a_1+1)+\frac{1}{\omega_2}(a_2^2+(a_2^{\dagger})^2+2a_2^{\dagger}a_2+1)) \end{aligned} \end{equation}</span></p> <p>Is this correct? Choosing each QHO to have ladder operators of dimension <span class="math-container">$d$</span>, I express them using</p> <pre><code>def a(d): a = np.zeros((d,d)) for i in range(d-1): a[i][i+1] = np.sqrt(i+1) return a def adag(d): return np.conjugate(a(d).T) a1 = np.kron(a(d), np.identity(d)) a2 = np.kron(np.identity(d), a(d)) a1dag = np.kron(adag(d), np.identity(d)) a2dag = np.kron(np.identity(d), adag(d)) ident = np.kron(np.identity(d), np.identity(d)) om1 = np.sqrt(omega**2+4*alpha/m) om2 = omega tildeq1=1/(np.sqrt(2*m*om1))*(a1+a1dag) tildeq2=1/(np.sqrt(2*m*om2))*(a2+a2dag) q1=1/np.sqrt(2)*(tildeq1+tildeq2) q2=1/np.sqrt(2)*(-tildeq1+tildeq2) H= om1*(a1dag@a1+1/2*ident)+om2*(a2dag@a2+1/2*ident) H_check = 1/(2m)(p1@p1+p2@p2)+(m*omega**2/2+alpha)(q1@q1+q2@q2)-2alpha*q1@q2 HI=-2*alpha*q1@q2 </code></pre> <p>Somehow H and H_check are not the same, using my method. What is going wrong?</p> https://physics.stackexchange.com/q/782893 2 Does it take more energy to bring a car to a halt if it is still accelerating on impact than travelling at constant speed? Thomas Bates https://physics.stackexchange.com/users/379537 2023-10-03T07:44:14Z 2023-10-03T18:34:50Z <p>So my physics is quite rusty, been out of varsity for a while.</p> <p>A friend asked me this and I am still pondering. Here is the scenario:</p> <ul> <li>2 Cars are travelling towards a wall, and make impact with the wall at the same speed, e.g. 10m/s.</li> <li>Assume the wall is unbreakable and the cars have the same mass, e.g. 500kg</li> <li>Car 1 is travelling at a constant speed of 10m/s when it hits the wall, i.e. no acceleration.</li> <li>Car 2 is accelerating towards the wall, and hits the wall at 10m/s, i.e. it accelerates all the way until it hits the wall.</li> <li>So both cars hit the wall with the same speed, but car 1 has no acceleration, and car 2 has acceleration.</li> </ul> <p>Does the wall exert more energy to bring Car 2 to a stop, because Car 1 only has momentum, but Car 2 has both momentum AND force?</p> <p>This makes sense logically to me but not sure how to explain it in equations.</p> https://physics.stackexchange.com/q/782764 0 Why is light emitted by an atom different to the light absorbed? Why do we not see absorbed light? lemonmeringue https://physics.stackexchange.com/users/379440 2023-10-02T11:28:26Z 2023-10-03T17:00:38Z <p>I’m confused about why we don’t see absorbed light. The way I understand it is if an atom absorbs a photon then the electrons move up to an energy level corresponding to the energy of the photon. The electrons move down and emit a photon with the same energy so the emitted light would be of the same or lower wavelength as the absorbed light? When hydrogen is excited the colour seen is a mixture of the emitted wavelengths which are the same as the wavelengths absorbed. So why is it that objects that absorb red light reflect blue light? Is red light still emitted but we don’t see it, or does this energy absorbed completely disappear with no emission? Is emission of photons from excited atoms different to how colour is seen from other objects? I’d really appreciate if someone could explain this to me</p> https://physics.stackexchange.com/q/782643 1 Time derivative term in Navier Stokes equation for fluid in porous media user134613 https://physics.stackexchange.com/users/290895 2023-10-01T09:51:53Z 2023-10-03T15:47:37Z <p>I was reading the research paper <a href="https://arxiv.org/abs/2304.05393" rel="nofollow noreferrer">Homogenization of peristaltic flows in piezoelectric porous media</a> and came across the hydrodynamic equation:</p> <p><span class="math-container">$$\mu \nabla^2 v^f -\underline{ \rho_f (\dot{v}^f + w \cdot \nabla) v^f)} = \nabla p -f$$</span></p> <p>where <span class="math-container">$\mu$</span> is the dynamic viscosity, <span class="math-container">$v^f$</span> is the fluid velocity, <span class="math-container">$w$</span> is the <a href="https://en.wikipedia.org/wiki/Soil_mechanics#Seepage:_steady_state_flow_of_water" rel="nofollow noreferrer">seepage velocity</a>, <span class="math-container">$p$</span> is the pressure, and <span class="math-container">$f$</span> is the body force.</p> <p>What confuses me is the time derivative term of the velocity (the underlined term in the equation above), especially when I compare it with the general form of the <a href="https://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations" rel="nofollow noreferrer">Navier Stokes equation</a>:</p> <p><span class="math-container">$$\underline{\rho (\frac{\partial v^f}{\partial t} + (v^f \cdot \nabla) v^f)} = - \nabla p + \mu \nabla^2 v^f + \frac{1}{3} \mu \nabla(\nabla \cdot v^f) + \rho g$$</span></p> <p>where <span class="math-container">$u$</span> here is the fluid velocity, <span class="math-container">$\rho$</span> is its density, <span class="math-container">$p$</span> is pressure of fluid, <span class="math-container">$\mu$</span> is the dynamic viscosity of fluid, <span class="math-container">$\rho g$</span> is the body force (gravity).</p> <p>Comparing the time derivative terms from both equations (the underlined terms), I don't understand why the <em>seepage</em> velocity is used in the material time derivative in the first equation. The <em>seepage</em> velocity is the velocity of fluid passing between the pores: I don't understand why it is used in the material time derivative that its multiplied by the fluid velocity. Why are the two velocities are used together?</p> <p>Also, the term of partial time derivative w.r.t. time in the first equation is <span class="math-container">$\rho_f \dot{v}^f v^f$</span> where as in the second equation its <span class="math-container">$\rho \frac{\partial u}{\partial t}$</span>, and so I don't understand why in the first equation we have it multiplied with the fluid velocity.</p> <p><strong>Note:</strong> In both equations, the fluid is a Newtonian fluid.</p> https://physics.stackexchange.com/q/779987 1 Spin Operator in Massless QED schris38 https://physics.stackexchange.com/users/242254 2023-09-12T13:28:57Z 2023-10-03T16:25:37Z <p>I am reading <a href="https://arxiv.org/pdf/1907.05438.pdf" rel="nofollow noreferrer">Subleading soft dressings of asymptotic states in QED and perturbative quantum gravity</a> by Choi and Akhoury. I wish to understand how to derive the subleading soft theorem in massless QED, using amplitude methods.</p> <p>The subleading soft theorem states that amplitudes describing soft photon emission during any interaction, factorize into what is usually referred to as the soft factor with the amplitude of the interaction in which the soft photon emission does not occur. Let <span class="math-container">$i\mathcal{M}_{\text{tree}}$</span> be the amplitude describing soft emission and <span class="math-container">$i\mathcal{M}_0$</span> be the amplitude of the elastic interaction. Then, the two are related as follows:</p> <p><span class="math-container">$$i\mathcal{M}_{\text{tree}}=(\mathcal{S}^{(0)}+ \mathcal{S}^{(1)}) \ i\mathcal{M}_0$$</span></p> <p>where</p> <p><span class="math-container">$$\mathcal{S}^{(0)}=\sum_{i\in \text{out}} e_i\frac{p_i\cdot\epsilon^*_{\lambda}(k)}{p_i\cdot k}- \sum_{i\in \text{in}} e_i\frac{p_i\cdot\epsilon^*_{\lambda}(k)}{p_i\cdot k}\\ \mathcal{S}^{(1)}=i\Big(\sum_{i\in \text{out}} e_i\frac{\epsilon^*_{\mu\lambda}(k)k_{\nu}J^{\mu\nu}_i}{p_i\cdot k}- \sum_{i\in \text{in}} e_i\frac{\epsilon^*_{\mu\lambda}(k)k_{\nu}J^{\mu\nu}_i}{p_i\cdot k}\Big)$$</span></p> <p>with <span class="math-container">$J^{\mu\nu}_i=L_i^{\mu\nu}+S_i^{\mu\nu}$</span>, and (according to Appendix A.1.2)</p> <p><span class="math-container">$$L^{\mu\nu}=-i\Big( p^{\mu}\frac{\partial}{\partial p_{\nu}}- p^{\nu}\frac{\partial}{\partial p_{\mu}} \Big)$$</span> <span class="math-container">$$S_{\mu\nu}=\frac{h}{E}\pmatrix{0&amp;0&amp;0&amp;0\\0&amp;0&amp;p^z&amp;-p^y\\0&amp;-p^z&amp;0&amp;p^x\\0&amp;p^y&amp;-p^x&amp;0}\label{A.25}\tag{A.25}$$</span></p> <p>Here, <span class="math-container">$k$</span> is used to label the momentum of the emitted photon and <span class="math-container">$\epsilon^*_{\lambda}(k)$</span> its polarization vector. Also, <span class="math-container">$p^{\mu}=(E,p^x,p^y,p^z)$</span> is the four-momentum of the massless matter field, whereas <span class="math-container">$h$</span> denotes its helicity (<span class="math-container">$h=\pm1/2$</span> for massless electrons). Each component of the spin operator is scalar when it comes to Dirac indices (i.e. not a Dirac <span class="math-container">$\gamma$</span> matrix or a product of some Dirac <span class="math-container">$\gamma$</span> matrices).</p> <p><strong>MY ATTEMPT:</strong> To prove that the soft theorem holds, I consider an interaction between an electron and a source. Let the amplitude of that interaction be</p> <p><span class="math-container">$$i\mathcal{M}_0=\bar{u}_+(q)\gamma^au_+(p)$$</span></p> <p>where <span class="math-container">$p$</span> and <span class="math-container">$q$</span> are the initial and final momenta of the electron, and with <span class="math-container">$\gamma^a$</span> I denote the Dirac matrix that is about to couple with the source.</p> <p>Adding a photon in the final state particles alters the amplitude in the following way (assume only emission from initial electron) <span class="math-container">$$i\mathcal{M}_0\rightarrow i\mathcal{M}_{\text{tree}}= \bar{u}_+(q)\gamma^a\frac{e}{(-2p\cdot k)} (-\not{p}+\not{k})\not{\epsilon}^*_{\lambda}(k) u_+(p)$$</span></p> <p>I wish to focus on deriving the spin contribution. To do that, I neglect the contribution that is proportional to <span class="math-container">$\not{p}$</span> and therefore after using that <span class="math-container">$2\gamma^{\mu}\gamma^{\nu}=\{\gamma^{\mu},\gamma^{\nu}\}+ [\gamma^{\mu},\gamma^{\nu}]$</span>, we have</p> <p><span class="math-container">$$i\mathcal{M}_{\text{tree}}^{(\text{spin})}= -\frac{e}{2(-2p\cdot k)}\epsilon^*_{\mu\lambda}(k)k_{\nu} \bar{u}_+(q)\gamma^a\ [\gamma^{\mu},\gamma^{\nu}]\ u_+(p)$$</span></p> <p>So, my guess is that this will be the spin contribution to the subleading soft factor (hence the label) since it contains the commutator between two <span class="math-container">$\gamma$</span> matrices! However, the spin operator appearing in the subleading soft factor in my case would be <span class="math-container">$S^{\mu\nu}_p=\frac{1}{2} \bar{u}_+(p)\ (\gamma^0[\gamma^{\mu},\gamma^{\nu}]+ [\gamma^{\mu},\gamma^{\nu}]\gamma^0)\ u_+(p)$</span> because this is the covariant spin operator in momentum space (and it has the proper values for each <span class="math-container">$\mu$</span> and <span class="math-container">$\nu$</span>). So, <strong>how do I make this operator appear in my subleading contribution associated with the spin</strong>?</p> https://physics.stackexchange.com/q/754708 0 Work done by free expansion of gas and the energy of the piston Akshay Venugopalan https://physics.stackexchange.com/users/361423 2023-03-10T18:14:46Z 2023-10-03T18:14:42Z <p>So this has been bothering me for a while. I know that for a free expansion of a gas, work done by it is zero. However, I have a doubt regarding the kinetic energy gained by a piston during this free expansion of a gas.</p> <p>So, let there be a cylindrical container of infinite length. <a href="https://i.stack.imgur.com/IKw56.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/IKw56.png" alt="enter image description here" /></a></p> <p>In the above picture, we can see that there is a fixed piston and a gas with pressure 'P', volume 'V' and temperature 'T'. The remaining volume of the cylinder is vacuum.</p> <p>Now, the piston is released. As a result, free expansion of gas occurs and the piston moves towards the right. <a href="https://i.stack.imgur.com/76OEY.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/76OEY.png" alt="enter image description here" /></a></p> <p>Now, if the piston is massless, I can understand that no work is done by the gas as the kinetic energy of the piston is zero, due to mass being zero.</p> <p>However, if the piston has a mass 'M', it will have a kinetic energy. However, if we again assume that work done by the gas is again zero, then where is the piston's kinetic energy coming from?</p> <p>Also, in both cases, we are assuming that there is no gravity and that the movement of the piston in the cylinder is frictionless.</p> https://physics.stackexchange.com/q/750037 0 Is there (emergent) higher form spontaneous symmetry breaking in classical statistical field theory? Nandagopal Manoj https://physics.stackexchange.com/users/227270 2023-02-14T02:17:53Z 2023-10-03T16:45:12Z <p>I was wondering if there are examples of (emergent) higher form spontaneous symmetry breaking (SSB) in classical statistical physics (finite temperature). I believe the deconfined phase of gauge theories can be interpreted this way, but I am interested in systems with local degrees of freedom (no gauge structure), and without fine-tuning, as lattice gauge theory without gauge structure will also show a phase transition, but it is very fine-tuned.</p> <p>Can the BKT transition be understood this way?</p> <p>I would prefer to think about spin models with discrete phase space per unit volume, but examples with continuous symmetry are also accepted.</p> https://physics.stackexchange.com/q/611408 51 Why does carbon dioxide not sink in air if other dense gases do? chasly - supports Monica https://physics.stackexchange.com/users/85871 2021-01-31T11:45:23Z 2023-10-03T17:59:28Z <p>Why does carbon dioxide not sink in air if other dense gases do?</p> <p>We evidently do not suffocate by carbon dioxide sinking to the bottom of the atmosphere and displacing oxygen and yet there are gases that do sink. This is commonly a problem in coal mines. Lower layers can fill up with gas that is unbreathable.</p> <p>Here is a demonstration showing <a href="https://youtu.be/i8sHyexy4WY?t=10" rel="noreferrer">a 'boat' floating on sulphur hexafluoride.</a></p> <p><strong>Question</strong></p> <p>Given a mixture of two mutually non-reactive gases, what property determines whether the denser gas sinks to the bottom?</p> https://physics.stackexchange.com/q/486963 0 Euler force for pendulum Q.stion https://physics.stackexchange.com/users/211093 2019-06-19T16:15:44Z 2023-10-03T17:07:05Z <p>Hello I have a question related to the Euler force. Why is this force never considered for a simple pendulum? </p> <p>As far as I understand, Euler force is given by (assume I would consider the 2d pendulum in a 3D space, that the quantities are vectors) <span class="math-container">\begin{equation} \boldsymbol{\dot{\omega}} \times \mathbf{r} \end{equation}</span> This means for the force to vanish, <span class="math-container">$\boldsymbol{\dot{\omega}} = 0$</span>, or <span class="math-container">$\mathbf{r} = 0$</span>, or the vectors must point in the same direction. I do not see why one of these conditions is satisfied. </p> https://physics.stackexchange.com/q/345092 0 Diode-Resistor-Capacitor Circuit Equations Ben https://physics.stackexchange.com/users/154435 2017-07-11T02:59:45Z 2023-10-03T18:01:15Z <p>So I took the time to measure the current dependency on voltage of a diode I have. I applied an exponential fit to it, and have a pretty reliable equation (within 1%). </p> <p>I'm interested in how a diode-resistor-capacitor series circuit response to different signals. Naturally, I'm starting with just DC voltage. </p> <p>The equation that I have for the voltage/current dependency for the diode is of the form </p> <p><span class="math-container">$$I=ae^{bV_D} \tag{1}$$</span></p> <p>where <span class="math-container">$V_D$</span> is the voltage across the diode. </p> <p>Using Kirchhoff's law, I get the following differential equation with an initial condition:</p> <p><span class="math-container">$$V = RQ' + \frac{1}c Q + \frac{1}b \ln\left(\frac{Q'}a\right)$$</span></p> <p><span class="math-container">$$Q(0)=0$$</span></p> <p>where <span class="math-container">$R$</span> is the resistance of the resistor, <span class="math-container">$c$</span> is the capacitance of the capacitor, <span class="math-container">$a$</span> and <span class="math-container">$b$</span> are the exponential regression constants from equation (1), and <span class="math-container">$V$</span> is the applied DC voltage.</p> <p>Does anyone know if it's possible to analytically solve this equation?</p> https://physics.stackexchange.com/q/100414 7 Vertex operator - state mapping in Polchinski's book Han Yan https://physics.stackexchange.com/users/20626 2014-02-23T09:57:51Z 2023-10-03T18:16:26Z <p>In Polchinski's textbook String Theory, section 2.8, the author argues that the unit operator $1$ corresponds to the vacuum state, and $\partial X^\mu$ is holomorphic inside couture $Q$ in figure 2.6(b), so operators $\alpha_m^\mu$ with $m\ge0$ vanishes.</p> <p>I am a bit confused about why $\partial X^\mu$ has no pole inside the contour. Before this section $\partial X^\mu$ always has the singularity part ($1/z^m$). Therefore would it be possible for you to give a more mathematical argument what condition requires $\partial X^\mu$ having no poles in this case?</p>